Ace Your Physics: 100 MCQs on Work, Energy & Power (Class 11)

Work, Energy & Power

Interactive Learning Module & 100 MCQs for Class 11 Physics

Introduction

Welcome! This module explores the fundamental concepts of Work, Energy, and Power in physics. These concepts are crucial for understanding how forces cause motion and how energy is transferred and transformed in various physical processes. Mastering these ideas forms the bedrock for many advanced topics in mechanics and other areas of physics.

Here, you'll find detailed explanations, important formulas, illustrative examples, and an extensive multiple-choice quiz to test your understanding. Let's dive in!

Work Explained

What is Work in Physics?

In physics, **Work** is done on an object when a force acting on the object causes it to move through a certain displacement. It's a measure of energy transfer. Importantly, for work to be done, three conditions must be met:

A force must be applied to the object.
The object must move (undergo displacement).
There must be a component of the force acting along the direction of the displacement (or opposite to it).

Calculating Work Done by a Constant Force

When a constant force F acts on an object, causing a displacement s, the work done (W) is defined as the scalar (dot) product of the force vector and the displacement vector:

W = F ⋅ s = Fs cos(θ)

Where:

W is the work done (scalar quantity).
F is the magnitude of the constant force.
s is the magnitude of the displacement.
θ is the angle between the force vector F and the displacement vector s.

Work = (Component of Force along displacement) × Displacement = F cos(θ) × s

Units and Nature of Work

The SI unit of work is the **Joule (J)**. 1 Joule = 1 Newton-meter (1 J = 1 N⋅m).
The CGS unit of work is the **erg**. 1 Joule = 10⁷ ergs.
Work is a **scalar** quantity, even though it's calculated from two vector quantities (Force and Displacement).
The dimensional formula for work is [ML²T⁻²], the same as energy.

Types of Work Done

Based on the angle θ between force and displacement:

Positive Work (0° ≤ θ < 90°): When cos(θ) is positive. Force has a component in the direction of motion. This typically increases the object's energy (e.g., lifting a box upwards by an applied upward force). Work done by gravity when an object falls.
Zero Work (θ = 90°): When cos(90°) = 0. The force is perpendicular to the displacement. No energy is transferred by this force (e.g., work done by centripetal force in uniform circular motion, work done carrying a suitcase horizontally at constant velocity by the supporting force).
Negative Work (90° < θ ≤ 180°): When cos(θ) is negative. Force has a component opposite to the direction of motion. This typically decreases the object's energy (e.g., work done by friction, work done by gravity when an object is lifted).

Work Done by a Variable Force

If the force acting on the object is not constant but varies with position, the work done is calculated by integrating the force over the displacement:

W = ∫_x₁^x₂ F(x) dx (For motion along x-axis)
In general: W = ∫_r₁^r₂ F ⋅ ds

Graphically, the work done by a variable force is represented by the **area under the Force-Displacement (F-x) graph**.

Area under Force-Displacement graph represents Work Done

The shaded area gives the work done.

Energy Explained

What is Energy?

Energy is defined as the **capacity to do work**. It is a fundamental scalar quantity associated with the state of one or more objects. Energy can exist in various forms (mechanical, thermal, chemical, electrical, nuclear, etc.) and can be transformed from one form to another.

The SI unit of energy is the **Joule (J)**, the same as work. Its dimensional formula is [ML²T⁻²].

Mechanical Energy

Mechanical energy is the energy associated with the motion and position of an object. It is typically divided into two forms: Kinetic Energy and Potential Energy.

Kinetic Energy (KE)

Kinetic Energy is the energy possessed by an object by virtue of its **motion**. An object in motion can do work on other objects upon collision.

The kinetic energy (K or KE) of an object of mass m moving with a speed v is given by:

KE = ½ mv²

KE is always non-negative (zero or positive).
KE is directly proportional to the mass and the square of the speed.

Relationship between KE and Momentum (p)

Linear momentum p = mv. We can express KE in terms of momentum:

KE = ½ mv² = ½ m (p/m)² = ½ m (p²/m²) = p²/2m
Conversely: p = √(2m KE)

This relationship is useful when comparing objects with equal KE or equal momentum.

