Work, Power and Energy
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Direct answer
The work-energy theorem states that the net work done on a body equals its change in kinetic energy: W(net) = (1/2)mv^2 − (1/2)mu^2, because it is the integral of F = ma along the path. Work itself is W = F s cos(theta), negative when force opposes displacement, and its sign bookkeeping matters more than its magnitude. For conservative forces the work is path-independent and storable as potential energy — mg h near the earth, (1/2)kx^2 in a spring — with F = −dU/dx. Power is the rate P = dW/dt = F v cos(theta), and the practical pair of numbers to remember is 1 horsepower = 746 W and 1 kWh = 3.6 × 10^6 J.
What you must remember
- Work basics: W = F s cos(theta); work by friction or a retarding force is negative; area under an F–s graph gives work even when F varies.
- Kinetic energy forms: KE = (1/2)mv^2 = p^2/(2m), so doubling momentum quadruples KE at fixed mass — a standard one-liner in Main.
- Conservative force test: work over a closed loop is zero; equivalently F = −dU/dx, which is how you recover spring force from (1/2)kx^2.
- Spring energy: U = (1/2)kx^2 is always positive and depends only on extension, never on how it was reached; in series, springs share the same force, and the combination constant obeys 1/k(series) = 1/k1 + 1/k2.
- Power: P = F v cos(theta) for instantaneous power; a pump lifting volume Q per second against height h delivers P = rho g Q h, the classic Main numerical.
- Units: 1 kWh = 3.6 × 10^6 J; 1 hp = 746 W; electricity meters in India bill in units of 1 kWh.
- Vertical circle checkpoints: a particle on a string just completes the loop when v(top) = sqrt(gL); the matching bottom speed is sqrt(5gL); tension difference between bottom and top is 6mg.
- Pattern note: Main favours W–E theorem numericals and pump power; Advanced layers variable forces, loop-the-loop checks and energy of systems with internal friction.
Can this pendulum loop the loop?
Energy conservation answers capability questions faster than forces. Release a pendulum bob of string length L from the horizontal: at the bottom, mg( L ) = (1/2)mv^2 gives v = sqrt(2gL). To complete a vertical circle on a string, the bob needs at least sqrt(5gL) at the bottom, because the top requires sqrt(gL) (gravity alone supplies the centripetal force there) and the climb to the top costs 2mgL more energy. Since sqrt(2gL) falls short, the string goes slack before the top and the bob executes projectile motion — the exact scenario Advanced describes without ever using the word "impossible".
Pump power uses the same bookkeeping in industry dress. Lifting 1200 litres of water per minute through 10 m — density 1000 kg/m^3 — means 20 kg/s: P = 20 × 9.8 × 10 = 1960 W, and with 70% efficiency the motor must be rated 1960/0.7 = 2800 W. The efficiency division, not the physics, is where the mark is usually lost.
Where students slip
Signs and systems cause the damage. Candidates compute work by gravity while a body moves up as positive; it is −mgh, and the W–E theorem only balances when negative work is entered negative. They treat the spring energy as (1/2)kx even when the force itself is kx, and they apply (1/2)mv^2 = mgh across a path containing friction, where the honest equation is (1/2)mv^2 = mgh − f s. A quieter slip is the work done by a variable force: the area under F–s is the only general method, and for a spring it gives exactly the (1/2) that constant-force intuition misses. Finally, remember power ratings: an engine delivering constant power accelerates with decreasing force as speed grows — P = F v means the force cannot stay constant.
Frequently asked questions
When is the work-energy theorem preferable to F = ma?
Whenever the question links speeds at two positions and skips the time between them — friction, inclines, curved tracks — because net work integrates out the intermediate forces.
What makes a force conservative?
Zero net work around any closed path, equivalently F = −dU/dx; gravity, spring force and electrostatic force qualify, friction does not.
Why does doubling momentum quadruple kinetic energy?
KE = p^2/(2m), so KE scales with the square of momentum at fixed mass; p to 2p sends p^2 to 4p^2.
What is the minimum speed at the top of a vertical circle on a string?
v(top) = sqrt(gL), where gravity alone provides the centripetal force and tension just touches zero; the corresponding bottom speed is sqrt(5gL).
How is pump power computed?
P = rho g Q h for the mechanical part (Q the volume flow rate, h the lift), then divide by efficiency to get the electrical input power.