JEE Physics Work Energy and Power Notes Set 04

Work Done by a Constant Force

Work is the scalar product of force and displacement: \( W=\vec F\cdot\vec d=Fd\cos\theta \), where \( \theta \) is the angle between them. Work is positive for \( \theta<90^\circ \), zero for \( \theta=90^\circ \) (for example the centripetal force in circular motion, or the normal reaction) and negative for \( \theta>90^\circ \) (friction, air resistance). With no displacement there is no work, so a man pushing a fixed wall does no work. The SI unit is the joule; its dimensions are \( ML^2T^{-2} \).
In terms of components, \( W=F_xd_x+F_yd_y+F_zd_z \). Work done by gravity is \( mgh \) for a descent of h, independent of the path. The work done against friction on a horizontal surface is \( \mu mgd \), and on an incline of inclination \( \theta \) it is \( \mu mg\cos\theta\,d \). When a body is lifted slowly, the work done by the applied force equals the gain in potential energy.
For a chain of length \( \ell \) and mass m pulled slowly onto a table from a hanging position, the work done is \( \frac{mg\ell}{2} \) (the rise of the centre of mass is \( \ell/2 \)).

Work Done by a Variable Force

For a force varying with position, \( W=\int F\,dx \), which is the area under the force-displacement graph (areas below the axis count as negative). For a spring, the force is \( F=-kx \) and the work done in stretching it by x is \( \frac12kx^2 \); the work done by the spring force is the negative of that. If the friction coefficient varies as \( \mu=\mu_0x \), the friction work is found by integration.

Work-Energy Theorem and Kinetic Energy

The net work done on a body by all forces equals the change in its kinetic energy: \( W_{net}=\Delta K=\frac12mv^2-\frac12mu^2 \). Kinetic energy is \( K=\frac12mv^2=\frac{p^2}{2m} \) and \( \frac{dK}{dv}=p \). The theorem holds for any forces, including friction and variable forces, and does not require the motion to be along a straight line.
For a retarding force F, the stopping distance is \( d=\frac{K}{F} \): it is proportional to the kinetic energy and to the square of the speed, so doubling the speed makes it four times as large. A bullet piercing a wall loses kinetic energy equal to (average resistance) times (thickness).

Potential Energy and Conservative Forces

A force is conservative if the work done by it depends only on the end points, or equivalently if the work around any closed path is zero. Gravity, the spring force and electrostatic forces are conservative; friction and air resistance are not. For a conservative force the potential energy is defined by \( F=-\frac{dU}{dx} \) (in three dimensions \( \vec F=-\nabla U \)), and \( W=-\Delta U \). A force is conservative if \( \frac{\partial F_y}{\partial x}=\frac{\partial F_x}{\partial y} \) and so on.
Gravitational potential energy near the earth is \( U=mgh \) and spring potential energy is \( U=\frac12kx^2 \). The graph of potential energy against position shows equilibrium at the minimum (stable) and maximum (unstable) points, where \( F=0 \). A particle of total energy E turns back where \( U=E \).

Conservation of Mechanical Energy

If only conservative forces do work, \( K+U \) remains constant. With non-conservative forces, \( W_{nc}=\Delta(K+U) \). A projectile from a cliff of height h with initial speed u strikes the ground at \( v=\sqrt{u^2+2gh} \) whatever the angle. A pendulum released from a horizontal position of length L has speed \( \sqrt{2gL} \) at the bottom. In an Atwood machine, the tension is \( \frac{2m_1m_2g}{m_1+m_2} \).

Power

Power is the rate of doing work. Average power is \( P=\frac{W}{t} \) and instantaneous power is \( P=\vec F\cdot\vec v \). The SI unit is the watt (1 W = 1 J/s); 1 horse-power = 746 W (taken as 750 W in many problems). Dimensions: \( ML^2T^{-3} \). The energy used is power times time; 1 kilowatt-hour = \( 3.6\times10^6 \) J.
A motor lifting a load m at a steady speed v needs \( P=mgv \); to raise a mass m through height h in time t it needs \( P=\frac{mgh}{t} \).

Worked Examples

1. A pump motor of power 2 kW raises water through 10 m: mass per minute \( =\frac{2000\times60}{10\times10}=1200 \) kg.
2. A block slides from rest down a smooth track of height 1 m onto a rough surface with \( \mu=0.2 \): the distance moved is \( \frac{h}{\mu}=5 \) m.
3. The potential energy \( U=2x^2+3y^3+2z \) gives the force \( -(4x,\,9y^2,\,2) \), which at (1, 2, 3) is \( -4\hat i-36\hat j-2\hat k \) N.