Force and Motion 9

 

1. Definition of Force

Force is a physical quantity that causes a change in the state of motion or the shape of an object. It is a vector quantity, meaning it has both magnitude and direction.

Basic Definition: Force is a push or a pull that can change the state of motion or shape of an object.

Physics Perspective: Force is any interaction that, when unopposed, will change the motion of an object. It has both magnitude and direction, and is measured in Newtons (N).

Mechanical Definition: Force is a physical quantity that causes an object to accelerate, slow down, or change direction when applied.

Everyday Understanding: Force is what you use when you push a door to open it, pull a cart, or kick a ball.

Force in Nature: Force is the influence that causes objects to move or stop moving, like the force of gravity pulling things toward the ground or the force of magnetism attracting iron objects.

➡️ A force can:

• Cause a stationary object to move.
• Change the speed or direction of a moving object.
• Deform or change the shape of an object.

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2. SI Unit of Force

The SI unit of force is the Newton (N).

• 1 Newton (N) = The amount of force required to accelerate a 1 kg mass by 1 m/s².

$$1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2$$

The CGS (centimeter-gram-second) system is a metric system of units that defines the unit of force as the dyne.

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3. Dimensional Formula of Force

The dimensional formula of force is derived from its definition:

$$F = m \cdot a$$

where:

• $m$ = mass → [M]
• $a$ = acceleration → [L T^{-2}]

$$\text{Dimensional formula of Force} = [M L T^{-2}]$$

4. Formula of Force

The general formula of force is based on Newton's Second Law of Motion:

$$F = m \cdot a$$

where:

• $F$ = Force
• $m$ = Mass of the object
• $a$ = Acceleration

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5. Formula derivation;

Newton’s Second Law:

According to Newton's second law, the rate of change of momentum of a body is directly proportional to the applied force and acts in the direction of the force.

$$F \propto \frac{dP}{dt}$$

where $P$ = momentum = $m \cdot v$

Taking derivative:

$$F = m \frac{dv}{dt}$$

Since $\frac{dv}{dt} = a$:

$$F = m \cdot a$$

or,

The equation $F = ma$ represents Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Here's how it's derived:

1. Starting with the definition of acceleration:

Acceleration is defined as the rate of change of velocity with respect to time:

$$a = \frac{dv}{dt}$$

where:

• $a$ is acceleration,
• $v$ is velocity,
• $t$ is time.

2. Velocity as a function of time:

Now, if we rearrange the equation to express velocity change over time, we have:

$$dv = a \, dt$$

3. Relation between momentum and force:

Momentum ($p$) is defined as the product of mass ($m$) and velocity ($v$):

$$p = mv$$

The rate of change of momentum (i.e., how momentum changes with time) is what we call force:

$$F = \frac{dp}{dt}$$

4. Differentiating momentum with respect to time:

Applying the chain rule to the equation $p = mv$, where mass $m$ is constant, we get:

$$\frac{dp}{dt} = m \frac{dv}{dt}$$

Since $\frac{dv}{dt} = a$, the equation becomes:

$$F = ma$$

Conclusion:

Thus, the force $F$ is equal to the mass $m$ of an object multiplied by its acceleration $a$, which is the expression of Newton's second law of motion:

$$F = ma$$

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6. Example of Force

Example 1:

• A force of 10 N is applied to a body of mass 2 kg. Find the acceleration.

$$F = m \cdot a$$ $$10 = 2 \cdot a$$ $$a = \frac{10}{2} = 5 \, m/s^2$$

Example 2:

• If a body of mass 5 kg is accelerating at 3 m/s², the force acting on it is:

$$F = 5 \times 3 = 15 \, N$$

✅ Summary:

Property	Description
Definition	Force is an external cause that changes the state of motion or shape of an object.
SI Unit	Newton (N)
Dimensional Formula	$[M L T^{-2}]$
Formula	$F = m \cdot a$
Example	If $m = 2 \, kg$ and $a = 5 \, m/s^2$, then $F = 10 \, N$
Types	Contact and Non-contact forces

Distance and Displacement

Distance and displacement are two fundamental concepts in physics that describe the motion of an object. While they may seem similar, they are distinctly different in terms of their definitions and properties.

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1. Distance

• Definition: Distance is the total path length traveled by an object, regardless of the direction.
• Scalar Quantity: Distance is a scalar quantity, meaning it only has magnitude (size) and no direction.
• Units: It is typically measured in meters (m), kilometers (km), etc.
• Characteristics:
• It is always positive.
• It depends on the actual path taken, so the value is always equal to or greater than the displacement.
• Distance is not concerned with the starting or ending points, but with how far the object has traveled.

Distance is the total path length traveled by an object, regardless of direction.

Formula:

$$\text{Distance} = \text{Sum of the total path lengths}$$

• Unit: Meter (m)
• Note: Distance is always a positive quantity and does not consider direction.

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2. Displacement

• Definition: Displacement is the shortest straight-line distance from the initial position to the final position of an object, along with the direction of motion.
• Vector Quantity: Displacement is a vector quantity, meaning it has both magnitude (size) and direction.
• Units: Measured in meters (m), similar to distance.
• Characteristics:
• Can be positive, negative, or zero, depending on the direction.
• The displacement depends on the initial and final positions, not on the path taken.
• Displacement is the straight-line distance, and it’s the shortest path between two points.

$$\text{Displacement}=\text{Final Position}-\text{Initial Position}$$

• Unit: Meter (m)
• Note: Displacement is a vector quantity, meaning it has both magnitude and direction.

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Key Differences:

Property	Distance	Displacement
Type	Scalar quantity	Vector quantity
Direction	Does not consider direction	Has both magnitude and direction
Magnitude	Always positive and can be large	Can be positive, negative, or zero
Path Taken	Dependent on the path taken	Independent of the path taken
Example	If an object moves 10 meters in a circle, the distance is 10 meters	If an object moves 10 meters in a circle, the displacement is 0 meters (same start and end point)

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Diagram:

Let's consider an object moving along a path:

1. Initial Position (A): Start point.
2. Final Position (B): End point.

If the object moves in a curved path from point A to B, the distance is the total length of the path, while the displacement is the straight-line distance between point A and point B.

Simple Diagram:

          B
         /|
        / |
       /  | Displacement (d)
      /   |
     /    |
 A -------|
    Distance (D)

• A to B: The curved path represents the distance traveled.
• A to B (Straight line): The straight line represents the displacement.

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Example:

• If an object moves 5 meters east and then 5 meters west:
  • Distance = 5 + 5 = 10 meters (total path traveled).
  • Displacement = 0 meters (since the object ends up at the same position it started).

Here is the diagram showing the difference between distance and displacement. The curved path represents the distance, while the straight line between the points A and B represents the displacement. The direction of movement is indicated with arrows.

Speed:

• Definition: Speed is the rate at which an object covers distance. It is a scalar quantity, meaning it only has magnitude and no direction.
• Formula: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$
• SI Unit: Meter per second (m/s)
• Fact: Speed can be constant (uniform speed) or variable (non-uniform speed). For example, when a car travels at a constant rate of 60 km/h, it has uniform speed.

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Velocity:

• Definition: Velocity is the rate at which an object changes its position. It is a vector quantity, meaning it has both magnitude and direction.
• Formula: $$\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$$
• SI Unit: Meter per second (m/s)
• Fact: The direction of velocity is important. For example, an object moving at 60 km/h eastward has a different velocity than one moving at the same speed westward.

✅ Detailed Explanation of Scalar and Vector Quantities

🔹 Scalar Quantity

A scalar quantity is fully described by magnitude (size or value) only. It does not have a direction. Scalars are represented by numerical values along with their units.