Potential Energy (PE)

Potential Energy is the energy stored in an object or system due to its **position or configuration**. This stored energy has the potential to be converted into kinetic energy or do work.

Potential energy is defined only for **conservative forces**. A force is conservative if the work done by it in moving an object between two points is independent of the path taken (e.g., gravity, ideal spring force). Change in Potential Energy (ΔPE) is defined as the negative of the work done by the conservative force:

ΔPE = PE_final - PE_initial = -W_conservative

Or, it's the work done by an external agent against the conservative force to change the configuration without changing kinetic energy.

Types of Potential Energy:

Gravitational Potential Energy (GPE): Energy stored due to an object's position in a gravitational field. Near the Earth's surface, for an object of mass m at height h above a reference level:
PE_gravity = mgh
where g is the acceleration due to gravity.
Elastic Potential Energy: Energy stored in an elastic object (like a spring) when it is deformed (stretched or compressed). For a spring with spring constant k deformed by a distance x from its equilibrium position:
PE_elastic = ½ kx²

Energy stored is proportional to the square of displacement.

Total Mechanical Energy

The total mechanical energy (E) of a system is the sum of its kinetic energy and potential energy:

E = KE + PE = ½ mv² + PE

Conservation Laws

Work-Energy Theorem

This is a fundamental theorem linking work and kinetic energy. It states that the **net work done by all forces (conservative and non-conservative) acting on an object is equal to the change in its kinetic energy**.

W_net = ΔKE = KE_final - KE_initial = ½ mv_f² - ½ mv_i²

This theorem is universally applicable, regardless of the nature of the forces involved.

Law of Conservation of Mechanical Energy

This law is a special case applicable only when all the forces doing work on a system are **conservative**. It states that:

If only conservative forces are doing work within an isolated system, the total mechanical energy (KE + PE) of the system remains constant.

KE_initial + PE_initial = KE_final + PE_final
E = KE + PE = Constant (when only conservative forces act)

Examples:

Freely falling body (neglecting air resistance): As the body falls, its PE decreases, but its KE increases such that the sum (KE + PE) remains constant.
Simple pendulum oscillation (neglecting air resistance): Energy continuously transforms between KE (maximum at the bottom) and PE (maximum at the extremes), but the total mechanical energy is conserved.

Simple Pendulum conserving mechanical energy

Energy conversion in a simple pendulum.

Conservative vs. Non-Conservative Forces

Conservative Forces

Work done is path independent.
Work done over a closed loop is zero.
Associated with a Potential Energy function (F = -∇PE).
Conserve mechanical energy.
Examples: Gravity, Ideal Spring Force, Electrostatic Force.

Non-Conservative Forces

Work done depends on the path taken.
Work done over a closed loop is generally non-zero (usually negative).
Not associated with a Potential Energy function.
Dissipate mechanical energy (often as heat, sound).
Examples: Friction, Air Resistance, Viscous Drag, Tension (can be complex).

Law of Conservation of Energy (General)

This is a broader principle: **Energy can neither be created nor destroyed, only transformed from one form to another or transferred from one system to another.** The total energy of an isolated system remains constant.

When non-conservative forces (like friction) are present, mechanical energy is *not* conserved, but the lost mechanical energy is converted into other forms (like heat), so the *total* energy is still conserved.

ΔKE + ΔPE + ΔE_internal + ... = 0 (For an isolated system)

Power Explained

What is Power?

Power is the **rate at which work is done** or the **rate at which energy is transferred or transformed**. It measures how quickly work is performed or energy is used.

Average Power

The average power (P_avg) delivered over a time interval Δt during which work ΔW is done is:

P_avg = ΔW / Δt = Total Work Done / Total Time Taken

Instantaneous Power

The instantaneous power (P) is the power delivered at a specific moment in time. It's the limiting value of the average power as the time interval Δt approaches zero:

P = dW / dt

Power in terms of Force and Velocity

Since dW = F ⋅ ds, the instantaneous power can also be expressed as the scalar product of the force vector F and the instantaneous velocity vector v:

P = dW/dt = (F ⋅ ds) / dt = F ⋅ (ds/dt) = F ⋅ v
P = Fv cos(θ)

Where θ is the angle between the force and velocity vectors at that instant.

