Chapter: Work, Energy and Power

A particle is released from height \(\begin{align}S\end{align}\) from the surface of the Earth. At a certain height its kinetic energy is three times its potential energy. The height from the surface of earth and the speed of the particle at that instant are respectively

The energy required to break one bond in \(\begin{align}DNA\end{align}\) is \(\begin{align}{{10}^{{-20}}}\end{align}\)\(\begin{align}J\end{align}\). This value \(\begin{align}\left( {in\text{ }eV} \right)\end{align}\) is nearly

A force \(\begin{align}F=20+10y\end{align}\) acts on a particle in y-direction, where \(\begin{align}F\end{align}\) is in newton and \(\begin{align}y\end{align}\) in meter. Work done by this force to move the particle from \(\begin{align}y=0\end{align}\)to \(\begin{align}y=1m\end{align}\) is

A particle of mass \(\begin{align}10\text{ }g\end{align}\) moves along a circle of radius \(\begin{align}6.4\text{ }cm\end{align}\) with a constant tangential acceleration. What is the magnitude of this acceleration, if the kinetic energy of the particle becomes equal to \(\begin{align}8\times {{10}^{{-4}}}J\end{align}\) by the end of the second revolution after the beginning of the motion?

A particle moves from a point \(\begin{align}\left( {-2\hat{i}+5\hat{j}} \right)\end{align}\) to \(\begin{align}\left( {4\widehat{j}+3\widehat{k}} \right)\end{align}\) when a force of \(\begin{align}\left( {4\widehat{i}+3\widehat{j}} \right)N\end{align}\) is applied. How much work has been done by the force?

Two similar springs \(\begin{align}P\end{align}\) and \(\begin{align}Q\end{align}\) have spring constants \(\begin{align}{{K}_{P}}\end{align}\) and \(\begin{align}{{K}_{Q}}\end{align}\) , such that \(\begin{align}{{K}_{P}}>{{K}_{Q}}\end{align}\) . They are stretched, first by the same amount \(\begin{align}\left( {case\text{ }a} \right)\end{align}\), then by the same force \(\begin{align}\left( {case\text{ }b} \right)\end{align}\). The work done by the springs \(\begin{align}{{W}_{P}}\end{align}\) and \(\begin{align}{{W}_{Q}}\end{align}\) are related as, in \(\begin{align}\left( {case\text{ }a} \right)\end{align}\) and \(\begin{align}\left( {case\text{ }b} \right)\end{align}\) , respectively

A block of mass \(\begin{align}10\text{ }kg\end{align}\), moving in \(\begin{align}x\end{align}\)-direction with a constant speed of \(\begin{align}10\text{ }m{{s}^{{-1}}}\end{align}\), is subjected to a retarding force \(\begin{align}F=0.1\times J/m\end{align}\)during its travel from \(\begin{align}x=20\text{ }m\text{ }to30\,m\end{align}\). Its final \(\begin{align}KE\end{align}\) will be

Two particles of masses \(\begin{align}{{m}_{1}},{{m}_{2}}\end{align}\) move with initial velocities \(\begin{align}{{u}_{1}}\end{align}\) and \(\begin{align}{{u}_{2}}\end{align}\) . On collision, one of the particles get excited to higher level, after absorbing energy \(\begin{align}\varepsilon \end{align}\). If final velocities of particles be \(\begin{align}{{v}_{1}}\end{align}\) and \(\begin{align}{{v}_{2}}\end{align}\) , then we must have

A mass \(\begin{align}m\end{align}\) moves in a circle on a smooth horizontal plane with velocity \(\begin{align}{{v}_{0}}\end{align}\) at a radius \(\begin{align}{{R}_{0}}\end{align}\) . The mass is attached to a string which passes through a smooth hole in the plane as shown. The tension in the string is increased gradually and finally m moves in a circle of radius \(\begin{align}\frac{{{{R}_{0}}}}{2}\end{align}\) . The final value of the kinetic energy is

A ball is thrown vertically downwards from a height of \(\begin{align}20m\end{align}\) with an initial velocity \(\begin{align}{{v}_{0}}\end{align}\) . It collides with the ground, loses \(\begin{align}50\%\end{align}\) of its energy in collision and rebounds to the same height. The initial velocity \(\begin{align}{{v}_{0}}\end{align}\) is \(\begin{align}\left( {Take,\text{ }g=10m{{s}^{{-2}}}} \right)\end{align}\)

