Work done by a force when the displacement of its point of application is zero

Four examples are described where a force acts on an object to accelerate the object and where the displacement of the point of application of the force is zero. Work is done to accelerate the object but the total work done by the force is zero.

Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of the centre of mass is given by F/m.In that case, Fs is the work done by F to change the translational kinetic energy of the body (ie 12 mv 2 ), where v is the velocity of the centre of mass and s is the displacement of the centre of mass rather than the displacement of the point of application of F. The two separate displacements might be the same, but sometimes they are not.
In some situations where F is applied to an object, the displacement of the point of application of F is zero even though the displacement of the centre of mass is not zero.In that case, it is the total work done by F that is zero, summed over changes in translational, rotational and possibly elastic or thermal energy.Four different examples are illustrated in figure 1, all showing that the work done by a force can be difficult to interpret [1,2].All four cases have been analysed previously by various authors, but a brief summary is presented below to highlight the fact that the total work done is zero in each case.

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Figure 1.different examples where there is no displacement of the point of application of a force.In all four cases, the total work done by each force is zero.
Figure 1(a) shows an example where a ball is rolling without slipping down an incline.The static friction force exerted on the bottom of the ball prevents the ball from slipping.The bottom of the ball comes to rest on the incline, so there is no displacement at the point where F is acting.The friction force has two effects.It reduces the acceleration down the incline compared with the acceleration on a frictionless incline, and it exerts a torque on the ball which increases its angular velocity.The work done to increase the rotational kinetic energy of the ball is equal and opposite the work done to reduce the translational kinetic energy.The total work done by the friction force is therefore zero [1][2][3].Energy is conserved since the gravitational potential energy at the top of the incline is converted to translational and rotational kinetic of the ball.There is no dissipation of energy by the friction force.
Figure 1(b) shows an example involving the acceleration of a car.If the drive wheel rolls without slipping, then the static friction force (F) at the bottom of the wheel acts to accelerate the car but it also exerts a torque on the drive wheel that acts to prevent the wheels spinning too fast.
The work done by the torque is equal and opposite the work done by F to accelerate the car, so the total work done by F is zero [4].
Figure 1(c) shows the bounce of a ball that is dropped vertically onto a horizontal surface.If the ball is perfectly elastic then the ball rebounds at the same speed as the incident speed so energy is conserved.During the impact, a vertical force (N) is exerted on the bottom of the ball.There is no displacement of N but the centre of mass moves downward by a distance y as the ball compresses, then rises as the ball expands.The equation of motion is N = −md 2 y/dt 2 where m is the mass of the ball.The ball decelerates to a temporary stop before bouncing off the surface.The initial kinetic energy is converted to elastic energy due to compression of the ball as it comes to a stop.The work done by N to decrease the kinetic energy is equal and opposite the work done to increase the stored elastic energy, so the total work done by N is zero while the ball compresses and also when the ball expands.
A more complicated and more puzzling example is shown in figure 1(d) where a ball rolls without slipping on a horizontal surface.The ball

Work done by a force when the displacement of its point of application is zero
eventually rolls to a stop due to rolling friction.The friction force is given by F = µ R N where µ R is the coefficient of rolling friction and N = mg is the normal reaction force, where m is the mass of the ball.The equation of motion in the horizontal direction is mdv/dt = −F = −µ R mg, where v is the velocity of the centre of mass, so the acceleration, a = dv/dt, is given by a = −µ R g.The velocity of the bottom of the ball is zero.Since there is no slipping, there is no relative motion between the bottom of the ball and the rolling surface, so the friction force is a static force.The friction force has two effects.It acts to reduce the velocity of the ball, and it acts in a direction to increase the angular velocity of the ball.The work done by the friction torque is equal and opposite the work done to decrease the velocity of the ball, so the total work done by the friction force is zero [5].
When a rolling ball rolls to a stop on a horizontal surface, the linear and angular velocities of the ball both decrease to zero, despite the fact that the friction force acts in a direction to increase the angular velocity.This result can be explained by the fact that the normal reaction force does not act through the centre of the ball.Instead, it acts at a distance D ahead of the centre, as shown in figure 1(d), and it exerts an opposing torque that is larger than the friction torque.Consequently, the angular velocity decreases with time as the ball rolls to a stop.No work is done by the normal force to accelerate the ball vertically, but work is done by the opposing torque and it is equal to the total decrease in rotational and translational kinetic energy of the ball.In that case, the total kinetic energy is not conserved since the ball rolls to a stop.However, the total work done by N is zero, since the decrease in rotational kinetic energy due to the opposing torque is equal to the increase in thermal energy of the ball [5].
A common feature in all four cases in figure 1 is that it is the total work done by a force that is zero if the displacement of the point of application is zero.The rolling ball example is more complicated since the total work done by the static friction force is zero, and the total work done by the normal reaction force is also zero.The normal force acts to reduce the angular velocity of the ball, with the result that kinetic energy is not conserved.The loss of energy can be accounted for by compression and expansion of the bottom of the ball, resulting in hysteresis losses and a rise in temperature of the ball.If a soft ball rolls along a hard surface, the leading edge of the ball compresses vertically and the trailing edge expands vertically.The compression force is larger than the expansion force, which is why the net vertical force is shifted towards the front of the ball and why energy is dissipated in the ball.