📌 Key Characteristics of Scalars:

• Only magnitude is required to describe them.
• No direction involved.
• Scalars follow ordinary arithmetic for addition and subtraction.

🚀 Examples of Scalar Quantities:

Quantity	Definition	Example	Unit
Speed	Rate of motion without direction	60 km/h	m/s, km/h
Distance	Total path covered by an object	5 meters	m, km
Time	Duration of an event	2 hours	s, min, hr
Mass	Quantity of matter in a body	10 kg	kg, g
Temperature	Degree of hotness or coldness	30°C	°C, K

🔎 Example of Scalar Calculation:

If you drive a car at 60 km/h for 2 hours, the total distance covered is:

$$\text{Distance}=\text{Speed}\times \text{Time}=60\times 2=120\text{km}$$

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🔹 Vector Quantity

A vector quantity requires both magnitude and direction to be fully defined. It is represented by an arrow, where:

• Length of the arrow = Magnitude
• Direction of the arrow = Direction of the quantity

📌 Key Characteristics of Vectors:

• Defined by both magnitude and direction.
• Follows vector addition rules (triangle law or parallelogram law).
• Can be represented in terms of components (x, y, z).

🚀 Examples of Vector Quantities:

Quantity	Definition	Example	Unit
Velocity	Rate of change of displacement with direction	60 km/h north	m/s, km/h
Displacement	Shortest distance between two points with direction	5 meters east	m, km
Force	Push or pull with direction	10 N at 30°	N (Newton)
Acceleration	Rate of change of velocity with direction	9.8 m/s² downward	m/s²
Momentum	Mass × velocity

Scalar and Vector Quantities

Type Definition Example Scalar Quantity A scalar quantity is defined by only its magnitude (size or value) and has no direction. - Speed (e.g., 60 km/h) - Distance (e.g., 5 meters) - Time (e.g., 2 hours) - Mass (e.g., 10 kg) Vector Quantity A vector quantity is defined by both magnitude and direction. - Velocity (e.g., 60 km/h north) - Displacement (e.g., 5 meters east) - Force (e.g., 10 N at 30°) - Acceleration (e.g., 9.8 m/s² downward)

Differences Between Scalar and Vector Quantities

Property Scalar Vector Definition Only magnitude Magnitude and direction Representation Single value Arrow (length = magnitude, direction = direction) Addition Rule Simple arithmetic addition Follows the vector addition rule (triangle/parallelogram law) Example Speed = 50 km/h Velocity = 50 km/h north

Diagram Explanation

Scalar Quantity: A scalar quantity is represented as a simple value. Example: If you walk 5 meters, the distance is 5 meters (no direction involved). Vector Quantity: A vector quantity is represented as an arrow, where: Length = Magnitude Arrowhead = Direction Example: If you walk 5 meters towards the east, it becomes a displacement vector.

🚀 Speed vs Velocity

Term             Definition Speed The rate at which an object covers distance. It is a scalar quantity that has only magnitude (size) but no direction. Velocity The rate at which an object changes its position in a specific direction. It is a vector quantity that has both magnitude and direction.

🔎 About Speed

Speed measures how fast an object is moving, irrespective of the direction. It only considers the distance traveled, not the displacement. Since speed has no direction, it is a scalar quantity.

✅ Examples of Speed:

A car moving at 60 km/h on a highway. A person running at 5 m/s on a track. A train traveling at 100 km/h.

📏 Formula for Speed:

$$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$$

🌍 SI Unit of Speed:

Meters per second (m/s) Other units: km/h, miles/hour (mph)

🔎 Example Calculation:

If a car travels 150 km in 3 hours, the speed is: $$\text{Speed}=\frac{150\text{km}}{3\text{hours}}=50\text{km/h}$$

🔎 About Velocity

Velocity measures how fast an object is moving AND in which direction. It considers the displacement (shortest path) rather than distance. Velocity is a vector quantity.

✅ Examples of Velocity:

A car moving at 60 km/h north. A plane flying at 800 km/h west. A ball rolling at 5 m/s down a slope.

📏 Formula for Velocity:

$$\text{Velocity}=\frac{\text{Displacement}}{\text{Time}}$$

🌍 SI Unit of Velocity:

Meters per second (m/s) Other units: km/h, miles/hour (mph)

🔎 Example Calculation:

If a person moves 120 m east in 20 seconds, the velocity is: $$\text{Velocity}=\frac{120m}{20s}=6m\mathrm{/}s\text{east}$$

🔥 Difference Between Speed and Velocity

Property Speed Velocity Definition Rate of change of distance with time Rate of change of displacement with time Type Scalar (no direction) Vector (magnitude and direction) Formula $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$ $\text{Velocity}=\frac{\text{Displacement}}{\text{Time}}$ Direction No Yes Magnitude Always positive Can be positive or negative (depending on direction) Example Car traveling at 60 km/h Car traveling at 60 km/h north

🔎 Graphical Representation

✅ Speed: If a car moves 60 km in 1 hour, the speed is: bash |----|----|----|----|----> 60 km/h ✅ Velocity: If a car moves 60 km north in 1 hour, the velocity is represented by an arrow: bash ↑ 60 km/h

💡 Key Points:

✔️ Speed = Magnitude only (no direction) ✔️ Velocity = Magnitude + Direction ✔️ Velocity can be zero if the displacement is zero (even if distance is non-zero) ✔️ If an object returns to its starting point, average velocity = zero, but average speed ≠ zero

🚀 Velocity

✅ Definition of Velocity

Velocity is the rate of change of displacement of an object with respect to time in a particular direction. It is a vector quantity because it includes both: Magnitude (how fast the object is moving) Direction (the path the object follows) $$\text{Velocity}=\frac{\text{Displacement}}{\text{Time Taken}}$$

🌍 SI Unit of Velocity:

Meters per second (m/s) Other common units: Kilometer per hour (km/h) Miles per hour (mph)

🏆 Types of Velocity

🔹 1. Uniform Velocity

When an object covers equal displacement in equal intervals of time in a straight line. 📌 Example: A car moving at 60 km/h toward the north without changing speed or direction. 📏 Formula: $$\text{Velocity}=\frac{\text{Displacement}}{\text{Time}}$$ ✅ Graph: A straight line on a velocity-time graph indicates uniform velocity. markdown | ------ v | / e | / l |_______/_________ o t

🔹 2. Non-Uniform (Variable) Velocity

When an object covers unequal displacement in equal intervals of time or changes direction while moving. 📌 Example: A car moving through traffic, changing speed and direction frequently. 📏 Formula: Average Velocity: $$\bar{v}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}$$ ✅ Graph: A curved line on a velocity-time graph indicates non-uniform velocity. lua | ------ v | / e | / l |______/ o t

🔹 3. Instantaneous Velocity

The velocity of an object at a specific moment of time. 📌 Example: The speedometer of a car shows the instantaneous velocity at a given time. 📏 Formula: $$v_{\text{inst}}=\frac{dx}{dt}$$ ✅ Graph: The slope of a displacement-time graph gives the instantaneous velocity. perl | / x | / | / |__/ t

🔹 4. Average Velocity

The total displacement divided by the total time taken. 📌 Example: A person walks 10 m east and then 10 m west in 10 seconds. Total displacement = 0 m → Average velocity = 0 m/s 📏 Formula: $$\bar{v}=\frac{\text{Total Displacement}}{\text{Total Time}}$$ ✅ Graph: If displacement is zero, the average velocity is zero. markdown |----|----|----| d | |_________________ t

🔹 5. Relative Velocity

The velocity of an object as observed from another moving object. 📌 Example: Two cars moving toward each other at 60 km/h each will have a relative velocity of 120 km/h. 📏 Formula: If two objects A and B are moving at velocities $v_A$ and $v_B$, then: Same direction: $$v_{\text{rel}}=v_A-v_B$$ Opposite direction: $$v_{\text{rel}}=v_A+v_B$$ ✅ Graph: If both objects move toward each other, their velocities are added.