Units and Dimension of Power

The SI unit of power is the **Watt (W)**. 1 Watt = 1 Joule per second (1 W = 1 J/s).
Another common unit is **horsepower (hp)**. 1 hp ≈ 746 W.
Power is a **scalar** quantity.
The dimensional formula for power is [ML²T⁻³].

Energy vs. Power

It's important to distinguish between energy and power:
- Energy is the total amount of work that can be done.
- Power is how fast that work is done.
A low-power device can do a large amount of work if given enough time. A high-power device can do the same work much faster.
Electrical energy is often measured in kilowatt-hours (kWh), which is a unit of energy (Power × Time): 1 kWh = (1000 W) × (3600 s) = 3.6 × 10⁶ J.

Collisions

A collision is a brief interaction between two or more bodies causing a change in their motion. During a collision, impulsive forces act, significantly changing the momenta of the interacting bodies.

Conservation Laws in Collisions

Conservation of Linear Momentum: In *any* type of collision (elastic or inelastic), if the net external force on the system of colliding bodies is zero, the total linear momentum of the system is conserved.
p_{initial_total} = p_{final_total}
Conservation of Kinetic Energy: Total kinetic energy is conserved *only* in perfectly elastic collisions. In inelastic collisions, some KE is converted into other forms (heat, sound, deformation).

Types of Collisions

Elastic Collision: A collision in which both total linear momentum AND total kinetic energy of the system are conserved. Collisions between ideal hard spheres or subatomic particles are often approximated as elastic.
Inelastic Collision: A collision in which total linear momentum is conserved, but total kinetic energy is *not* conserved. Some KE is lost. Most real-world macroscopic collisions are inelastic.
Perfectly Inelastic Collision: A type of inelastic collision where the colliding objects stick together after the collision and move with a common final velocity. Maximum loss of KE (consistent with momentum conservation) occurs in this type.

Coefficient of Restitution (e)

The coefficient of restitution provides a measure of the elasticity of a collision between two objects in one dimension. It is defined as the ratio of the relative speed of separation after the collision to the relative speed of approach before the collision:

e = |v₂ - v₁| / |u₁ - u₂|

Where u₁, u₂ are initial velocities and v₁, v₂ are final velocities along the line of impact.

For a **perfectly elastic** collision: e = 1 (Relative speed of separation equals relative speed of approach).
For a **perfectly inelastic** collision: e = 0 (Relative speed of separation is zero as objects stick together).
For **inelastic** collisions: 0 < e < 1.

Practice Quiz (100 Questions)

Test your understanding with the following multiple-choice questions. Click the button below each question to reveal the explanation and correct answer.

1. Work done is defined as:

A) Force × Time
B) Force × Displacement
C) Force × Velocity
D) Force × Acceleration

Explanation: Work done (W) by a constant force (F) is defined as the product of the component of the force in the direction of the displacement (s) and the magnitude of the displacement. Mathematically, W = F ⋅ s = Fs cos(θ), where θ is the angle between the force and displacement vectors.
Correct Answer: B

2. The SI unit of work is:

A) Watt (W)
B) Newton (N)
C) Joule (J)
D) Pascal (Pa)

Explanation: Work is measured in Joules (J) in the SI system. 1 Joule is defined as the work done when a force of 1 Newton displaces an object by 1 meter in the direction of the force (1 J = 1 N⋅m).
Correct Answer: C

3. When is the work done by a force considered zero? (Select the best answer)

A) When the force is zero.
B) When the displacement is zero.
C) When the force is perpendicular to the displacement.
D) All of the above.

Explanation: Work done W = Fs cos(θ). Work is zero if F=0, s=0, or cos(θ)=0 (meaning θ=90°, force perpendicular to displacement). Therefore, all the listed conditions result in zero work done.
Correct Answer: D

4. A coolie lifts a luggage of 15 kg from the ground and puts it on his head 1.5 m above the ground. Calculate the work done by him on the luggage (g = 10 m/s²).

