Potential energy
Potential energy (U) is energy stored in a body because of its position or shape.
Gravitational PE near the Earth: lifting mass m by height h needs work mgh against gravity, so U = mgh (taking U = 0 at the ground). Only changes in U matter, so we may pick any level as zero.
For any conservative force: F = −dU/dx. The force points the way U decreases (a ball rolls downhill).
Spring potential energy
A spring pulls back with F = −kx (Hooke's law), where x is the stretch or squeeze and k is the spring constant (N/m).
Work done by us to stretch it from 0 to x is the triangle area under the F–x graph: ½ × x × kx. So U = ½kx².
- Stretch or squeeze by the same x → same U (x² is positive either way).
- Work by the spring force from x₁ to x₂: W = −(½kx₂² − ½kx₁²).
- Block on a spring on a smooth floor: ½kx² + ½mv² = constant, so maximum speed v = x√(k/m) at the natural length.
Conservative and non-conservative forces
A force is conservative if:
- its work between two points depends only on those points, not on the path, and
- its work around any closed path is zero, and
- it can be written as F = −dU/dx.
Examples: gravity, spring force, electrostatic force.
A force is non-conservative if its work depends on the path. Examples: friction, air drag, viscous force. These turn mechanical energy into heat or sound, so K + U decreases.
Conservation of mechanical energy
If only conservative forces do work: W = −ΔU. The work–energy theorem says W = ΔK. So ΔK + ΔU = 0, which means K + U = constant.
With friction: K_i + U_i = K_f + U_f + heat. Total energy (including heat) is always conserved.
Example: a ball dropped from height H has speed v = √(2gh) after falling h, because mgh = ½mv².
Try it at home
Push down a ball-point pen's click spring and let go on a table: stored ½kx² sends the pen up. Push harder (more x) and it jumps higher. Then slide a book once on a smooth table and once on a cloth. On the cloth it stops sooner: friction took the energy away as heat.
Exam corner
Typical CBSE questions: derive U = ½kx² from the F–x graph (2–3 marks), state two properties of conservative forces, show K + U is constant for a freely falling body (3 marks), and spring–block numericals.
Key formulas and definitions
- U = mgh (near Earth)
- F = −dU/dx
- Spring: F = −kx, U = ½kx²
- Only conservative forces: K + U = constant
- With friction: K_i + U_i = K_f + U_f + heat
- Free fall: v = √(2gh)
Worked examples
1. Find the potential energy of a 2 kg pot on a 1.5 m shelf (g = 10 m/s²).
Step 1: U = mgh. Step 2: U = 2 × 10 × 1.5 = 30 J.
2. A spring with k = 400 N/m is squeezed 5 cm. Find the stored energy.
Step 1: x = 0.05 m. Step 2: U = ½ × 400 × 0.0025 = 0.5 J.
3. A 0.2 kg stone falls from 20 m. Find its speed just before it hits the ground (no air drag).
Step 1: mgh = ½mv². Step 2: v = √(2gh) = √(2 × 10 × 20) = √400. Answer: 20 m/s. (Mass cancels.)
4. A 1 kg block moving at 2 m/s hits a spring (k = 100 N/m). Find the maximum squeeze.
Step 1: ½mv² = ½kx². Step 2: 1 × 4 = 100 x². Step 3: x² = 0.04, x = 0.2 m.
5. A 2 kg box slides down a 5 m high rough slope and reaches the bottom at 8 m/s. How much energy became heat?
Step 1: U at top = 2 × 10 × 5 = 100 J. Step 2: K at bottom = ½ × 2 × 64 = 64 J. Step 3: heat = 100 − 64 = 36 J.
6. U(x) = 3x² − 12x J. Find the force at x = 1 m and the point where the force is zero.
Step 1: F = −dU/dx = −(6x − 12) = 12 − 6x. Step 2: at x = 1, F = 6 N (towards +x). Step 3: F = 0 when x = 2 m (bottom of the U curve, stable balance).
7. A spring is stretched from 2 cm to 6 cm (k = 500 N/m). Find the extra work done.
Step 1: W = ½k(x₂² − x₁²). Step 2: = ½ × 500 × (0.0036 − 0.0004) = 250 × 0.0032. Answer: 0.8 J.
Common mistakes
- Thinking U = mgh is an absolute value. Only differences matter; the zero level is your choice.
- Writing spring energy as kx or ½kx. It is ½kx² (area of a triangle).
- Using K + U = constant when friction acts. Then heat must be added on the final side.
- Saying friction is conservative because it is "always there". It is non-conservative because its work depends on the path length.