JEE Physics Work Energy and Power Notes Set 02

Conservative and Non-Conservative Forces

A force is conservative if the work it does between two points does not depend on the path, or equivalently if the work done over any closed loop is zero. Gravity, the electrostatic force, and the spring force are conservative; all central forces are conservative. A force is non-conservative if the work depends on the path; friction, air resistance and viscous drag are examples, and these depend on velocity. Work done against a conservative force can be fully recovered, whereas work done against friction is lost as heat.

Potential Energy

Potential energy is the energy stored by a system because of the arrangement of its parts, and it is defined only for conservative forces. The change in potential energy is the negative of the work done by the conservative force: \( \Delta U = -W_c \), and \( F = -\dfrac{dU}{dx} \) (in three dimensions \( \vec F = -\nabla U \)). Near the earth's surface \( U = mgh \) relative to the reference level. For a body at its centre of mass height, such as a uniform rod or chain, use the height of the centre of mass.

Equilibrium and the Potential Energy Curve

At equilibrium \( F = -\dfrac{dU}{dx} = 0 \), i.e. the slope of the U-x graph is zero.
Stable equilibrium: U is a minimum, \( \dfrac{d^2U}{dx^2} > 0 \). A small displacement produces a restoring force.
Unstable equilibrium: U is a maximum, \( \dfrac{d^2U}{dx^2} < 0 \). A small displacement grows.
Neutral equilibrium: U is constant, both derivatives are zero.

Conservation of Mechanical Energy

If only conservative forces do work, the total mechanical energy \( E = K + U \) stays constant: \( K_1 + U_1 = K_2 + U_2 \), or \( \Delta K + \Delta U = 0 \). If non-conservative forces act, the net work done by them equals the change in mechanical energy, \( W_{nc} = \Delta E \). For free fall from rest through height x, \( U = mg(h - x) \) and \( K = mgx \), always adding to mgh.

Springs

A spring exerts a restoring force \( F = -kx \). The work done by an external agent in stretching it from x₁ to x₂ is \( \tfrac{1}{2}k(x_2^2 - x_1^2) \), and the stored energy is \( U = \tfrac{1}{2}kx^2 \), the area under the F-x graph.
Block hitting a spring: \( \tfrac{1}{2}mv^2 = \tfrac{1}{2}kx_{max}^2 \).
Mass dropped from height h on a spring: \( mg(h + x) = \tfrac{1}{2}kx^2 \).
Mass hung on a spring: equilibrium extension \( x = mg/k \), but if released from the natural length the maximum extension is \( 2mg/k \).

Power and Efficiency

Average power \( P_{av} = \dfrac{W}{t} \). Instantaneous power \( P = \dfrac{dW}{dt} = \vec F\cdot\vec v \). The unit is watt (J/s); 1 hp = 746 W. The area under a power-time graph is the work done, and the slope of a work-time graph is the power.
Efficiency \( \eta = \dfrac{\text{useful output}}{\text{input}} \).
For constant power, \( v \propto t^{1/2} \) and \( x \propto t^{3/2} \) starting from rest. For a pump or jet moving a fluid of density \( \rho \) with speed v through area A, mass flow per second is \( \rho Av \), force is \( \rho Av^2 \), power is \( \tfrac{1}{2}\rho Av^3 \).

Worked Examples

Worked Example 1. A spring of constant 200 N/m is compressed by 10 cm. Find the speed it gives to a 0.5 kg ball placed against it on a smooth surface.
Answer:
\( \tfrac{1}{2}kx^2 = \tfrac{1}{2}mv^2 \Rightarrow 100\times 0.01 = 0.25\,v^2 \).
\( v^2 = 4 \Rightarrow v = 2 \) m/s.

Worked Example 2. A motor lifts 120 kg of water per minute from a 30 m deep well at 60% efficiency (g = 10 m/s²). Find the input power.
Answer:
Output \( = mgh/t = 120\times 10\times 30/60 = 600 \) W.
Input \( = 600/0.6 = 1000 \) W.

Worked Example 3. A 2 kg block slides from rest down a smooth curved track of height 5 m onto a rough horizontal floor (\( \mu = 0.25 \)). How far does it travel on the floor?
Answer:
Energy balance (it starts and ends at rest): \( mgh = \mu mg\,s \).
\( s = h/\mu = 5/0.25 = 20 \) m.