A) 150 J
B) 225 J
C) 15 J
D) 0 J

Explanation: The force applied by the coolie is against gravity, equal to the weight of the luggage (mg). The displacement is vertical (h). Work done W = Force × Displacement = (mg) × h = (15 kg × 10 m/s²) × 1.5 m = 150 × 1.5 = 225 J.
Correct Answer: B

5. What type of energy is possessed by a moving car?

A) Potential Energy
B) Kinetic Energy
C) Chemical Energy
D) Sound Energy

Explanation: Kinetic energy is the energy possessed by an object due to its motion. Since the car is moving, it possesses kinetic energy.
Correct Answer: B

6. The formula for kinetic energy (KE) is:

A) mgh
B) mv
C) ½ mv²
D) ma

Explanation: The kinetic energy of an object of mass 'm' moving with velocity 'v' is given by the formula KE = ½ mv².
Correct Answer: C

7. If the velocity of a body is doubled, its kinetic energy becomes:

A) Half
B) Double
C) Four times
D) Unchanged

Explanation: KE = ½ mv². If velocity v becomes 2v, the new KE' = ½ m(2v)² = ½ m(4v²) = 4 (½ mv²) = 4 KE. The kinetic energy becomes four times the original value.
Correct Answer: C

8. The energy possessed by a body due to its position or configuration is called:

A) Kinetic Energy
B) Potential Energy
C) Mechanical Energy
D) Thermal Energy

Explanation: Potential energy is the stored energy an object has due to its position (like height above ground - gravitational PE) or its state/configuration (like a stretched spring - elastic PE).
Correct Answer: B

9. What is the potential energy of an object of mass 10 kg kept at a height of 5 m? (g = 9.8 m/s²)

A) 49 J
B) 50 J
C) 490 J
D) 500 J

Explanation: Gravitational Potential Energy (PE) = mgh. PE = 10 kg × 9.8 m/s² × 5 m = 490 J.
Correct Answer: C

10. Work done by a conservative force:

A) Depends on the path taken
B) Is always zero
C) Depends only on the initial and final positions
D) Is always positive

Explanation: A conservative force (like gravity, elastic spring force) is one for which the work done in moving an object between two points is independent of the path taken and depends only on the initial and final positions. The work done over a closed path is zero for a conservative force.
Correct Answer: C

11. Which of the following is a non-conservative force?

A) Gravitational force
B) Electrostatic force
C) Frictional force
D) Elastic spring force

Explanation: Frictional force is a non-conservative force because the work done by friction depends on the path taken (longer path = more work done by friction) and energy is dissipated as heat. Work done by friction over a closed path is not zero.
Correct Answer: C

12. The Work-Energy Theorem states that:

A) Work done = Change in Potential Energy
B) Work done = Change in Kinetic Energy
C) Power = Work done / Time
D) Force = Mass × Acceleration

Explanation: The Work-Energy Theorem states that the net work done on an object by all forces is equal to the change in its kinetic energy (W_net = ΔKE = KE_final - KE_initial).
Correct Answer: B

13. An object of mass 5 kg is moving with a velocity of 10 m/s. A force is applied which brings it to rest after covering a distance of 25 m. Calculate the work done by the force.

A) -250 J
B) 250 J
C) -500 J
D) 500 J

Explanation: Using the Work-Energy Theorem, W_net = ΔKE. Initial KE = ½ mv² = ½ × 5 × (10)² = 250 J. Final KE = 0 (since it comes to rest). Work done W = KE_final - KE_initial = 0 - 250 J = -250 J. The negative sign indicates work done against the motion.
Correct Answer: A

14. Power is defined as:

A) The total energy consumed
B) The rate of doing work
C) The force applied per unit area
D) The capacity to do work

Explanation: Power (P) is the rate at which work (W) is done or energy is transferred. P = W / t, where 't' is the time taken.
Correct Answer: B

15. The SI unit of power is:

A) Joule (J)
B) Newton (N)
C) Watt (W)
D) Kilogram (kg)

Explanation: The SI unit of power is the Watt (W), named after James Watt. 1 Watt is defined as 1 Joule of work done per second (1 W = 1 J/s).
Correct Answer: C

16. 1 horsepower (hp) is approximately equal to:

A) 500 W
B) 746 W
C) 1000 W
D) 100 W

Explanation: Horsepower (hp) is another unit of power, commonly used for engines. 1 horsepower is approximately equal to 746 Watts.
Correct Answer: B

