Work and Energy
We will cover following topics
Work And Energy
The work done by a force of magnitude \(F\) units in displacing an object \(s\) units along its direction is given by
\[W=Fs\]It follows from Newton’s second law that in the absence of fields (such as gravity),
\[W=\Delta K E\]Similarly, when an object is displaced in the presence of field forces, without change in velocity,
\[W=-\Delta P E\]Conservation Of Energy
According to the law of conservation of energy, energy can neither be created nor be destroyed, but can change its form from kinetic to potential or vice-versa.
PYQs
Work and Energy
1) A fixed wire is in the shape of the cardiod \(r=a(1+\cos \theta)\), the initial line being the downward vertical. A small ring of mass \(m\) can slide on the wire and is attached to point \(r=0\) of the cardiod by an elastic string of natural length \(a\) and modulus of elasticity 4 \(\mathrm{mg}\). The string is released from rest when the string is horizontal. Show by laws of conservation of energy that \(a \theta^{2}(1+\cos \theta)-g \cos \theta(1-\cos \theta)=0\), \(g\) being the acceleration due to gravity.
[2017, 10M]
2) A spherical shot of \(W\) gm weight and radius \(r\) cm, lies at the bottom of cylindrical bucket of radius \(R\) cm. The bucket is filled with water up to a depth of \(h\) cm \((h>2r)\). Show that the minimum amount of work done in lifting the shot just clear of the water must be \(\left[W\left(h-\dfrac{4 r^{3}}{3 R^{2}}\right)\right]+W^{\prime}\left(r-h+\dfrac{2 r^{3}}{3 R^{2}}\right) \mathrm{cm} \mathrm{gm}\). \(\mathrm{W}^{\prime} \mathrm{gm}\) is the weight of water displaced by the shot.
[2017, 17M]
3) A heavy ring of mass \(m\), slides on a smooth vertical rod and is attached to a light string which passes over a small pulley distant \(a\) from the rod and has a mass \(M(>m)\) fastened to its other end. Show that if the ring be dropped from a point in the rod in the same horizontal plane as the pulley, it will descend a distance \(\dfrac{2 M m a}{M^{2}-m^{2}}\) before coming to rest.
[2012, 20M]
4) A mass of 560 kg, moving with a velocity of 240 m/sec strikes a fixed target and is brought to rest in \(\dfrac{1}{100}\) sec. Find the impulse of the blow on the target and assuming the resistance to be uniform throughout the time taken by the body in coming to rest, find the distance through which it penetrates.
[2011, 20M]
5) A shell lying in a straight smooth horizontal tube suddenly breaks into two portions of masses \(m_{1}\) and \(m_{2}\). If \(s\) be the distance between the two masses inside the tube after time the show that the work done by the explosion can be written as equal to \(\dfrac{1}{2} \dfrac{m_{1} m_{2}}{m_{1}+m_{2}} \cdot \dfrac{s^{2}}{t^{2}}\).
[2008, 15M]
6) A body of mass \(\left(\mathrm{m}_{1}+\mathrm{m}_{2}\right)\) moving in a straight line is split into two parts of masses \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\) by an internal explosion, which generates kinetic energy \(E\). If after the explosion the two parts move in the same line as before, find their relative velocity.
[2005, 15M]
7) A car of mass 750 \(\mathrm{kg}\) is running up a hill of 1 \(\mathrm{in}\) 30 at a steady speed of 36 \(\mathrm{km} / \mathrm{hr}\), the friction is equal to the weight of 40 \(\mathrm{kg}\). Find the work done in 1 \(\mathrm{second}\).
[2004, 15M]