Work, Energy and Power - Class 11 Physics

Work, Energy and Power

Overview: This chapter connects the concepts of force and motion through Work and Energy. It encompasses the Work-Energy Theorem, Conservation of Energy, and Power.

1. Work

A. Work done by a Constant Force

Work is defined as the product of the component of force in the direction of displacement and the magnitude of displacement.

W = F . d = Fd cos(θ)

F: Constant Force
d: Displacement
θ: Angle between Force and Displacement vectors
Scalar Quantity: Work has no direction.
SI Unit: Joule (J). 1 J = 1 N.m
Dimensions: [ML²T^-2]

Cases:

θ = 0° (Force || Displacement) → W = Fd (Max Positive Work)
θ = 90° (Force ⊥ Displacement) → W = 0 (Zero Work) e.g., Centripetal force.
θ = 180° (Force opposite) → W = -Fd (Max Negative Work) e.g., Friction.

B. Work done by a Variable Force

When force varies with position, work is calculated by integration (area under F-x graph).

W = ∫ F(x) dx

From position x_i to x_f.

2. Kinetic Energy (K)

The energy possessed by a body by virtue of its motion.

K = (1/2)mv²

Relation with Momentum (p): p = mv

K = p² / 2m

3. Work-Energy Theorem

The work done by the net force on a particle is equal to the change in its kinetic energy.

W_net = ΔK = K_f - K_i

Valid for both constant and variable forces.

4. Potential Energy (U)

The energy stored in a body by virtue of its position or configuration.

Defined only for Conservative Forces.

ΔU = - W_conservative

F = - dU/dx

A. Gravitational Potential Energy

For a body of mass m at height h near Earth's surface:

U = mgh

B. Potential Energy of a Spring

For a spring with spring constant k stretched/compressed by x:

U = (1/2)kx²

Restoring Force (Hooke's Law): F = -kx

5. Conservative & Non-Conservative Forces

Conservative Forces

Work done depends only on initial and final positions, not on the path taken.
Work done in a closed loop is zero.
Examples: Gravitational force, Electrostatic force, Spring force.

Non-Conservative Forces

Work done depends on the path taken.
Work done in a closed loop is not zero.
Examples: Friction, Viscous drag, Air resistance.

6. Conservation of Mechanical Energy

If only conservative forces act on a system, the total mechanical energy (E) remains constant.

E = K + U = Constant

ΔK + ΔU = 0

7. Motion in a Vertical Circle

A particle of mass m tied to a string of length L moving in a vertical circle.

At Lowest Point (L): Velocity v_L ≥ √(5gL) to complete the circle. Tension is max.
At Highest Point (H): Velocity v_H ≥ √(gL) to just cross the top. Tension is min (zero).
Condition for oscillation: v_L ≤ √(2gL).

8. Collisions

A. Elastic Collision

Both Momentum and Kinetic Energy are conserved.

1D Elastic Collision:

v₁ = [(m₁-m₂)/(m₁+m₂)]u₁ + [2m₂/(m₁+m₂)]u₂

B. Inelastic Collision

Only Momentum is conserved. Kinetic Energy is lost (converted to heat/sound).

Perfectly Inelastic: Bodies stick together and move with common velocity.

v = (m₁u₁ + m₂u₂) / (m₁ + m₂)

Coefficient of Restitution (e)

e = |Relative velocity of separation| / |Relative velocity of approach|

e = 1 (Elastic)
0 < e < 1 (Inelastic)
e = 0 (Perfectly Inelastic)

9. Power

Rate of doing work.

P = dW/dt = F . v

SI Unit: Watt (W). 1 W = 1 J/s.
Horsepower (hp): 1 hp = 746 W.

Numericals - Work, Energy and Power

Numericals

Vector Product

Q1. A force F = (5i + 3j + 2k) N is applied on a particle which displaces it from origin to point r = (2i - j) m. Calculate the work done.

Given:

F = 5i + 3j + 2k

d = r₂ - r₁

d = (2i - j) - (0) = 2i - j

W = F ċ d

W = (5i + 3j + 2k) ċ (2i - 1j + 0k)

W = (5)(2) + (3)(-1) + (2)(0)

W = 10 - 3 + 0

W = 7 J

Error Analysis

Q2. The momentum of a body is increased by 50%. What is the percentage increase in its Kinetic Energy?