17. A machine does 1960 J of work in 2 minutes. What is its power?

A) 16.33 W
B) 980 W
C) 3920 W
D) 0 W

Explanation: Time t = 2 minutes = 2 × 60 seconds = 120 s. Work W = 1960 J. Power P = W / t = 1960 J / 120 s ≈ 16.33 W.
Correct Answer: A

18. Instantaneous power can also be expressed as:

A) F / v
B) Force × Velocity (scalar product: F ⋅ v)
C) F × a
D) m × v

Explanation: Instantaneous power P = dW/dt. Since dW = F ⋅ ds, P = (F ⋅ ds) / dt = F ⋅ (ds/dt) = F ⋅ v, where F is the force and v is the instantaneous velocity. It's the scalar (dot) product of force and velocity vectors.
Correct Answer: B

19. The law of conservation of energy states that:

A) Energy can be created but not destroyed.
B) Energy can be destroyed but not created.
C) Energy can neither be created nor destroyed, only transformed.
D) Energy is always constant in kinetic form.

Explanation: The law of conservation of energy is a fundamental principle stating that the total energy of an isolated system remains constant over time. Energy can change forms (e.g., potential to kinetic, mechanical to heat), but the total amount remains the same.
Correct Answer: C

20. For a freely falling body, which quantity remains constant throughout the fall (neglecting air resistance)?

A) Kinetic Energy
B) Potential Energy
C) Total Mechanical Energy
D) Velocity

Explanation: When air resistance is neglected, gravity is the only force doing work, and it's a conservative force. Therefore, the total mechanical energy (sum of kinetic and potential energy, KE + PE) remains constant throughout the fall. PE converts into KE as it falls.
Correct Answer: C

21. A ball is dropped from a height 'h'. Just before hitting the ground, its velocity is 'v'. What is its velocity when it is at height h/2?

A) v/2
B) v/√2
C) v
D) v/4

Explanation: By conservation of energy: Initial energy (at height h) = mgh (KE=0). Final energy (at height 0) = ½ mv² (PE=0). So, mgh = ½ mv². At height h/2: Total energy = mgh. PE = mg(h/2). KE = Total Energy - PE = mgh - mg(h/2) = mg(h/2). Let velocity at h/2 be v'. KE = ½ mv'². So, ½ mv'² = mg(h/2) => v'² = gh. Since mgh = ½ mv², we have v² = 2gh or gh = v²/2. Substituting gh in v'²: v'² = v²/2 => v' = v/√2.
Correct Answer: B

22. The potential energy stored in a spring is given by:

A) ½ kx
B) kx
C) ½ kx²
D) kx²

Explanation: The elastic potential energy stored in a spring with spring constant 'k' when it is stretched or compressed by a distance 'x' from its equilibrium position is given by PE_elastic = ½ kx².
Correct Answer: C

23. If a spring with spring constant 'k' is stretched by a distance 'x', the work done by the spring force is:

A) ½ kx²
B) -½ kx²
C) kx²
D) -kx²

Explanation: The spring force (F = -kx) acts opposite to the displacement when stretched. The work done *by* the spring force is the negative change in its potential energy or can be calculated by integrating the force. Work done by spring force = -ΔPE = -(½ kx² - 0) = -½ kx². Alternatively, work done *by the external agent* to stretch the spring is +½ kx².
Correct Answer: B

24. A body of mass 'm' is moving in a circle of radius 'r' with constant speed 'v'. The work done by the centripetal force in one complete revolution is:

A) mv²/r
B) 2πr × (mv²/r)
C) ½ mv²
D) Zero

Explanation: The centripetal force always acts towards the center of the circle, perpendicular to the instantaneous velocity (and displacement) of the object, which is tangential. Since the angle between force and displacement is always 90°, cos(90°) = 0. Therefore, the work done by the centripetal force is always zero.
Correct Answer: D

25. A light body and a heavy body have the same kinetic energy. Which one has greater momentum?

A) The light body
B) The heavy body
C) Both have the same momentum
D) Cannot be determined

Explanation: Kinetic Energy KE = p²/2m, where p is momentum (p=mv). So, p = √(2m KE). If KE is the same for both bodies, then momentum 'p' is proportional to √m. The heavier body (larger 'm') will have greater momentum.
Correct Answer: B