🔢 Numerical Examples

Example 1:

A car moves 120 km toward the east in 2 hours. Find the velocity of the car. 📏 Solution: Given: Displacement = 120 km (east) Time = 2 hours Using the formula: $$v=\frac{\text{Displacement}}{\text{Time}}=\frac{120\text{km}}{2\text{h}}=60\text{km/h (east)}$$ ✅ Answer: The velocity of the car is 60 km/h east.

Example 2:

A train travels 500 m north in 20 seconds. Find the velocity of the train. 📏 Solution: Given: Displacement = 500 m (north) Time = 20 s Using the formula: $$v=\frac{500\text{m}}{20\text{s}}=25\text{m/s (north)}$$ ✅ Answer: The velocity of the train is 25 m/s north.

Example 3:

A car is moving at 80 km/h toward the east. Another car is moving at 60 km/h toward the west. Find their relative velocity. 📏 Solution: Given: $v_1=80\text{km/h (east)}$ $v_2=60\text{km/h (west)}$ Since they are moving toward each other: $$v_{\text{rel}}=80+60=140\text{km/h}$$ ✅ Answer: The relative velocity is 140 km/h.

📈 Graphical Representation

✅ Uniform Velocity Graph:

Straight line indicates constant velocity. lua | v | ----- e | / l |_______/ o t

✅ Non-Uniform Velocity Graph:

Curved line indicates changing velocity. perl | v | / e | / l | / o |__/ t

✅ Instantaneous Velocity Graph:

Slope at a specific point gives instantaneous velocity. perl | / x | / | / |___/ t

💡 Key Takeaways:

✔️ Velocity measures both how fast and in what direction an object moves. ✔️ Uniform velocity = constant speed and direction. ✔️ Non-uniform velocity = changing speed or direction. ✔️ Instantaneous velocity = velocity at a particular instant. ✔️ Relative velocity = velocity of an object compared to another moving object.

🚀 Acceleration

✅ Definition of Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. It measures how quickly an object speeds up, slows down, or changes direction. It is a vector quantity because it has both: Magnitude (how much the velocity changes) Direction (the direction of change) $$a=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$$ where: $a$ = acceleration $\mathrm{\Delta }v$ = change in velocity = $v_f-v_i$ $\mathrm{\Delta }t$ = time taken for the change in velocity

🌍 SI Unit of Acceleration:

Meters per second squared (m/s²)

🏆 Types of Acceleration

🔹 1. Uniform Acceleration

When an object changes its velocity by an equal amount in equal intervals of time. 📌 Example: A freely falling object under gravity experiences uniform acceleration of 9.81 m/s² downward. 📏 Formula: $$a=\frac{v_f-v_i}{t}$$ ✅ Graph: A straight line on a velocity-time graph shows uniform acceleration. markdown v | | / | / |______/ t

🔹 2. Non-Uniform (Variable) Acceleration

When an object changes its velocity by an unequal amount in equal intervals of time. 📌 Example: A car accelerating at different rates in traffic. 📏 Formula: Instantaneous Acceleration: $$a_{\text{inst}}=\frac{dv}{dt}$$ ✅ Graph: A curved line on a velocity-time graph shows non-uniform acceleration. markdown v | | / | / |___/ t

🔹 3. Positive Acceleration

When an object's velocity increases over time. 📌 Example: A car speeding up from 20 km/h to 60 km/h. ✅ Graph: A positive slope indicates positive acceleration. markdown v | | / | / |______/ t

🔹 4. Negative Acceleration (Deceleration or Retardation)

When an object's velocity decreases over time. 📌 Example: A car slowing down when approaching a red light. ✅ Graph: A negative slope indicates negative acceleration. markdown v | | / | / |____/ t

🔹 5. Centripetal Acceleration

When an object moves in a circular path at a constant speed but continuously changes direction toward the center of the circle. 📌 Example: A satellite orbiting the Earth. 📏 Formula: $$a_c=\frac{v^2}{r}$$ where: $a_c$ = centripetal acceleration $v$ = velocity $r$ = radius of the circular path

📏 Formula for Acceleration

✅ Basic Formula:

$$a=\frac{v_f-v_i}{t}$$ where: $a$ = acceleration $v_f$ = final velocity $v_i$ = initial velocity $t$ = time

🔢 Numerical Examples

Example 1:

A car starts from rest and reaches a speed of 25 m/s in 5 seconds. Find the acceleration. 📏 Solution: Given: $v_i=0$ (car starts from rest) $v_f=25m\mathrm{/}s$ $t=5s$ Using the formula: $$a=\frac{v_f-v_i}{t}=\frac{25-0}{5}=5m\mathrm{/}{s}^2$$ ✅ Answer: The acceleration is 5 m/s².

Example 2:

A train is moving at 30 m/s. It comes to rest in 10 seconds. Find the acceleration. 📏 Solution: Given: $v_i=30m\mathrm{/}s$ $v_f=0m\mathrm{/}s$ $t=10s$ Using the formula: $$a=\frac{0-30}{10}=-3m\mathrm{/}{s}^2$$ ✅ Answer: The negative sign indicates deceleration of 3 m/s².

Example 3:

An object starts from rest and accelerates uniformly at 2 m/s² for 4 seconds. Find the final velocity and distance traveled. 📏 Solution: Given: $v_i=0$ $a=2m\mathrm{/}{s}^2$ $t=4s$ (a) Final velocity: Using the first equation of motion: $$v_f=v_i+at=0+(2)(4)=8m\mathrm{/}s$$ (b) Distance traveled: Using the second equation of motion: $$s=v_{i}t+\frac{1}{2}a{t}^2$$ $$s=0+\frac{1}{2}(2)(4^2)=\frac{1}{2}(2)(16)=16m$$ ✅ Answer: Final velocity = 8 m/s Distance traveled = 16 m

Example 4:

An object moving at 20 m/s comes to rest over a distance of 50 m. Find the acceleration. 📏 Solution: Given: $v_i=20m\mathrm{/}s$ $v_f=0m\mathrm{/}s$ $s=50m$ Using the third equation of motion: $$v_f^2=v_i^2+2as$$ $$0=(20{)}^2+2(a)(50)$$ $$0=400+100a$$ $$100a=-400$$ $$a=-4m\mathrm{/}{s}^2$$ ✅ Answer: The acceleration is −4 m/s² (negative sign indicates deceleration).

💡 Key Takeaways:

✔️ Acceleration measures how quickly velocity changes. ✔️ Positive acceleration → Speed increases. ✔️ Negative acceleration → Speed decreases (deceleration). ✔️ Constant slope = Uniform acceleration. ✔️ Changing slope = Non-uniform acceleration.