Let initial momentum = p

Final momentum p' = p + 50% of p

p' = p + 0.5p = 1.5p

Relation between KE and Momentum:

K = p² / 2m

K' = (p')² / 2m

K' = (1.5p)² / 2m

K' = 2.25 (p² / 2m)

K' = 2.25 K

Percentage Increase:

% Inc = [(K' - K) / K] × 100

% Inc = [(2.25K - K) / K] × 100

% Inc = 1.25 × 100

% Inc = 125%

Work-Energy Theorem

Q3. A bullet of mass 10g traveling horizontally with a velocity of 150 m/s strikes a stationary wooden block and comes to rest in 0.03 s. Calculate the distance of penetration.

Given: m = 10g = 0.01 kg, u = 150 m/s, v = 0, t = 0.03 s

Using Equation of Motion:

v = u + at

0 = 150 + a(0.03)

a = -150 / 0.03

a = -5000 m/s²

Calculating Retarding Force:

F = ma

F = 0.01 × (-5000)

F = -50 N

Using Work-Energy Theorem:

W = ΔK = K_f - K_i

W = 0 - (1/2)mu²

W = -0.5 × 0.01 × (150)²

W = -112.5 J

Relating to Distance:

W = F ċ s

-112.5 = -50 × s

s = 112.5 / 50

s = 2.25 m

Spring Energy

Q4. A spring with spring constant k = 100 N/m is compressed by 10 cm. How much potential energy is stored in it?

Given: k = 100 N/m, x = 10 cm = 0.1 m

U = (1/2) kx²

U = 0.5 × 100 × (0.1)²

U = 50 × 0.01

U = 0.5 J

Conservation of Energy

Q5. A ball is dropped from a height of 20m. Calculate its velocity just before it hits the ground. (g = 10 m/s²)

Using Conservation of Mechanical Energy:

PE_top = KE_bottom

mgh = (1/2)mv²

v² = 2gh

Substituting values:

v = √(2gh)

v = √(2 × 10 × 20)

v = √(400)

v = 20 m/s

Power & Efficiency

Q6. An engine pumps up 100 kg of water through a height of 10 m in 5 s. Efficiency is 60%. Calculate the power of the engine.

Output Power (Useful Work):

P_out = Work / time = mgh / t

P_out = (100 × 10 × 10) / 5

P_out = 2000 W

Using Efficiency Formula:

η = P_out / P_in

0.60 = 2000 / P_in

P_in = 2000 / 0.6

P_in = 3333.33 W

P_in ≈ 3.33 kW

Inelastic Collision

Q7. A 2kg ball moving at 6 m/s collides head on with a 4kg ball at rest. If the collision is perfectly inelastic, find the common velocity.

Given: m₁ = 2kg, u₁ = 6 m/s, m₂ = 4kg, u₂ = 0

By Conservation of Momentum:

m₁u₁ + m₂u₂ = (m₁ + m₂)v

(2)(6) + (4)(0) = (2 + 4)v

12 + 0 = 6v

12 = 6v

v = 2 m/s

Vertical Circular Motion

Q8. What is the minimum speed required at the bottom to complete a vertical circle of radius 2m? (g = 10 m/s²)

Condition for completing the circle:

v_bottom ≥ √(5gR)

Calculation:

v = √(5 × 10 × 2)

v = √(100)

v = 10 m/s

Integration

Q9. A force F = 2x + 1 acts on a particle. Calculate the work done to move it from x = 1 to x = 2.

Work involved with Variable Force:

W = ∫ F dx

W = ∫₁² (2x + 1) dx

Integrating:

W = [x² + x]₁²

W = (2² + 2) - (1² + 1)

W = (4 + 2) - (1 + 1)

W = 6 - 2

W = 4 J

Restitution

Q10. A ball falls from a height of 10m and rebounds to 2.5m. Calculate the coefficient of restitution.