26. A light body and a heavy body have the same momentum. Which one has greater kinetic energy?

A) The light body
B) The heavy body
C) Both have the same kinetic energy
D) Cannot be determined

Explanation: Kinetic Energy KE = p²/2m. If momentum 'p' is the same for both bodies, then KE is inversely proportional to mass 'm' (KE ∝ 1/m). The lighter body (smaller 'm') will have greater kinetic energy.
Correct Answer: A

27. Energy is a:

A) Scalar quantity
B) Vector quantity
C) Tensor quantity
D) Dimensionless quantity

Explanation: Energy is a scalar quantity. It has magnitude but no direction. Work and power are also scalar quantities.
Correct Answer: A

28. Work is a:

A) Scalar quantity
B) Vector quantity
C) Tensor quantity
D) Sometimes scalar, sometimes vector

Explanation: Although work is calculated using force and displacement (vectors) via the dot product (W = F ⋅ s), the result (work) is a scalar quantity. It represents energy transfer and has magnitude only.
Correct Answer: A

29. Power is a:

A) Scalar quantity
B) Vector quantity
C) Tensor quantity
D) Dimensionless quantity

Explanation: Power, being the rate of doing work (a scalar) or energy transfer (a scalar), is also a scalar quantity. Although instantaneous power can be calculated as P = F ⋅ v (dot product of two vectors), the result is scalar.
Correct Answer: A

30. In an elastic collision:

A) Only momentum is conserved
B) Only kinetic energy is conserved
C) Both momentum and kinetic energy are conserved
D) Neither momentum nor kinetic energy is conserved

Explanation: An elastic collision is defined as a collision in which both the total momentum and the total kinetic energy of the system are conserved.
Correct Answer: C

31. In an inelastic collision:

A) Only momentum is conserved
B) Only kinetic energy is conserved
C) Both momentum and kinetic energy are conserved
D) Neither momentum nor kinetic energy is conserved

Explanation: In any collision (elastic or inelastic), total momentum is conserved (assuming no external forces). However, in an inelastic collision, some kinetic energy is lost (converted into heat, sound, deformation, etc.), so only momentum is conserved. In a perfectly inelastic collision, the objects stick together after impact.
Correct Answer: A

32. A force F = (2i + 3j - k) N acts on a body, producing a displacement s = (i - j + 2k) m. The work done is:

A) 3 J
B) -3 J
C) 7 J
D) -7 J

Explanation: Work done W = F ⋅ s (dot product). W = (2i + 3j - k) ⋅ (i - j + 2k) W = (2)(1) + (3)(-1) + (-1)(2) W = 2 - 3 - 2 = -3 J.
Correct Answer: B

33. The area under the Force-displacement (F-x) graph represents:

A) Power
B) Impulse
C) Change in momentum
D) Work done

Explanation: For a variable force, the work done is given by the integral W = ∫ F dx. This integral represents the area under the curve on a Force versus displacement graph.
Correct Answer: D

34. Kilowatt-hour (kWh) is the unit of:

A) Power
B) Work/Energy
C) Time
D) Force

Explanation: Kilowatt (kW) is a unit of power. Hour (h) is a unit of time. Power × Time = Energy (or Work). Therefore, kWh is a unit of energy, commonly used for electrical energy consumption (1 kWh = 1 'unit' of electricity). 1 kWh = (1000 W) × (3600 s) = 3.6 × 10⁶ J.
Correct Answer: B

35. When you stretch a rubber band, you store:

A) Kinetic energy
B) Gravitational potential energy
C) Elastic potential energy
D) Chemical energy

Explanation: Stretching or deforming an elastic object like a rubber band stores energy due to its change in configuration. This stored energy is called elastic potential energy.
Correct Answer: C

36. A block is pulled across a rough horizontal surface at constant velocity. The work done by friction is:

A) Positive
B) Negative
C) Zero
D) Equal to the applied force

Explanation: Frictional force always opposes motion. Therefore, the angle between the frictional force and the displacement is 180°. Work done W = F_friction * s * cos(180°) = - (F_friction * s). Work done by friction is negative, representing energy dissipation.
Correct Answer: B

37. A car engine delivers a constant power P. If the resistive forces are negligible, how does its velocity 'v' depend on time 't' (starting from rest)?