🚀 Retardation (Deceleration)

✅ Definition of Retardation

Retardation (or deceleration) is the rate of decrease of velocity of an object with respect to time. It is essentially negative acceleration because the velocity decreases over time. It is a vector quantity since it has both: Magnitude (how much the velocity decreases) Direction (opposite to the motion of the object) $$a=\frac{\mathrm{\Delta }v}{t}$$ where: $a$ = retardation (negative acceleration) $\mathrm{\Delta }v$ = change in velocity $t$ = time taken for the change in velocity

🌍 SI Unit of Retardation:

Meters per second squared (m/s²)

🏆 Difference Between Acceleration and Retardation

Property Acceleration Retardation Definition Rate of increase of velocity Rate of decrease of velocity Sign Positive (+) Negative (−) Effect on Object Speeds up the object Slows down the object Example A car increasing speed from 20 km/h to 60 km/h A car slowing down from 60 km/h to 20 km/h Graph Positive slope on velocity-time graph Negative slope on velocity-time graph

🧪 Formula for Retardation

Since retardation is negative acceleration, the basic formula becomes: $$a=\frac{v_f-v_i}{t}$$ where: $a$ = retardation (negative value) $v_f$ = final velocity $v_i$ = initial velocity $t$ = time taken If the velocity decreases, the value of $v_f-v_i$ becomes negative, resulting in negative acceleration (retardation).

🔢 Numerical Examples

Example 1:

A car moving at 30 m/s slows down uniformly to 10 m/s in 5 seconds. Find the retardation. 📏 Solution: Given: $v_i=30m\mathrm{/}s$ $v_f=10m\mathrm{/}s$ $t=5s$ Using the formula: $$a=\frac{v_f-v_i}{t}=\frac{10-30}{5}=\frac{-20}{5}=-4m\mathrm{/}{s}^2$$ ✅ Answer: Retardation = −4 m/s² (negative sign indicates a decrease in speed).

Example 2:

A train traveling at 40 m/s stops in 20 seconds. Find the retardation. 📏 Solution: Given: $v_i=40m\mathrm{/}s$ $v_f=0m\mathrm{/}s$ (since the train stops) $t=20s$ Using the formula: $$a=\frac{0-40}{20}=\frac{-40}{20}=-2m\mathrm{/}{s}^2$$ ✅ Answer: Retardation = −2 m/s²

Example 3:

A car is traveling at 25 m/s and comes to rest after covering a distance of 100 m. Find the retardation. 📏 Solution: Given: $v_i=25m\mathrm{/}s$ $v_f=0m\mathrm{/}s$ $s=100m$ Using the third equation of motion: $$v_f^2=v_i^2+2as$$ $$0=(25{)}^2+2(a)(100)$$ $$0=625+200a$$ $$200a=-625$$ $$a=-3.125m\mathrm{/}{s}^2$$ ✅ Answer: Retardation = −3.125 m/s²

💡 Key Takeaways:

✔️ Retardation is negative acceleration. ✔️ It occurs when an object slows down over time. ✔️ It always acts in the opposite direction to the motion of the object. ✔️ Negative slope in velocity-time graph = Retardation.

🚀 Average Velocity and Relative Velocity

✅ 1. Average Velocity

➡️ Definition of Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. $$\bar{v}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}$$ where: $\bar{v}$ = average velocity $\mathrm{\Delta }x$ = total displacement $\mathrm{\Delta }t$ = total time taken

🌍 SI Unit of Average Velocity:

Meters per second (m/s)

🏆 Example of Average Velocity:

A car travels 120 km north in 2 hours and then returns 60 km south in 1 hour. Find the average velocity. 📏 Solution: Total displacement = $120\text{km}-60\text{km}=60\text{km}$ (north) Total time = $2+1=3\text{hours}$ $$\bar{v}=\frac{60\text{km}}{3\text{h}}=20\text{km/h}$$ ✅ Answer: The average velocity = 20 km/h (north)

🏆 Average Speed vs Average Velocity:

Property Average Speed Average Velocity Definition Total distance traveled per unit time Total displacement per unit time Formula $\frac{\text{Total distance}}{\text{Total time}}$ $\frac{\text{Total displacement}}{\text{Total time}}$ Value Always positive Can be positive, negative, or zero Example A car moving in a circle → High average speed, zero average velocity A car moving in a straight line → Both are equal

✅ 2. Relative Velocity

➡️ Definition of Relative Velocity

Relative velocity is the velocity of one object with respect to another object. It represents how fast one object is moving compared to another. $${\vec{v}}_{AB}={\vec{v}}_A-{\vec{v}}_B$$ where: ${\vec{v}}_{AB}$ = velocity of object A relative to object B ${\vec{v}}_A$ = velocity of object A ${\vec{v}}_B$ = velocity of object B

🌍 SI Unit of Relative Velocity:

Meters per second (m/s)

🏆 Types of Relative Velocity:

🔹 (a) Objects Moving in the Same Direction

If two objects are moving in the same direction, the relative velocity is the difference of their velocities: $$v_{AB}=v_A-v_B$$ 📌 Example: Object A = 40 m/s Object B = 30 m/s $$v_{AB}=40-30=10m\mathrm{/}s$$ ✅ Answer: The relative velocity = 10 m/s in the direction of object A.

🔹 (b) Objects Moving in Opposite Directions

If two objects are moving in opposite directions, the relative velocity is the sum of their velocities: $$v_{AB}=v_A+v_B$$ 📌 Example: Object A = 40 m/s (right) Object B = 30 m/s (left) $$v_{AB}=40+30=70m\mathrm{/}s$$ ✅ Answer: The relative velocity = 70 m/s

🔹 (c) Object at Rest

If one object is at rest, the relative velocity is simply the velocity of the moving object. 📌 Example: Object A = 50 m/s Object B = 0 m/s (rest) $$v_{AB}=50-0=50m\mathrm{/}s$$ ✅ Answer: The relative velocity = 50 m/s

🏆 Example of Relative Velocity:

Two cars A and B are moving in opposite directions. Car A is moving at 60 km/h east and car B is moving at 40 km/h west. Find the relative velocity of A with respect to B. 📏 Solution: Since they are moving in opposite directions: $$v_{AB}=v_A+v_B=60+40=100\text{km/h}$$ ✅ Answer: The relative velocity = 100 km/h

🔢 Numerical Problems

Example 1:

A train moving at 100 m/s overtakes another train moving at 60 m/s in the same direction. What is the relative velocity of the faster train with respect to the slower one? 📏 Solution: $v_A=100m\mathrm{/}s$ $v_B=60m\mathrm{/}s$ Since they are moving in the same direction: $$v_{AB}=100-60=40m\mathrm{/}s$$ ✅ Answer: Relative velocity = 40 m/s

Example 2:

Two cars A and B are moving towards each other. Car A is moving at 80 m/s and car B is moving at 60 m/s. Find the relative velocity. 📏 Solution: Since they are moving in opposite directions: $$v_{AB}=80+60=140m\mathrm{/}s$$ ✅ Answer: Relative velocity = 140 m/s

🔬 Difference Between Average Velocity and Relative Velocity

Property Average Velocity Relative Velocity Definition  Total displacement per unit time Velocity of one object relative to another Formula $\bar{v}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}$ $v_{AB}=v_A-v_B$ Direction Same as the direction of displacement Depends on the motion of both objects Value A single value over a time interval Can be positive, negative, or zero Example Car covering 100 km in 2 hours → Average velocity = 50 km/h Two trains moving in opposite directions at 40 km/h and 30 km/h → Relative velocity = 70 km/h

💡 Key Takeaways:

✔️ Average velocity → Total displacement over time. ✔️ Relative velocity → How fast one object appears to move relative to another. ✔️ Relative velocity depends on the direction of motion. ✔️ Average velocity depends on displacement and time.

1. Linear Motion

Definition:

Linear motion is the movement of an object in a straight line or along a single dimension.

Difference:

Linear motion can occur horizontally, vertically, or at an angle. Vertical motion is a type of linear motion that happens specifically in the up or down direction due to the effect of gravity.

