Work, Energy and Power

1. Work Done

Scalar product of Force and Displacement.

Constant Force: W = F ⋅ s = Fs cosθ.
Variable Force: W = ∫ F ⋅ dx (Area under F-x graph).
Positive Work: θ < 90° (Force helps motion).
Negative Work: θ > 90° (Force opposes motion, e.g., Friction).
Zero Work: θ = 90° (Normal force, Centripetal force).

(Asked in NEET 2016, 2019)

2. Energy

Capacity to do work.

Kinetic Energy (K): ½mv² = p²/2m.
Potential Energy (U):
Gravitational: mgh.
Spring: ½kx².
Work-Energy Theorem: W_all = ΔK = K_f - K_i.
Conservation of Mechanical Energy: If only conservative forces act, K + U = Constant.

(Asked in NEET 2018, 2020, 2022)

3. Collisions

Interaction where momentum is conserved.

Elastic Collision: Momentum & KE conserved. (e=1).
Inelastic Collision: Momentum conserved, KE lost. (0 < e < 1).
Perfectly Inelastic: Bodies stick together. Max KE loss. (e=0).
Coefficient of Restitution (e): Velocity of Sep / Velocity of App.
e = |v₂ - v₁| / |u₁ - u₂|.

4. Power

Rate of doing work. P = dW/dt.
P = F ⋅ v.
Unit: Watt (W). 1 hp = 746 W.

(Asked in NEET 2017, 2021, 2023)

Numericals: Work, Power, Energy

Q1. A bullet of mass 10g moving at 500 m/s penetrates 10 cm into a wooden block and stops. Find average resistive force.

Solution:
Change in KE = Work done by Resistance.
K_f - K_i = -F × s
0 - ½(0.01)(500)² = -F(0.1)
-0.5 × 0.01 × 250000 = -0.1 F
1250 = 0.1 F
F = 12,500 N.

Q2. An engine pumps 2000 kg of water to height 20 m in 60 sec. Efficiency is 80%. Find electric power required.

Solution:
Output Power P_out = mgh / t = (2000 × 10 × 20) / 60 = 400000/60 = 6666.7 W.
Efficiency η = P_out / P_in.
0.8 = 6666.7 / P_in
P_in = 6666.7 / 0.8 ≈ 8333 W.

Q3. A body of mass 2 kg moving at 10 m/s hits a stationary body of mass 3 kg. They stick together. Find common velocity.

Solution:
Conservation of Momentum:
m₁u₁ + m₂u₂ = (m₁ + m₂)v.
(2)(10) + (3)(0) = (2 + 3)v.
20 = 5v → v = 4 m/s.

Q4. A ball drops from 20 m and rebounds to 5 m. Find coefficient of restitution e.

Solution:
Rebound height h₂ = e² h₁.
5 = e² (20) → e² = 5/20 = 1/4.
e = √(1/4) = 0.5.

Q5. A spring (k=500 N/m) is compressed by 10 cm. How much energy is stored?

Solution:
U = ½kx².
x = 0.1 m.
U = 0.5 × 500 × (0.1)²
U = 250 × 0.01 = 2.5 J.

Q6. A bucket of water is rotated in vertical circle of radius 1 m. What is minimum speed at top so water doesn't spill?

Solution:
Condition: Centripetal force provided by Weight (Tension=0).
mg = mv²/r → v = √(rg).
v = √(1 × 10) = √10 m/s ≈ 3.16 m/s.

Q7. KE of a body is increased by 300%. Find % increase in Momentum.

Solution:
K_f = K_i + 3K_i = 4K_i.
p = √(2mK). Since K becomes 4 times, p becomes √4 = 2 times.
p_f = 2p_i.
% change = (2-1)/1 × 100 = 100%.

Q8. Uniform chain of length L, mass M hangs L/3 over table edge. Work done to pull it up?

Solution:
Mass hanging m' = M/3. CoM of hanging part is at L/6 below table.
Work = m'gh = (M/3) × g × (L/6).
Work = MgL / 18.

Q9. Force F = (2 + x) acts on particle moving from x=0 to x=2. Find Work Done.

Solution:
W = ∫ F dx from 0 to 2.
W = ∫(2 + x) dx = [2x + x²/2] from 0 to 2.
W = [2(2) + 2²/2] - 0.
W = 4 + 2 = 6 J.

Q10. A car accelerates from rest at constant power P. Velocity at time t?

Solution:
P = Fv = mav = m(dv/dt)v.
v dv = (P/m) dt.
integrate: v²/2 = (P/m)t.
v = √(2Pt / m).

Important Formulae

1. Work & Power

Work:

W = Fs cosθ

Power:

P = W/t = F ⋅ v

Spring Work W = ½k(x_f² - x_i²)

2. Energy

Kinetic Energy:

K = ½mv² = p²/2m

Potential Energy:

U_g = mgh | U_s = ½kx²

Work-Energy Theorem: W_net = K_f - K_i

3. Collisions

Coefficient of Restitution:

e = (v₂ - v₁) / (u₁ - u₂)

Elastic (1D) Velocities:

v₁ = [(m₁-m₂)u₁ + 2m₂u₂] / (m₁+m₂)

4. Vertical Circle

Critical Velocities:

Bottom: √(5gr) | Top: √(gr)

Tension:

T_bot - T_top = 6mg

20 NEET Golden Facts

1. Conservative Force: Work done depends only on initial and final positions (Gravity, Electrostatic, Spring). Path independent.
2. Non-Conservative: Work done depends on path (Friction, Viscous drag). Energy lost as heat.
3. KE vs Momentum: A lighter body and heavier body have same Momentum → Lighter body has more KE. (K = p²/2m).
4. Same KE: If Light and Heavy body have same KE → Heavy body has more Momentum. (p = √(2mK)).
5. Work by Friction: Can be positive, negative, or zero. (Usually negative, but positive for block on moving truck).
6. Equilibrium: dU/dx = 0. Stable (Min U), Unstable (Max U), Neutral (Const U).
7. Vertical Circle: Work done by Tension is always zero (Force perpendicular to displacement).
8. Elastic Collision 1D: If masses are equal, velocities separate after collision (Exchange velocities).
9. Power-Velocity: If Power is constant, v ∝ t^1/2 and s ∝ t^3/2.
10. Spring Constant: If spring is cut into n equal parts, new constant is nk. Stiffer.
11. Explosion: Internal forces only. Momentum conserved. KE increases (Chemical → Kinetic).
12. Chain on Table: Work done to pull hanging part (length l/n) onto table is Mgl / 2n².
13. Oblique Collision: Component of velocity parallel to common tangent remains unchanged.
14. Stopping Distance: Work = Change in KE. Fs = ½mv². s ∝ v². Double speed → 4x distance.
15. Rebound: Height reached after n rebounds on floor: h_n = e²ⁿ h.
16. KE Loss Inelastic: Max loss occurs in perfectly inelastic collision.
17. Conservative Loop: Work done by conservative force in a closed loop is zero.
18. Graph Area: Area under Power-Time graph gives Work Done.
19. Head-on Collision: Very heavy body hits light body at rest → Heavy continues same v, Light moves with 2v.
20. Kilowatt-hour: Unit of Energy (not Power). 1 kWh = 3.6 × 10⁶ J.

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