A) v ∝ t
B) v ∝ √t
C) v ∝ t²
D) v is constant

Explanation: Power P = Force × Velocity = Fv. Also, F = ma = m(dv/dt). So P = m(dv/dt)v. Rearranging: P dt = mv dv. Integrating both sides (∫P dt = ∫mv dv): Pt = ½ mv² (assuming v=0 at t=0). Thus, v² = (2P/m)t => v = √(2P/m) * √t. Therefore, v ∝ √t.
Correct Answer: B

38. Work done by the tension in the string of a simple pendulum during one complete oscillation is:

A) Positive
B) Negative
C) Zero
D) Depends on the amplitude

Explanation: The tension in the string of a simple pendulum always acts along the string, towards the point of suspension. The instantaneous displacement of the bob is always perpendicular to the string (along the tangent to the arc). Since force (tension) is perpendicular to displacement, the work done by tension is zero (W = T⋅s = Ts cos(90°) = 0).
Correct Answer: C

39. What is the dimensional formula for energy?

A) [MLT⁻²]
B) [ML²T⁻²]
C) [ML²T⁻³]
D) [MLT⁻¹]

Explanation: Energy has the same dimensions as work. Work = Force × Displacement. Dimension of Force = [MLT⁻²]. Dimension of Displacement = [L]. Therefore, Dimension of Work/Energy = [MLT⁻²] × [L] = [ML²T⁻²].
Correct Answer: B

40. What is the dimensional formula for power?

A) [MLT⁻²]
B) [ML²T⁻²]
C) [ML²T⁻³]
D) [MLT⁻³]

Explanation: Power = Work / Time. Dimension of Work = [ML²T⁻²]. Dimension of Time = [T]. Therefore, Dimension of Power = [ML²T⁻²] / [T] = [ML²T⁻³].
Correct Answer: C

100. A spring stores 10 J of energy when compressed by 1 cm. How much energy will it store if compressed by 2 cm?

A) 10 J
B) 20 J
C) 40 J
D) 80 J

Explanation: Energy stored in a spring PE = ½ kx². So, PE is proportional to x² (PE ∝ x²). If the compression x is doubled (from 1 cm to 2 cm), the energy stored will become (2)² = 4 times the original energy. New Energy = 4 × Initial Energy = 4 × 10 J = 40 J.
Correct Answer: C

Key Takeaways

Work (W) is energy transfer via force causing displacement: W = Fs cos(θ). It's scalar, measured in Joules (J).
Work-Energy Theorem: Net work done equals change in kinetic energy: W<0xE2><0x82><0x99><0xE1><0xB5><0x8A><0xE1><0xB5><0x9C> = ΔKE.
Kinetic Energy (KE): Energy of motion: KE = ½ mv² = p²/2m.
Potential Energy (PE): Stored energy due to position/configuration (defined for conservative forces). E.g., PE<0xE1><0xB5><0x8D><0xE1><0xB5><0xA3><0xE1><0xB5><0x92><0xE1><0xB5><0x9B> = mgh, PE<0xE2><0x82><0x91><0xE1><0xB5><0x85><0xE1><0xB5><0x92><0xE1><0xB5><0xA0><0xE1><0xB5><0x9C><0xE1><0xB5><0x96><0xE1><0xB5><0x84> = ½ kx².
Conservative Forces: Work done is path-independent (e.g., gravity). Mechanical energy (KE + PE) is conserved if only these forces act.
Non-Conservative Forces: Work done is path-dependent (e.g., friction). They dissipate mechanical energy.
Conservation of Energy (Total): Energy is never created/destroyed, only transformed. Total energy of an isolated system is constant.
Power (P): Rate of doing work or energy transfer: P = W/t = dW/dt = F ⋅ v. Unit is Watt (W).
Collisions: Momentum is conserved if F<0xE2><0x82><0x91><0xE2><0x82><0x9F><0xE1><0xB5><0x9C> = 0. KE is conserved only in elastic collisions (coefficient of restitution e=1). For perfectly inelastic collisions, e=0.

Interactive Guide: Work, Energy & Power (Class 11 Physics)