Examples:

A car moving on a straight road. A person walking down a hallway. A train traveling on a straight track.

SI Unit:

Displacement (distance): meter (m) Velocity: meters per second (m/s) Acceleration: meters per second squared (m/s²)

Formulas:

Displacement: $$s = v t + \frac{1}{2} a t^2$$ Velocity: $$v = u + a t$$ Final velocity: $$v^2 = u^2 + 2 a s$$ Where: $s$ = displacement $u$ = initial velocity $v$ = final velocity $a$ = acceleration $t$ = time

2. Vertical Motion

Definition:

Vertical motion is a type of linear motion where an object moves along the vertical axis under the influence of gravity.

Difference:

Vertical motion is always influenced by gravity ($g = 9.81 \,$$m/s^2$). Gravity acts downward, causing objects to accelerate or decelerate in the vertical direction.

Examples:

A ball thrown up in the air. A stone dropped from a building. A person jumping off a ledge.

SI Unit:

Same as linear motion: meter (m), meters per second (m/s), meters per second squared (m/s²)

Formulas:

Displacement: $$h = u t + \frac{1}{2} g t^2$$ Final velocity: $$v = u + g t$$ Final velocity (alternate): $$v^2 = u^2 + 2 g h$$ Where: $h$ = height (displacement) $u$ = initial velocity $v$ = final velocity $g$ = acceleration due to gravity ($9.81 \, m/s^2$) $t$ = time

Key Differences:

Feature Linear Motion Vertical Motion Direction Any direction (horizontal, vertical, angled) Only along the vertical axis Effect of Gravity Not necessarily involved Always influenced by gravity Acceleration Any constant value Constant value due to gravity = 9.81 m/s² Example Car moving on a road Ball thrown upward

1. Linear Motion

Definition:

Linear motion refers to the motion of an object along a straight path in any direction.

Symbols and Units:

Quantity   Symbol    SI Unit Displacement $s$        meter(m) Initial velocity $u$        meters per second (m/s) Final velocity $v$        meters per second (m/s) Acceleration $a$        meters per second squared (m/s²) Time $t$        second (s)

Formulas:

Displacement: $$s = u t + \frac{1}{2} a t^2$$ Final velocity: $$v = u + a t$$ Final velocity (without time): $$v^2 = u^2 + 2 a s$$

Example:

A car moving along a straight road with constant acceleration. A person walking in a straight line.

2. Vertical Motion

Definition:

Vertical motion refers to the motion of an object along the vertical axis under the influence of gravity.

Symbols and Units:

Quantity Symbol SI Unit Height (displacement) $h$ meter (m) Initial velocity $u$ meters per second (m/s) Final velocity $v$ meters per second (m/s) Acceleration due to gravity $g$ meters per second squared (m/s²) Time $t$ second (s)

Formulas:

Displacement: $$h = u t + \frac{1}{2} g t^2$$ Final velocity: $$v = u + g t$$ Final velocity (without time): $$v^2 = u^2 + 2 g h$$ For upward motion, use $g = -9.81 \, m/s^2$ (since gravity opposes upward motion). For downward motion, use $g = +9.81 \, m/s^2$.

Example:

A ball thrown upwards. An object dropped from a height.

3. Key Differences with Symbols

Feature Linear Motion Vertical Motion Direction Any direction Only vertical (up or down) Effect of gravity Not necessarily involved Always involved ($g = 9.81 \, m/s^2$) Displacement $s$ $h$ Acceleration $a$ $g$ Formulas $s = u t + \frac{1}{2} a t^2$ $h = u t + \frac{1}{2} g t^2$

✅ Summary:

Linear motion = General straight-line motion. Vertical motion = Linear motion influenced by gravity in the vertical direction. Same formulas, but in vertical motion, $a$ is replaced with $g$.

1. Definition of Inertia

Inertia is the property of an object to resist any change in its state of motion or rest. According to Newton's First Law of Motion: "An object will remain at rest or continue in uniform motion in a straight line unless acted upon by an external unbalanced force."

✅ Key Points about Inertia:

Inertia is a natural tendency of objects. An object at rest remains at rest unless a force acts on it. An object in motion continues moving in the same direction and speed unless acted upon by a force. The greater the mass of an object, the greater its inertia (more resistance to change).

2. Types of Inertia

Inertia is classified into three types based on the type of motion:

(i) Inertia of Rest

The tendency of an object to stay at rest unless acted upon by an external force. Example: A person sitting in a stationary bus falls backward when the bus suddenly starts moving. A book on a table remains at rest until someone pushes it.

(ii) Inertia of Motion

The tendency of an object to remain in motion unless acted upon by an external force. Example: A person moving forward when a running bus suddenly stops. A ball rolling on the ground continues to roll unless stopped by friction or an obstacle.

(iii) Inertia of Direction

The tendency of an object to keep moving in the same direction unless acted upon by an external force. Example: A car takes a sharp turn, and passengers tend to move outward due to the inertia of direction. Water spilling out of a moving container when it turns.

3. Examples of Inertia

Type of Inertia Example Inertia of Rest A coin remains on a card when the card is pulled quickly. Inertia of Motion A cyclist continues to move forward after stopping pedaling. Inertia of Direction Passengers move sideways when a car turns sharply.

4. Relation Between Mass and Inertia

Inertia is directly proportional to mass — Greater the mass, greater the inertia. Mass ($m$) is a measure of an object’s resistance to change in motion. Mathematically, inertia can be represented as: $$I \propto m$$ Where: $I$ = Inertia $m$ = Mass Example: A heavier truck has greater inertia than a small car, so it is harder to start or stop its motion. A larger rock is harder to move than a small pebble because it has greater mass and inertia.

5. Difference Between Mass and Inertia

Feature Mass Inertia Definition Quantity of matter in a body Resistance to change in motion SI Unit Kilogram (kg) No unit (it’s a property) Depends on Matter present in an object Mass of the object Example A truck has more mass than a car A truck resists more change in motion than a car

6. Diagram of Inertia

(a) Inertia of Rest

----------------- | | | COIN | ----------------- | | | GLASS | -----------------

(b) Inertia of Motion

[BUS] → → → 🧍🧍 → → → (Bus stops) 🧍🧍 → → → (Passengers keep moving forward)

(c) Inertia of Direction

→ (Car turning right) 🧍🧍 → → (Passengers lean left)

7. Mathematical Understanding of Inertia

From Newton's First Law: $$F = m a$$ Where: $F$ = Force applied $m$ = Mass of the object $a$ = Acceleration produced Greater mass → Greater inertia → More force required to change the motion.

✅ Summary:

Inertia = Resistance to change in motion. Directly proportional to mass. Three types of inertia = Rest, Motion, and Direction. Heavier objects = Greater inertia = Harder to change their state.

1. Newton's Laws of Motion

Sir Isaac Newton formulated three fundamental laws that describe the relationship between the motion of an object and the forces acting on it. These are known as Newton's Laws of Motion.