A uniform force of \(\begin{align}\left( {3\widehat{i}+\widehat{j}} \right)N\end{align}\)acts on a particle of mass \(\begin{align}2\,kg\end{align}\). Hence, the particle is displaced from position \(\begin{align}\left( {2i+\widehat{k}} \right)\end{align}\)\(\begin{align}m\end{align}\) to position \(\begin{align}\left( {4\widehat{i}+3\widehat{j}-\widehat{k}} \right)\end{align}\)\(\begin{align}m\end{align}\). The work done by the force on the particle is

The potential energy of a particle in a force field \(\begin{align}U=\frac{A}{{{{r}^{2}}}}-\frac{B}{r}\end{align}\), where \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\)are positive constants and \(\begin{align}r\end{align}\) is the distance of particle from the centre of the field. For stable equilibrium, the distance of the particle is

The potential energy of a system increases, if work is done

Force \(\begin{align}F\end{align}\)on a particle moving in a straight line varies with distance d as shown in the figure. The work done on the particle during its displacement of \(\begin{align}12\,m\end{align}\) is

A block of mass \(\begin{align}M\end{align}\) is attached to the lower end of a vertical spring. The spring is hung from a ceiling and has force constant value \(\begin{align}k\end{align}\). The mass is released from rest with the spring initially unstretched. The maximum extension produced in the length of the spring will be

A body of mass \(\begin{align}1\,kg\end{align}\) is thrown upwards with a velocity \(\begin{align}20\,m{{s}^{{-1}}}\end{align}\) . It momentarily comes to rest after attaining a height of \(\begin{align}18\,m\end{align}\). How much energy is lost due to air friction? \(\begin{align}\left( {Take\,g=10\text{ }m{{s}^{{-2}}}} \right)\end{align}\)

\(\begin{align}300\,J\end{align}\) of work is done in sliding a \(\begin{align}2\,kg\end{align}\) block up an inclined plane of height \(\begin{align}10\,m\end{align}\). Taking \(\begin{align}g=10\,m/{{s}^{2}}\end{align}\) , work done against friction is

A body of mass \(\begin{align}3\text{ }kg\end{align}\) is under a constant force, which causes a displacement s in metre in it, given by the relation \(\begin{align}s=\frac{1}{3}{{\text{t}}^{2}}\end{align}\), where \(\begin{align}\text{t}\end{align}\) is in second. Work done by the force in \(\begin{align}\text{2}\,\text{s}\end{align}\) is

A force \(\begin{align}F\end{align}\) acting on an object varies with distance \(\begin{align}x\end{align}\) as shown here. The force is in newton and \(\begin{align}x\end{align}\) is in metre. The work done by the force in moving the object from \(\begin{align}x\text{ }=\text{ }0\end{align}\) to \(\begin{align}x\text{ }=6\text{ }m\end{align}\)is

A bomb of mass \(\begin{align}30\text{ }kg\end{align}\) at rest explodes into two pieces of masses \(\begin{align}18\text{ }kg\end{align}\)and \(\begin{align}12\text{ }kg\end{align}\). The velocity of \(\begin{align}18\,\,kg\end{align}\)mass is \(\begin{align}6\,m{{s}^{{-1}}}\text{ }\end{align}\). The kinetic energy of the other mass is

A particle of mass \(\begin{align}{{m}_{1}}\end{align}\) is moving with a velocity \(\begin{align}{{v}_{1}}\end{align}\) and another particle of mass \(\begin{align}{{m}_{2}}\end{align}\) is moving with a velocity \(\begin{align}{{v}_{2}}\end{align}\) . Both of them have the same momentum, but their different kinetic energies are \(\begin{align}{{E}_{1}}\end{align}\) and\(\begin{align}{{E}_{2}}\end{align}\) respectively. If \(\begin{align}{{m}_{1}}>{{m}_{2}}\end{align}\) , then

A ball of mass \(\begin{align}2\,kg\end{align}\) and another of mass \(\begin{align}4\text{ }kg\end{align}\) are dropped together from a \(\begin{align}60\text{ }ft\end{align}\) tall building. After, a fall of \(\begin{align}30\text{ }ft\end{align}\) each towards earth, their respective kinetic energies will be in the ratio of

If kinetic energy of a body is increased by \(\begin{align}300\,\%\end{align}\), then percentage change in momentum will be