Given: H₁ = 10m, H₂ = 2.5m

Formula:

e = √(H₂ / H₁)

Calculation:

e = √(2.5 / 10)

e = √(1 / 4)

e = 1/2

e = 0.5

(Coefficient of restitution is unitless)

Formulas & Facts - Work, Energy and Power

Formulas

Physical Qty / Concept	Formula	SI Unit
Work (Constant Force)	W = F . d = Fd cosθ	Joule (J)
Work (Variable Force)	W = ∫ F(x) dx	Joule (J)
Kinetic Energy	K = (1/2)mv² = p²/2m	Joule (J)
Potential Energy (Gravitational)	U = mgh	Joule (J)
Spring Potential Energy	U = (1/2)kx²	Joule (J)
Work-Energy Theorem	W_net = ΔK	Joule (J)
Power	P = W/t = F . v	Watt (W)
Elastic Collision (v1)	[(m1-m2)/(m1+m2)]u1 + [2m2/(m1+m2)]u2	m/s
Vertical Circle (Bottom Velocity)	v ≥ √(5gL)	m/s
Coeff. of Restitution	e = v_sep / v_app	Dimensionless

50 NEET Facts

Key concepts and quick revision points for NEET aspirants. (Selection of key facts)

1. Work by Centripetal Force The work done by centripetal force is always zero because the centripetal force is always perpendicular (90 degrees) to the instantaneous displacement of the particle. Calculating W = Fd cos(90) yields zero. This applies to satellites orbiting planets or electrons orbiting nuclei.

2. Conservative Force Definition A force is conservative if the work done by it in moving a particle between two points is independent of the path taken. Equivalently, the work done by a conservative force in a closed loop is zero. Electrostatic and Gravitational forces are prime examples.

3. Friction Work Friction is a non-conservative force. The work done by kinetic friction is always negative because friction always opposes the relative motion between surfaces (Angle is 180 degrees). This work dissipates mechanical energy into heat.

4. Potential Energy Reference The zero level of potential energy is arbitrary. We can choose any level as U=0. However, the change in potential energy between two points is unique and physically significant. Typically, ground is taken as U=0 for gravity.

5. Spring Force The spring force is a variable conservative force. It follows Hooke's Law F = -kx. The negative sign indicates that the force is restoring, always directed towards the equilibrium position. The work done by spring force depends only on initial and final extensions.

6. Energy vs Power Energy is the capacity to do work, while Power is the rate at which work is done or energy is transferred. A machine with high power can do a large amount of work in a short time. 1 Horsepower = 746 Watts.

7. KE and Momentum Graph Since K = p^2 / 2m, for a constant mass, the graph of Kinetic Energy (K) versus Momentum (p) is a parabola (y proportional to x^2). If we plot sqrt(K) vs p, it will be a straight line passing through the origin.

8. Perfectly Inelastic Collision In a perfectly inelastic collision, the two bodies stick together after impact and move with a common velocity. This type of collision results in the maximum possible loss of kinetic energy, although momentum is still perfectly conserved.

9. Work done by Internal Forces The work done by internal forces of a system is not necessarily zero. For example, in an explosion, internal chemical energy is converted into kinetic energy of fragments, so internal forces do positive work increasing the system's KE.

10. Elastic Collision Equal Mass When a moving body collides elastically head-on with a stationary body of the same mass, they exchange velocities. The moving body comes to rest, and the stationary body moves with the initial velocity of the first body.

11. Vertical Circle Critical Points To complete a vertical circle, a particle attached to a string must have a minimum speed of sqrt(5gL) at the bottom. At the top, the minimum speed must be sqrt(gL) to keep the string taut (Tension >= 0).

12. Area under F-x Graph The area under the Force vs Displacement (F-x) graph gives the total work done by the force. Area above the x-axis represents positive work, while area below represents negative work.

13. Power and Velocity Instantaneous power can be expressed as the dot product of Force and Velocity vectors (P = F.v). If force is constant and acting in the direction of motion, P = Fv. If power is constant, v is proportional to t^(1/2) (for constant mass starting from rest).

14. Stable Equilibrium A system is in stable equilibrium if its Potential Energy is at a local minimum (dU/dx = 0 and d^2U/dx^2 > 0). If displaced slightly, a restoring force brings it back to the equilibrium position.

15. Unstable Equilibrium A system is in unstable equilibrium if its Potential Energy is at a local maximum (dU/dx = 0 and d^2U/dx^2 < 0). If displaced slightly, the net force pushes it further away from equilibrium.