✅ 1st Law of Motion: The Law of Inertia

Statement: "An object will remain at rest or continue in uniform motion in a straight line unless acted upon by an external unbalanced force." Inertia: The property of objects to resist changes in their state of motion. Key Points: Objects at rest remain at rest. Objects in motion remain in motion with the same speed and direction unless a force acts on them. This law explains why seatbelts are important: without them, a person in a moving car would continue moving forward when the car stops suddenly. Example: A book on a table stays at rest until someone pushes it. A spaceship in space will continue moving indefinitely unless a force (like gravity or friction) acts on it. Diagram: [Book on table] ← (No force) → [Book at rest]

✅ 2nd Law of Motion: The Law of Acceleration

Statement: "The acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass." Mathematically: $$F = m \cdot a$$ Where: $F$ = Force applied (in Newtons, N) $m$ = Mass of the object (in kilograms, kg) $a$ = Acceleration produced (in meters per second squared, m/s²) Key Points: Acceleration depends on both the applied force and the object's mass. The greater the force, the greater the acceleration (for the same mass). The greater the mass, the smaller the acceleration (for the same force). Example: Pushing a car results in less acceleration than pushing a bicycle, even if the same force is applied, because the car has more mass. Diagram: [Car with force applied] →→→ [Smaller Acceleration]

✅ 3rd Law of Motion: Action and Reaction

Statement: "For every action, there is an equal and opposite reaction." This means that if an object exerts a force on another object, the second object exerts an equal force in the opposite direction on the first object. Key Points: Forces always come in pairs: action and reaction. These forces act on different objects. They are equal in magnitude but opposite in direction. Example: When you jump off a boat, you push the boat backward (action), and the boat pushes you forward (reaction). A rocket’s engine pushes down on the ground (action), and the rocket is pushed upward (reaction). Diagram: [Action] → Rocket engine → (Thrust down) [Reaction] ← Rocket pushed up ← (Equal force)

2. Brief History of Newton's Laws

Isaac Newton, an English mathematician and physicist, published his laws in 1687 in his groundbreaking book "Philosophiæ Naturalis Principia Mathematica" (Mathematical Principles of Natural Philosophy). His laws were the first to systematically explain motion and forces. Newton's Laws laid the foundation for classical mechanics, a key area of physics.

3. Key Points of Newton's Laws

Law Statement Example Key Concept 1st Law (Inertia) An object will remain at rest or in uniform motion unless acted upon by an external force. A book on a table stays at rest. Inertia - resistance to change in motion. 2nd Law (Acceleration) Force = Mass × Acceleration. The acceleration of an object is proportional to the force and inversely proportional to its mass. A heavy truck accelerates slower than a small car under the same force. Acceleration depends on force and mass. 3rd Law (Action-Reaction) For every action, there is an equal and opposite reaction. Jumping off a boat pushes the boat backward. Forces come in pairs (action and reaction).

4. Examples of Newton's Laws

1st Law (Inertia): Example: A ball at rest on the ground stays at rest unless a force (such as a kick) moves it. 2nd Law (Acceleration): Example: If you push a sled and a car with the same force, the sled (with less mass) will accelerate more than the car. 3rd Law (Action and Reaction): Example: When you push against a wall, the wall pushes back with an equal and opposite force.

5. Applications of Newton's Laws

Vehicles: The laws explain how vehicles accelerate (2nd law) and how seatbelts work (1st law). Rocket Launches: The rocket's engines produce action and reaction forces (3rd law) and accelerate due to the force applied (2nd law). Sports: Whether a soccer player kicks a ball (action and reaction), or a runner accelerates (2nd law), Newton’s Laws explain the motion.

6. Diagrams Summary

1st Law (Inertia): A book remains at rest. [Book on table] ← (No force) → [Book at rest] 2nd Law (Acceleration): A force causes an object to accelerate. [Car with force applied] →→→ [Smaller Acceleration] 3rd Law (Action-Reaction): A rocket’s engine pushes down, rocket is pushed up. [Action] → Rocket engine → (Thrust down) [Reaction] ← Rocket pushed up ← (Equal force)

✅ Summary:

1st Law: Objects resist changes in motion (inertia). 2nd Law: The force on an object equals its mass times acceleration. 3rd Law: Every action has an equal and opposite reaction. Newton's Laws revolutionized physics and laid the foundation for classical mechanics.

1. First Law of Motion (Inertia)

Problem: A stationary car of mass 1500 kg is suddenly pushed with a force of 3000 N. How much acceleration will the car experience? Solution: From Newton's Second Law of Motion: $$F = m \cdot a$$ Where: $F = 3000 \, \text{N}$ (Force applied) $m = 1500 \, \text{kg}$ (Mass of the car) $a = ?$ (Acceleration) Rearranging the formula to find acceleration: $$a = \frac{F}{m}$$ Substituting the given values: $$a = \frac{3000}{1500} = 2 \, \text{m/s}^2$$ Answer: The acceleration of the car is $2 \, \text{m/s}^2$.

2. Second Law of Motion (Force, Mass, Acceleration)

Problem: A 5 kg object is acted upon by a net force of 20 N. What is its acceleration? Solution: From Newton's Second Law of Motion: $$F = m \cdot a$$ Where: $F = 20 \, \text{N}$ (Net force) $m = 5 \, \text{kg}$ (Mass of the object) $a = ?$ (Acceleration) Rearranging the formula to find acceleration: $$a = \frac{F}{m}$$ Substituting the given values: $$a = \frac{20}{5} = 4 \, \text{m/s}^2$$ Answer: The acceleration of the object is $4 \, \text{m/s}^2$.

3. Third Law of Motion (Action and Reaction)

Problem: A person exerts a force of 50 N on a wall. What is the force exerted by the wall on the person? Solution: According to Newton's Third Law of Motion: "For every action, there is an equal and opposite reaction." The person exerts a force of 50 N on the wall. By Newton’s Third Law, the wall exerts an equal force of 50 N back on the person, but in the opposite direction. Answer: The force exerted by the wall on the person is 50 N in the opposite direction.

4. Second Law of Motion (Force, Mass, Acceleration, with friction)

Problem: A 10 kg box is pushed with a force of 60 N. The coefficient of friction between the box and the surface is $\mu = 0.4$. Find the acceleration of the box. Solution: First, calculate the frictional force using: $$f_{\text{friction}} = \mu \cdot N$$ Where: $\mu = 0.4$ (Coefficient of friction) $N = m \cdot g$ (Normal force, where $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity) So, the normal force $N$ is: $$N = 10 \cdot 9.8 = 98 \, \text{N}$$ Now, the frictional force is: $$f_{\text{friction}} = 0.4 \cdot 98 = 39.2 \, \text{N}$$ The net force $F_{\text{net}}$ is the applied force minus the frictional force: $$F_{\text{net}} = 60 \, \text{N} - 39.2 \, \text{N} = 20.8 \, \text{N}$$ Now, use Newton's Second Law to find the acceleration: $$F_{\text{net}} = m \cdot a$$ Rearranging to find $a$: $$a = \frac{F_{\text{net}}}{m}$$ Substitute the values: $$a = \frac{20.8}{10} = 2.08 \, \text{m/s}^2$$ Answer: The acceleration of the box is $2.08 \, \text{m/s}^2$.

5. First Law of Motion (Inertia of Rest)

Problem: A car is at rest on a flat surface. If a force of 5000 N is applied to it, and the mass of the car is 2000 kg, calculate the acceleration of the car. Solution: From Newton's Second Law of Motion: $$F = m \cdot a$$ Where: $F = 5000 \, \text{N}$ (Force applied) $m = 2000 \, \text{kg}$ (Mass of the car) $a = ?$ (Acceleration) Rearranging the formula to find acceleration: $$a = \frac{F}{m}$$ Substituting the given values: $$a = \frac{5000}{2000} = 2.5 \, \text{m/s}^2$$ Answer: The acceleration of the car is $2.5 \, \text{m/s}^2$.