A stone is thrown at an angle of \(\begin{align}45{}^\circ \end{align}\) to the horizontal with kinetic energy \(\begin{align}K\end{align}\). The kinetic energy at the highest point is

A child is swinging a swing. Minimum and maximum heights of swing from the earth's surface are \(\begin{align}0.75\text{ }m\end{align}\)and \(\begin{align}2\text{ }m\end{align}\) respectively. The maximum velocity of this swing is

Two bodies with kinetic energies in the ratio \(\begin{align}4:1\end{align}\) are moving with equal linear momentum. The ratio of their masses is

A force acts on a \(\begin{align}3.0\text{ }g\end{align}\) particle in such a way that the position of the particle as a function of time is given by \(\begin{align}x=3t-4{{t}^{2}}+{{t}^{3}}\end{align}\),where \(\begin{align}\text{x}\end{align}\) is in metre and \(\begin{align}t\end{align}\) in second. The work done during the first \(\begin{align}4\end{align}\) s is

A rubber ball is dropped from a height of \(\begin{align}\text{5}\,\text{m}\end{align}\) on a planet where the acceleration due to gravity is not known. On bouncing it rises to \(\begin{align}1.8\text{ }m\end{align}\). The ball loses its velocity on bouncing by a factor of

If the momentum of a body is increased by \(\begin{align}50\%\end{align}\), then the percentage increase in its kinetic energy is

The \(\begin{align}\text{KE}\end{align}\) acquired by a mass \(\begin{align}\text{m}\end{align}\) in travelling a certain distance \(\begin{align}\text{d}\end{align}\), starting from rest, under the action of a constant force is directly proportional to

Two masses \(\begin{align}1\text{ }g\end{align}\) and \(\begin{align}9\text{ }g\end{align}\) are moving with equal kinetic energies. The ratio of the magnitudes of their respective linear momenta is

A position dependent force \(\begin{align}\text{F=}\left( {7-2\times 3{{x}^{2}}} \right)\text{N}\end{align}\), acts on a small body of mass \(\begin{align}2\text{ }kg\end{align}\) and displaces it from \(\begin{align}x=0\end{align}\)to \(\begin{align}x=5m\end{align}\). Work done in joule is

A bullet of mass \(\begin{align}10\text{ }g\end{align}\) leaves a rifle at an initial velocity of \(\begin{align}1000\text{ }m/s\end{align}\)and strikes the earth at the same level with a velocity of \(\begin{align}500\text{ }m/s\end{align}\). The work done in joule to overcome the resistance of air will be

Water falls from a height of 60 m at the rate of \(\begin{align}15\,kg/s\end{align}\) to operate a turbine. The losses due to frictional force are \(\begin{align}10\%\end{align}\) of the input energy. How much power is generated by the turbine? \(\begin{align}\left( {g=10\,m/{{s}^{2}}} \right)\end{align}\)

A point mass \(\begin{align}m\end{align}\) is moved in a vertical circle of radius \(\begin{align}r\end{align}\) with the help of a string. The velocity of the mass is \(\begin{align}\sqrt{{7gr}}\end{align}\) at the lowest point. The tension in the string at the lowest point is

An object of mass \(\begin{align}500\text{ }g\end{align}\), initially at rest acted upon by a variable force whose \(\begin{align}X\end{align}\) component varies with X in the manner shown. The velocities of the object a point\(\begin{align}X=8\,m\end{align}\) and \(\begin{align}X=12\,m\text{ }\end{align}\), would be the respective values of (nearly)

A mass m is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

A body initially at rest and sliding along a frictionless track from a height \(\begin{align}h\end{align}\) (as shown in the figure) just completes a vertical circle of diameter \(\begin{align}AB=D\end{align}\). The height \(\begin{align}h\end{align}\)is equal to

Consider a drop of rain water having mass \(\begin{align}1\text{ }g\end{align}\) falling from a height of \(\begin{align}1\text{ }km\end{align}\). It hits the ground with a speed of \(\begin{align}50\text{ }m/s\end{align}\). Take g constant with a value of \(\begin{align}10\text{ }m/{{s}^{2}}\end{align}\). The work done by the (i) gravitational force and the (ii) resistive force of air is

A body of mass \(\begin{align}1\text{ }kg\end{align}\)begins to move under the action of a time dependent force\(\begin{align}F=\left( {2t\widehat{i}+3{{t}^{2}}\widehat{j}} \right)N\end{align}\), where \(\begin{align}\widehat{i}\end{align}\) and \(\begin{align}\widehat{j}\end{align}\)are unit vectors along \(\begin{align}X\end{align}\) and \(\begin{align}Y\end{align}\) axes. What power will be developed by the force at the time \(\begin{align}\left( t \right)\end{align}\) ?

