Class XI Physics: Work, Energy, and Power
The Currency of the Universe: Work, Energy, and Power
In the world of physics, “Work” isn’t what you do at a desk, and “Power” isn’t just about politics. These terms have very strict, mathematical definitions that act as the universal currency for every interaction in existence.
While Laws of Motion told us how things move, Work-Energy tells us what it costs for them to move. It’s the ultimate accounting system of nature where the books always balance.
The Core Pillars of Energy Dynamics
1. Work: The Dot Product Reality
Work is only done when a force causes a displacement. But there’s a catch: only the component of force in the direction of motion counts.
- Formula: W = F · d cos(θ)
- The “Zero Work” Trap: If you carry a heavy box horizontally at a constant velocity, you are doing zero work on the box in the direction of motion because your force (upward) is perpendicular to the displacement (horizontal).
2. The Work-Energy Theorem
This is the “Golden Rule.” It states that the total work done by all forces (conservative and non-conservative) on an object is exactly equal to the change in its Kinetic Energy.
- W_total = ΔK = ½mvᶠ² – ½mvᵢ²
3. Potential Energy: The Landscape of Force
Potential energy (U) is “stored” energy based on position. The most important concept here is the relationship between force and potential energy:
- F = -dU/dxThis means that objects always want to move toward a state of lower potential energy.
The Gauntlet: 10 Challenging Aptitude Questions
Question 1: The Variable Force Challenge
A particle moves along the x-axis under the influence of a force F = (3x² + 2x – 5) N. Calculate the work done by this force in moving the particle from x = 0 to x = 2m.
Question 2: The Vertical Loop-the-Loop
A small ball of mass m slides down a frictionless track and enters a circular loop of radius R. What is the minimum height H from which the ball must be released so that it completes the full vertical circle without losing contact with the track?
Question 3: The Hanging Chain Problem
A uniform chain of length L and mass M is lying on a smooth table and 1/3rd of its length is hanging vertically over the edge. How much work is required to pull the hanging part back onto the table?
Question 4: Constant Power Acceleration
An engine provides a constant power P to a car of mass M. If the car starts from rest, find the expression for its velocity v as a function of time t. (Assume no friction).
Question 5: The Spring-Block-Slope Combo
A block of mass m is pushed against a spring (force constant k) compressing it by x. When released, the block travels up a rough incline of angle θ. If the coefficient of friction is μ, how far along the incline will the block travel before stopping?
Question 6: Potential Energy Graph Analysis
The potential energy of a particle is given by U(x) = a/x² – b/x.
- Find the point of stable equilibrium.
- What is the force acting on the particle at this point?
Question 7: The Oblique Collision
Two identical billiard balls, A and B, undergo an elastic collision. Ball A is moving with velocity u while B is at rest. After the collision, they move off at different angles. Prove that if the collision is not head-on, the balls will always move at 90° to each other after the impact.
Question 8: The Bullet and the Block
A bullet of mass m moving at velocity v hits a wooden block of mass M suspended by a string of length L. The bullet gets embedded in the block. Find the minimum velocity v required so that the block-bullet system performs a full vertical circle.
Question 9: Water Pump Efficiency
A pump is required to lift 600 kg of water per minute from a well 25m deep and eject it with a speed of 50 m/s. What is the Power of the engine in Kilowatts? (Take g = 10 m/s²)
Question 10: The Conservative Force Test
A force is defined as F = (yî + xĵ). Is this force conservative? Calculate the work done by this force in moving a particle from (0,0) to (1,1) via two different paths:
- Path 1: (0,0) → (1,0) → (1,1)
- Path 2: The straight line y = x.
Detailed Explanations & Solutions
1. Variable Force
Work is the integral of F dx.
W = ∫ (3x² + 2x – 5) dx from 0 to 2.
W = [x³ + x² – 5x] from 0 to 2.
W = (8 + 4 – 10) – (0) = 2.
Result: 2 Joules.
2. Vertical Loop
At the top of the loop, the minimum velocity must be v = √(gR).
Using Conservation of Energy: mgH = mg(2R) + ½m(gR).
mgH = 2.5 mgR.
Result: H = 2.5R.
3. Hanging Chain
The mass of the hanging part is M/3. Its center of mass is at a distance L/6 below the table edge.
Work = Change in Potential Energy = (M_hanging) × g × (h_cm).
Work = (M/3) × g × (L/6).
Result: MgL / 18.
4. Constant Power
P = Fv = (ma)v = m(dv/dt)v.
P dt = mv dv.
Integrate both sides: Pt = ½mv².
Result: v = √(2Pt/M).
5. Spring and Friction
Energy supplied = ½kx².
Energy consumed = Work against Gravity + Work against Friction.
½kx² = (mg sinθ)d + (μ mg cosθ)d.
Result: d = (kx²) / [2mg(sinθ + μ cosθ)].
6. PE Graph Analysis
For equilibrium, dU/dx = 0.
-2a/x³ + b/x² = 0 → x = 2a/b.
Since d²U/dx² is positive at this point, it is Stable Equilibrium.
Result: x = 2a/b; Force = 0.
7. Oblique Collision
Conservation of Momentum: m u = m v₁ + m v₂ → u = v₁ + v₂.
Conservation of Kinetic Energy: u² = v₁² + v₂².
From vector addition: u² = v₁² + v₂² + 2v₁v₂ cosθ.
Comparing the two equations: 2v₁v₂ cosθ = 0. Since velocities are non-zero, cosθ = 0.
Result: θ = 90°.
8. Bullet-Block System
Cons. of Momentum: mv = (m+M)V_system.
For vertical circle: V_system = √(5gL).
Result: v = [(m+M)/m] × √(5gL).
9. Water Pump Power
Power = (mgh + ½mv²) / time.
Mass = 600 kg, time = 60 s.
P = [600 × 10 × 25 + ½ × 600 × 50²] / 60.
P = [150,000 + 750,000] / 60 = 900,000 / 60.
Result: 15,000 W or 15 kW.
10. Conservative Force
A force is conservative if ∂Fₓ/∂y = ∂Fᵧ/∂x.
Here, ∂(y)/∂y = 1 and ∂(x)/∂x = 1. They are equal.
Result: The force is conservative. The work done for both paths will be 1 Joule.
Key Summary Table: Energy Types
| Type | Formula | Depends On |
| Kinetic Energy | ½mv² | Speed and Mass |
| Gravitational PE | mgh | Height and Mass |
| Elastic (Spring) PE | ½kx² | Compression/Extension |