6. Third Law of Motion (Rocket Propulsion)

Problem: A rocket expels exhaust gases at a rate of 500 kg/s with a velocity of 2000 m/s. What is the thrust (force) produced by the rocket? Solution: From Newton's Third Law of Motion and the principle of conservation of momentum: $$F = \dot{m} \cdot v$$ Where: $\dot{m} = 500 \, \text{kg/s}$ (rate of mass ejected) $v = 2000 \, \text{m/s}$ (velocity of the exhaust gases) Substitute the values: $$F = 500 \cdot 2000 = 1,000,000 \, \text{N}$$ Answer: The thrust produced by the rocket is $1,000,000 \, \text{N}$.

Summary of Solutions:

1st Law: $a = 2 \, \text{m/s}^2$ 2nd Law: $a = 4 \, \text{m/s}^2$ 3rd Law: Force exerted by wall = $50 \, \text{N}$ (opposite direction) Frictional Force Problem: $a = 2.08 \, \text{m/s}^2$ Inertia of Rest Problem: $a = 2.5 \, \text{m/s}^2$ Rocket Propulsion Problem: $F = 1,000,000 \, \text{N}$

Elasticity and Plasticity

1. Elasticity

Definition: Elasticity is the property of a material to return to its original shape and size after the deforming force is removed. This means that the material regains its original shape when the applied stress is within a certain limit, known as the elastic limit. Elastic Limit: The maximum amount of stress that a material can endure while still returning to its original shape after the force is removed.

Key Points about Elasticity:

The material stretches or compresses when a force is applied, but once the force is removed, the material returns to its original state. Hooke's Law governs elasticity, stating that the force required to deform a material is directly proportional to the deformation (within the elastic limit). $$F = k \cdot x$$ Where: $F$ = Force applied $k$ = Spring constant or stiffness of the material $x$ = Displacement or deformation from the equilibrium position

Example of Elasticity:

A rubber band stretches when pulled, but when the force is removed, it returns to its original shape. A metal spring, when stretched, will return to its original length once the force is released (within its elastic limit).

2. Plasticity

Definition: Plasticity refers to the ability of a material to permanently deform under stress. Once the material exceeds its elastic limit, it undergoes plastic deformation and does not return to its original shape. Yield Point: The point at which a material starts to undergo permanent deformation (transition from elastic to plastic behavior).

Key Points about Plasticity:

Plastic deformation occurs when the material is stressed beyond its elastic limit. Once the material is deformed plastically, it cannot return to its original shape even after the stress is removed. In plastic materials, the stress applied is no longer proportional to the strain.

Example of Plasticity:

Clay: When clay is molded, it undergoes plastic deformation. Once shaped, it does not return to its original shape. Metal: When a metal rod is bent beyond its elastic limit, it will remain bent permanently.

3. Difference Between Elasticity and Plasticity

Feature Elasticity Plasticity Definition Ability to return to the original shape after deformation. Permanent deformation beyond the elastic limit. Deformation Reversible deformation. Irreversible deformation. Elastic Limit Deformation occurs within the elastic limit. Deformation occurs beyond the elastic limit. Behavior Follows Hooke's Law (linear relation between stress and strain). No proportionality between stress and strain after the yield point. Example Rubber bands, springs. Clay, metal rods (after bending). Energy Energy is stored and released. Energy is absorbed and not recovered.

4. Relation to Newton's Laws of Motion

Elasticity and Newton’s 3rd Law (Action and Reaction): When a force is applied to an elastic object (such as a spring), the object exerts an equal and opposite force back. This is consistent with Newton's Third Law, where every action has an equal and opposite reaction. For example, when you pull a spring, the spring pulls back with the same force. Elasticity and Newton’s 2nd Law (Force and Acceleration): The force applied to an elastic object causes a deformation (acceleration). According to Newton’s 2nd Law, the force is proportional to the acceleration or deformation, where $F = k \cdot x$. The deformation (strain) is related to the applied force (stress), following Hooke’s Law for elastic materials. Plasticity and Newton’s 2nd Law: When the force applied exceeds the elastic limit, the material will deform plastically. The relationship between stress and strain no longer follows Hooke's Law, and the material's behavior is non-linear. Plastic deformation involves the flow of material, and the forces involved are higher, leading to permanent changes in the object's structure.

5. Graph Representation

Elasticity (Hooke's Law)

Graph for Elastic Deformation: The force vs. deformation graph for an elastic material is linear, showing that the force is directly proportional to the displacement until the elastic limit is reached. Force ↑ | | / ← Hooke’s Law region | / | / | / |---------/------------ Elastic Limit | |_________________________________ Deformation In the elastic region, the material follows Hooke's Law, and the deformation is proportional to the applied force. After the elastic limit, the graph shows non-linear behavior.

Plasticity

Graph for Plastic Deformation: Once the elastic limit is exceeded, the graph shows a yield point, after which the material experiences permanent deformation. The slope flattens, and the material does not return to its original shape. Force ↑ | | / ← Elastic region | / | / | / |------------/---------- Yield Point | / | / | / |--------/-------------------- Plastic region | |_________________________________ Deformation In the plastic region, further force increases the deformation permanently without a significant increase in force.

6. Summary:

Elasticity refers to the temporary deformation where the material returns to its original shape, governed by Hooke’s Law. Plasticity refers to permanent deformation beyond the elastic limit. Both properties relate to Newton's Laws, with elasticity following proportional relationships between force and deformation and plasticity occurring after the material exceeds its elastic limit. Here is a compilation of important formulas related to force and motion, along with their derivations:

1. Newton's Second Law of Motion:

Formula: $$F = m \cdot a$$ Where: $F$ = Force $m$ = Mass $a$ = Acceleration Derivation: Newton's Second Law of Motion states that the force acting on an object is directly proportional to the acceleration produced and inversely proportional to its mass. The law can be derived from the basic definition of acceleration: $$a = \frac{\Delta v}{\Delta t}$$ Where: $\Delta v$ is the change in velocity, $\Delta t$ is the change in time. Now, the change in momentum ($\Delta p$) of an object is given by: $$\Delta p = m \cdot \Delta v$$ Since momentum $p = m \cdot v$, the rate of change of momentum is: $$\frac{d p}{dt} = \frac{d}{dt}(m \cdot v) = m \cdot \frac{dv}{dt} = m \cdot a$$ So, the force ($F$) is: $$F = m \cdot a$$ This is the equation of Newton’s Second Law.

2. Newton's Third Law of Motion:

Formula: $$F_{\text{action}} = -F_{\text{reaction}}$$ Derivation: Newton's Third Law of Motion states that for every action force, there is an equal and opposite reaction force. If two bodies interact, the force exerted by body A on body B is equal in magnitude but opposite in direction to the force exerted by body B on body A. There is no complex derivation for this law; it's a fundamental axiom. For example, when you push a wall, the wall pushes back on you with an equal and opposite force. The forces are equal in magnitude but opposite in direction.

3. Work-Energy Theorem:

Formula: $$W = F \cdot d \cdot \cos(\theta)$$ Where: $W$ = Work done $F$ = Force $d$ = Displacement $\theta$ = Angle between the force and displacement direction Derivation: Work is defined as the force applied on an object multiplied by the displacement over which the force is applied. If the force is not applied in the same direction as the displacement, the angle $\theta$ is taken into account. The formula can be derived from the basic definition of work: $$W = F \cdot \Delta x$$ Where: $\Delta x$ is the displacement of the object. When the force and displacement are not in the same direction, the angle between them is included: $$W = F \cdot d \cdot \cos(\theta)$$ If $\theta = 0^\circ$, then $\cos(0) = 1$, and the work is simply $W = F \cdot d$.