What is the minimum velocity with which a body of mass \(\begin{align}m\end{align}\) must enter a vertical loop of radius \(\begin{align}R\end{align}\) so that it can complete the loop?

A particle of mass \(\begin{align}m\end{align}\) is driven by a machine that delivers a constant power \(\begin{align}k\end{align}\) watts. If the particle starts from rest, the force on the particle at time \(\begin{align}t\end{align}\) is

The heart of a man pumps \(\begin{align}5\text{ }L\end{align}\) of blood through the arteries per minute at a pressure of \(\begin{align}150\text{ }mm\end{align}\)of mercury. If the density of mercury be \(\begin{align}13.6\times {{10}^{3}}\,kg/{{m}^{3}}\end{align}\) and \(\begin{align}g=10\,m/{{s}^{2}}\end{align}\), then the power of heart in watt is

An engine pumps water through a hose pipe. Water passes through the pipe and leaves it with a velocity of \(\begin{align}2m{{s}^{{-1}}}\end{align}\) . The mass per unit length of water in the pipe is \(\begin{align}100\text{ }kg\text{ }{{m}^{{-1}}}\end{align}\). What is the power of the engine?

An engine pumps water continuously through a hose. Water leaves the hose with a velocity \(\begin{align}v\end{align}\) and \(\begin{align}m\end{align}\) is the mass per unit length of the water jet. What is the rate at which kinetic energy is imparted to water?

Water falls from a height of \(\begin{align}60\,m\end{align}\) at the rate of \(\begin{align}15\text{ }kg/s\end{align}\) to operate a turbine. The losses due to frictional forces are \(\begin{align}10\%\end{align}\) of energy. How much power is generated by the turbine ? \(\begin{align}\left( {Take\,g=10m/{{s}^{2}}} \right)\end{align}\)

A stone is tied to a string of length \(\begin{align}l\end{align}\) and is whirled in a vertical circle with the other end of the string as the centre. At a certain instant of time, the stone is at its lowest position and has a speed \(\begin{align}u\end{align}\). The magnitude of the change in velocity as it reaches a position where the string is horizontal (\(\begin{align}g\end{align}\) being acceleration due to gravity) is

A stone is attached to one end of a string and rotated in a vertical circle. If string breaks at the position of maximum tension, it will break at

How much water a pump of \(\begin{align}2\text{ }kW\end{align}\)can raise in one minute to a height of \(\begin{align}10\text{ }m\end{align}\)? \(\begin{align}\left( {Take\,g=10m/{{s}^{2}}} \right)\end{align}\)

An object flying in air with velocity \(\begin{align}\left( {20\widehat{i}+25\widehat{j}-12\widehat{k}} \right)\end{align}\)suddenly breaks in two pieces whose masses are in the ratio \(\begin{align}1:5\end{align}\). The smaller mass flies off with a velocity \(\begin{align}\left( {100\widehat{i}+35\widehat{j}+8\widehat{k}} \right)\end{align}\). The velocity of the larger piece will be

A particle of mass \(\begin{align}5\text{ }m\end{align}\) at rest suddenly breaks on its own into three fragments. Two fragments of mass \(\begin{align}m\end{align}\) each move along mutually perpendicular direction with each speed \(\begin{align}v\end{align}\). The energy released during the process is

Body \(\begin{align}A\end{align}\) of mass \(\begin{align}4m\end{align}\) moving with speed \(\begin{align}u\end{align}\) collides with another body \(\begin{align}B\end{align}\)of mass \(\begin{align}2m\end{align}\), at rest. The collision is head on and elastic in nature. After the collision the fraction of energy lost by the colliding body \(\begin{align}A\end{align}\) is

A moving block having mass \(\begin{align}m\end{align}\), collides with another stationary block having mass\(\begin{align}4m\end{align}\). The lighter block comes to rest after collision. When the initial velocity of the lighter block is \(\begin{align}v\end{align}\), then the value of coefficient of restitution \(\begin{align}\left( e \right)\end{align}\) will be

