JEE Physics Work Energy and Power Notes Set 03

Work

1. Work done by a constant force: \( W = \vec F\cdot\vec s = Fs\cos\theta \). It is positive for \( \theta < 90^\circ \), zero for \( 90^\circ \) and negative for larger angles.
2. Variable force: \( W = \int \vec F\cdot d\vec r \). On a graph of F against x it is the area under the curve. For \( F = cx \) from 0 to \( x_1 \), \( W = \tfrac12cx_1^2 \).
3. Gravity: \( W_g = -mgh \) for a rise through h and depends only on the height change. Friction, being non-conservative, depends on the path.
4. Spring: \( W_{spring} = -\tfrac12k(x_2^2 - x_1^2) \). Work by an external agent that stretches it slowly from rest is \( \tfrac12kx^2 \).
5. Static friction can do positive, negative or zero work, depending on the situation.

Energy

1. Work-energy theorem: the work done by all forces (internal and external) equals the change in kinetic energy, \( W_{net} = \Delta K \).
2. If only conservative forces do work, \( K + U \) is constant. The work of other forces equals the change in total mechanical energy, \( W_{other} = \Delta(K+U) \).
3. Only external forces can change the momentum of a system, but internal forces can change its kinetic energy (explosions, springs).
4. Zero kinetic energy means zero momentum, but zero momentum does not mean zero kinetic energy.
5. A conservative force has work that depends only on end points, and around a closed path the work is zero. Only conservative forces have a potential energy.

Equilibrium from the Potential Energy Curve

1. Equilibrium is where \( dU/dx = 0 \).
2. Minimum of U (\( d^2U/dx^2 > 0 \)): stable. Maximum of U (\( d^2U/dx^2 < 0 \)): unstable. Flat region: neutral.
3. For \( U = x^2 - 3x \), \( F = 3 - 2x \) and the equilibrium is at x = 1.5 m.

Power

1. Average power \( P = W/t \), instantaneous power \( P = \vec F\cdot\vec v \). Unit watt; 1 horse power is 746 W.
2. Uniform acceleration from rest: \( P = ma\cdot at = ma^2t \), increasing linearly with time.
3. Constant power: \( v\propto\sqrt t \), distance \( s\propto t^{3/2} \). In terms of distance, \( v = (3Px/m)^{1/3} \).
4. Vehicle on a slope at constant speed: \( P = (f + mg\sin\theta)v \). Train at uniform speed on a level track: \( P = fv \).
5. The work done by a man lifting a mass slowly is \( mgh \) whatever the time, but the power depends on the time taken.

Useful Results

Solved Examples

1. A 2 kg body has \( x = t^4/4 + 3 \). Then \( v = t^3 \) and the work in the first 2 s equals \( \tfrac12\times2\times8^2 = 64 \) J.
2. A force \( F = 3x^2 - 12x \) acts on a 2 kg block with speed \( 4\sqrt5 \) m/s at the origin. Work from 0 to 2 m is \( 8 - 24 = -16 \) J, so \( \tfrac12\times2\,v^2 = 80 - 16 \) and \( v = 8 \) m/s.
3. A ball loses 35 percent of its energy at each impact. Dropped from 100 m it rises to 65 m.
4. A train of weight \( 10^7 \) N moves at 10 m/s against a friction of 0.5 kgf per quintal. Friction is \( 5\times10^4 \) N, so the power is 500 kW.