16. Neutral Equilibrium Detailed description of neutral equilibrium where PE is constant (dU/dx = 0, second derivative 0). Displacement places it in a new equilibrium.

17. Chain on Table Problem Work done to pull a hanging part of a chain (length l, mass m) back onto the table is mgl/2 (where center of mass rises by l/2).

18. Bullet Wood Block When a bullet enters a block, the resistive force is generally assumed uniform. Work done against resistance equals loss in KE.

19. Oblique Collision In 2D collisions, momentum is conserved along both X and Y axes independently.

20. Springs in Series Equivalent spring constant K_eq for series: 1/K_eq = 1/k1 + 1/k2. Force is same, extension adds up.

21. Springs in Parallel Equivalent spring constant K_eq for parallel: K_eq = k1 + k2. Extension is same, force adds up.

22. Coefficient of Restitution Range e is strictly between 0 and 1 for real world inelastic collisions. It depends on the material of the colliding bodies.

23. Mass Energy Equivalence E = mc^2. A small amount of mass can be converted into a huge amount of energy (Nuclear reactions).

24. Kilowatt Hour 1 kWh is a unit of Energy, not Power. It equals 3.6 Mega Joules.

25. Work by Static Friction Work done by static friction can be positive, negative, or zero. e.g., On a block placed on an accelerating truck, static friction does positive work.

26. Frame Reference Dependence Work and Kinetic Energy are frame-dependent quantities. They can have different values in different inertial frames.

27. Conservative Force Field F = -Gradient(U). In 3D: F = -(dU/dx i + dU/dy j + dU/dz k).

28. Ball Bouncing Factor If a ball bounces with coeff of restitution 'e', the height of nth rebound is H_n = e^(2n) * H.

29. Total Distance Bouncing Total distance traveled by a bouncing ball before stopping = H * (1+e^2)/(1-e^2).

30. Total Time Bouncing Total time taken for a bouncing ball to stop = T * (1+e)/(1-e), where T is time of first fall.

31. Work by Normal reaction Usually zero as N is perpendicular to displacement. But can be non-zero (e.g., in an elevator).

32. Efficiency Efficiency = Output / Input. Always < 100% for real machines due to friction.

33. Heavy vs Light Logic If Light and Heavy body have same KE, heavy body has more momentum. (p = sqrt(2mK)).

34. Heavy vs Light Momentum If Light and Heavy body have same Momentum, lighter body has more KE. (K = p^2/2m).

35. Variable Mass System Force F = v_rel * (dm/dt). Rocket propulsion principle.

36. Pseudo Force Work In non-inertial frames, we must consider work done by pseudo forces to apply Work-Energy theorem correctly.

37. Cutting a Spring If a spring (k) is cut into n equal parts, new stiffness of each part is nk.

38. Rod Vertical Circle For a rigid rod, min velocity at bottom to complete circle is sqrt(4gL), not sqrt(5gL), because velocity at top can be zero.

39. Man climbing stairs Work done by man against gravity = mgh. Work done by reaction of stairs on man = 0 (point of application doesn't move).

40. Stopping Distance Stopping distance = v^2 / 2ug. Proportional to square of velocity.

41. Area under Power-Time The area under P-t graph gives the Work done.

42. PE Curve Slope Negative of slope of U-r curve gives Force.

43. Collision in CM Frame In Center of Mass frame, total momentum is always zero before and after collision.

44. Impulse Impulse = Change in Momentum. Impulse is also integral of Force over time.

45. Explosion In an explosion of a projectile, the Center of Mass continues to follow the original parabolic path (if air resistance is ignored).

46. Friction on Inclined Plane Work by friction down a plane of length L: W = -u mg cos(theta) L.

47. Conservative Force Curl For a force to be conservative, Curl F must be zero. (del x F = 0).

48. Max Power Transfer Power delivered to a load is maximum when load resistance equals internal resistance.

49. Human Heart Power Average power of human heart is about 1.2 Watts.

50. Energy Units 1 erg = 10^-7 Joules. 1 eV = 1.6 x 10^-19 Joules. 1 Calorie = 4.18 Joules.

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