Two identical balls \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\)having velocities of \(\begin{align}0.5\text{ }m/s\end{align}\) and respectively collide elastically in one dimension. The velocities of \(\begin{align}B\end{align}\) and \(\begin{align}A\end{align}\) after the collision respectively will be

On a frictionless surface, a block of mass \(\begin{align}M\end{align}\) moving at speed \(\begin{align}v\end{align}\) collides elastically with another block of same mass \(\begin{align}M\end{align}\) which is initially at rest. After collision the first block moves at an angle \(\begin{align}\theta \end{align}\) to its initial direction and has a speed \(\begin{align}v/3\end{align}\). The second block's speed after the collision is

A body of mass \(\begin{align}\left( {4m} \right)\end{align}\) is lying in \(\begin{align}xy-plane\end{align}\) at rest. It suddenly explodes into three pieces. Two pieces each of mass \(\begin{align}\left( m \right)\end{align}\) move perpendicular to each other with equal speeds \(\begin{align}\left( v \right)\end{align}\) . The total kinetic energy generated due to explosion is

A ball moving with velocity \(\begin{align}2\text{ }m{{s}^{{-1}}}\end{align}\) collides head on with another stationary ball of double the mass. If the coefficient of restitution is \(\begin{align}0.5\end{align}\), then their velocities \(\begin{align}\left( {in\,m{{s}^{{-1}}}} \right)\end{align}\)after collision will be

An explosion blows a rock into three parts. Two parts go off at right angles to each other. These two are, \(\begin{align}1\text{ }kg\end{align}\) first part moving with a velocity \(\begin{align}12m{{s}^{{-1}}}\end{align}\)and \(\begin{align}2\text{ }kg\end{align}\)second part moving with a velocity of \(\begin{align}8m{{s}^{{-1}}}\end{align}\). If the third part flies off with a velocity of \(\begin{align}4m{{s}^{{-1}}}\end{align}\) , its mass would be

A shell of mass \(\begin{align}200\text{ }g\end{align}\) is ejected from a gun of mass \(\begin{align}4\text{ }kg\end{align}\)by an explosion that generates \(\begin{align}1.05\text{ }kJ\text{ }\end{align}\) of energy. The initial velocity of the shell is

A stationary particle explodes into two particles of masses \(\begin{align}{{m}_{1}}\end{align}\) and \(\begin{align}{{m}_{2}}\end{align}\), which move in opposite directions with velocities \(\begin{align}{{v}_{1}}\end{align}\) and \(\begin{align}{{v}_{2}}\end{align}\) . The ratio of their kinetic energies \(\begin{align}{{E}_{1}}/{{E}_{2}}\end{align}\) is

Two equal masses \(\begin{align}{{m}_{1}}\end{align}\) and \(\begin{align}{{m}_{2}}\end{align}\) moving along the same straight line with velocities \(\begin{align}+3\,m/s\end{align}\) and \(\begin{align}-5\,m/s\end{align}\)respectively collide elastically. Their velocities after the collision will be respectively

A metal ball of mass \(\begin{align}2\text{ }kg\end{align}\)moving with a velocity of \(\begin{align}36\text{ }km/h\end{align}\)has a head on collision with a stationary ball of mass \(\begin{align}3\text{ }kg\end{align}\). If after the collision, the two balls move together, the loss in kinetic energy due to collision is

A metal ball of mass \(\begin{align}2\text{ }kg\end{align}\) moving with a velocity of \(\begin{align}36\text{ }km/h\end{align}\) has a head on collision with a stationary ball of mass \(\begin{align}3\text{ }kg\text{ }\end{align}\). If after the collision, the two balls move together, the loss in kinetic energy due to collision is

A body of mass \(\begin{align}m\end{align}\) moving with velocity \(\begin{align}3km/h\end{align}\) collides with a body of mass \(\begin{align}2\text{ }m\end{align}\) at rest. Now, the coalesced mass starts to move with a velocity

Two identical balls \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\)moving with velocities \(\begin{align}+0.5\text{ }m/s\end{align}\) and \(\begin{align}-0.3\text{ }m/s\end{align}\)respectively, collide head on elastically. The velocity of the balls \(\begin{align}A\end{align}\) and \(\begin{align}B\end{align}\) after collision will be respectively

The coefficient of restitution e for a perfectly elastic collision is