4. Kinematic Equations of Motion (for Uniform Acceleration):

The following are the standard kinematic equations used when acceleration is constant: First Equation of Motion: $$v = u + a \cdot t$$ Where: $v$ = Final velocity $u$ = Initial velocity $a$ = Acceleration $t$ = Time Derivation: This equation is derived from the definition of acceleration, which is the rate of change of velocity: $$a = \frac{\Delta v}{\Delta t} = \frac{v - u}{t}$$ Rearranging for $v$: $$v = u + a \cdot t$$ Second Equation of Motion: $$s = u \cdot t + \frac{1}{2} a \cdot t^2$$ Where: $s$ = Displacement Derivation: This equation is obtained by integrating the first equation of motion: $$v = u + a \cdot t$$ The displacement is the area under the velocity-time graph, which gives: $$s = \int_0^t (u + a \cdot t) \, dt = u \cdot t + \frac{1}{2} a \cdot t^2$$ Third Equation of Motion: $$v^2 = u^2 + 2as$$ Where: $v$ = Final velocity $u$ = Initial velocity $a$ = Acceleration $s$ = Displacement Derivation: This equation can be derived by eliminating time $t$ from the first two equations of motion: Start with: $$v = u + a \cdot t \quad \text{and} \quad s = u \cdot t + \frac{1}{2} a \cdot t^2$$ Solve for $t$ in terms of $v, u, a$, and substitute into the displacement equation: $$s = \frac{v^2 - u^2}{2a}$$ Rearranging: $$v^2 = u^2 + 2as$$

5. Impulse and Momentum:

Formula: $$F \cdot \Delta t = \Delta p$$ Where: $F$ = Force $\Delta t$ = Time interval $\Delta p$ = Change in momentum Derivation: Impulse is the change in momentum. Momentum ($p$) is defined as: $$p = m \cdot v$$ So, the change in momentum is: $$\Delta p = m \cdot (v_f - v_i)$$ According to Newton’s second law, the force is the rate of change of momentum: $$F = \frac{\Delta p}{\Delta t}$$ Multiplying both sides by $\Delta t$: $$F \cdot \Delta t = \Delta p$$ This gives the relationship between impulse and momentum.

6. Gravitational Force:

Formula: $$F = \frac{G \cdot m_1 \cdot m_2}{r^2}$$ Where: $F$ = Gravitational force between two objects $G$ = Gravitational constant ($6.674 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}$) $m_1, m_2$ = Masses of the two objects $r$ = Distance between the centers of the two objects Derivation: This is derived from Newton's Law of Universal Gravitation, which states that every particle of matter in the universe exerts an attractive force on every other particle. The force is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

7. Centripetal Force:

Formula: $$F_c = \frac{m \cdot v^2}{r}$$ Where: $F_c$ = Centripetal force $m$ = Mass of the object $v$ = Speed of the object moving in a circular path $r$ = Radius of the circular path Derivation: For an object moving in a circular path, the acceleration it experiences is centripetal acceleration: $$a_c = \frac{v^2}{r}$$ From Newton’s second law, force is the product of mass and acceleration: $$F_c = m \cdot a_c = \frac{m \cdot v^2}{r}$$ This is the formula for centripetal force.

8. Work-Energy Theorem:

Formula: $$W = \Delta K$$ Where: $W$ = Work done $\Delta K$ = Change in kinetic energy Derivation: The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. Kinetic energy ($K$) is given by: $$K = \frac{1}{2} m v^2$$ The change in kinetic energy is: $$\Delta K = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$$ By the work-energy theorem, the work done is equal to the change in kinetic energy: $$W = \Delta K$$

1. Example: Using Newton's Second Law

Problem: A car with a mass of 1000 kg accelerates at $2 \, \text{m/s}^2$. What is the force acting on the car? Solution: Use the formula for Newton's Second Law: $$F = m \cdot a$$ Where: $m = 1000 \, \text{kg}$ $a = 2 \, \text{m/s}^2$ Substituting the values: $$F = 1000 \cdot 2 = 2000 \, \text{N}$$ Answer: The force acting on the car is 2000 N.

2. Example: Kinematic Equation (First Equation of Motion)

Problem: A cyclist is initially traveling at a speed of $5 \, \text{m/s}$ and accelerates at $0.5 \, \text{m/s}^2$. How fast will the cyclist be traveling after 10 seconds? Solution: Use the first equation of motion: $$v = u + a \cdot t$$ Where: $u = 5 \, \text{m/s}$ (initial velocity) $a = 0.5 \, \text{m/s}^2$ (acceleration) $t = 10 \, \text{seconds}$ Substituting the values: $$v = 5 + (0.5 \cdot 10) = 5 + 5 = 10 \, \text{m/s}$$ Answer: The cyclist will be traveling at 10 m/s after 10 seconds.

3. Example: Work Done by a Force

Problem: A person applies a force of $50 \, \text{N}$ to move a box across the floor for a distance of $10 \, \text{m}$. The force is applied in the same direction as the motion. How much work is done? Solution: Use the formula for work: $$W = F \cdot d \cdot \cos(\theta)$$ Since the force is applied in the same direction as the motion, $\theta = 0^\circ$, so $\cos(0^\circ) = 1$. $$W = F \cdot d = 50 \cdot 10 = 500 \, \text{J}$$ Answer: The work done is 500 Joules.

4. Example: Gravitational Force

Problem: What is the gravitational force between two objects of mass $m_1 = 5 \, \text{kg}$ and $m_2 = 10 \, \text{kg}$ that are $2 \, \text{m}$ apart? Solution: Use Newton's Law of Universal Gravitation: $$F = \frac{G \cdot m_1 \cdot m_2}{r^2}$$ Where: $G = 6.674 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}$ (gravitational constant) $m_1 = 5 \, \text{kg}$ $m_2 = 10 \, \text{kg}$ $r = 2 \, \text{m}$ Substituting the values: $$F = \frac{6.674 \times 10^{-11} \cdot 5 \cdot 10}{2^2} = \frac{6.674 \times 10^{-11} \cdot 50}{4} = 8.3425 \times 10^{-10} \, \text{N}$$ Answer: The gravitational force between the two objects is $8.3425 \times 10^{-10} \, \text{N}$.

5. Example: Centripetal Force

Problem: A car of mass $800 \, \text{kg}$ is moving at a speed of $20 \, \text{m/s}$ around a curve with a radius of $50 \, \text{m}$. What is the centripetal force acting on the car? Solution: Use the formula for centripetal force: $$F_c = \frac{m \cdot v^2}{r}$$ Where: $m = 800 \, \text{kg}$ $v = 20 \, \text{m/s}$ $r = 50 \, \text{m}$ Substituting the values: $$F_c = \frac{800 \cdot 20^2}{50} = \frac{800 \cdot 400}{50} = \frac{320000}{50} = 6400 \, \text{N}$$ Answer: The centripetal force acting on the car is 6400 N.

6. Example: Impulse and Momentum

Problem: A soccer player kicks a ball of mass $0.5 \, \text{kg}$ and changes its velocity from $2 \, \text{m/s}$ to $10 \, \text{m/s}$. What is the impulse exerted on the ball? Solution: Impulse is the change in momentum. The momentum is given by $p = m \cdot v$, so the change in momentum is: $$\Delta p = m \cdot (v_f - v_i)$$ Where: $m = 0.5 \, \text{kg}$ $v_f = 10 \, \text{m/s}$ $v_i = 2 \, \text{m/s}$ Substituting the values: $$\Delta p = 0.5 \cdot (10 - 2) = 0.5 \cdot 8 = 4 \, \text{Ns}$$ Answer: The impulse exerted on the ball is 4 Ns.

7. Example: Work-Energy Theorem