Calculating work done on a rigid body

Question

How in general does one calculate the work done by some force acting by a rigid body? Do you have to take into account the torque and the translation? For example:

Suppose we have a ball rolling down an incline (not slipping, there is rolling friction), starting at rest at the top. We have only two forces acting on the ball: gravity and friction.

We have $W_{grav}+W_{fric}=K$

How do you calculate the work done by a force on a rigid body? The work done by gravity is $mgh$ but how can we cacluate the work done by friction? Naively I think it is $Fd$ where $F$ is the friction force, but friction also induces the rotational motion of the ball.

Answer 1

There is more than one approach one can use to calculate the work done by a contact force on a rigid object. All approaches are valid so long as you're careful about how you actually use your calculated work.

One way is to use the infinitesimal displacement of the point of application $d\vec l_\text{app}$:

$$W = \int \vec F \cdot d\vec l_\text{app} \tag{1}$$

In the absence of other forces (and forms of potential energy), this will correctly give you the quantity $1/2 \int v^2 dm$, i.e., the change in total kinetic energy.

In your object-down-the-ramp example, the static frictional force would do zero work.

Another way to approach the same problem is to use the center of mass displacement $d\vec l_\text{CM}$:

$$W = \int \vec F \cdot d\vec l_\text{CM} \tag{2}$$

In the absence of other forces (and forms of potential energy), this will give you the change in center-of-mass kinetic energy $\Delta K_\text{CM} \equiv \Delta\left(\frac{1}{2}mv_\text{CM}^2\right)$.

Yet another way is an extension of method #2, in which one considers the rotational aspects of the motion:

$$W = W_\text{CM} + W_\text{rot} = \int \vec F \cdot d\vec l_\text{CM} + \int \vec\tau \cdot d\vec\theta \tag{3}$$

This latter approach lends itself well to separating out work that causes changes in the center-of-mass kinetic energy and changes in the rotational energy of a rigid body:

$$\Delta K_\text{tot} = \Delta K_\text{CM} + \Delta K_\text{rot}$$

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More info on the different definitions of work: Article by Sherwood

Answer 2

The frictional force that keeps an object rolling without slipping does no work on the object. How can it? The point at which it operates is not moving with respect to the surface. That the velocity of the contact point is identically zero is the quintessential nature of rolling without slipping.

If the frictional force doesn't do work, what does it do? It is instead is a constraint force that acts to keep the object rolling without slipping.

The kinetic energy of an object that is translating and rotating is $KE = \frac12 mv^2 + \frac12 I\omega^2$ where $m$ is the mass of the object, $v$ is the velocity of the center of mass, $I$ is the moment of inertia about the center of mass, and $\omega$ is the angular velocity. In the case of an object that is rolling without slipping, the relation between angular velocity and center of mass velocity is given by $\omega = \frac v r$. The kinetic energy for such an object can thus be re-expressed as $KE = \frac12 m\bigl(1 + \frac I{mr^2}\bigr) v^2$.

In the case of an object rolling down a ramp starting from at rest and rolling down a vertical height $h$, the gravitational force does work $W=mgh$. The frictional force does no work, so the work-energy principle dictates that $mgh = \frac12 m\bigl(1 + \frac I{mr^2}\bigr) v^2$, or $$v = \sqrt{\frac 1 {1+\frac I{mr^2}} g h}$$

In the case of a solid sphere, $I=\frac 2 5 mr^2$ and thus the velocity is given by $v = \sqrt{\frac{10}7 gh}$. (See http://hyperphysics.phy-astr.gsu.edu/hbase/sphinc.html for an alternate derivation.) For a hollow sphere, $I=\frac 2 3 mr^2$, yielding $v = \sqrt{\frac65 gh}$.

Answer 3

For a rigid body, we have two different types of kinetic energy:

Translational KE

This is the usual $\frac{1}{2}mv^2$

Rotational KE

This is new, and takes the form $\frac{1}{2}J\omega^2$, where $J$ is the polar moment of inertia taken about the axis of rotation that passes through the centre of mass.

Work

To cause a change in translational KE, you need to do work by a resultant force, and the work done is the displacement of the centre of mass multiplied by the component of force that lies in the direction of the force.

However, we need to explain changes in rotational KE. This is caused by doing work by a resultant torque about the centre of mass. This work is equal to the rotation multiplied by the component of the resultant torque that lies in the plane of rotation.

Therefore, as an equation, the total work is given by:

$$W = \int \vec F \cdot d \vec r + \int \vec \tau \cdot d \vec \theta$$

Where $\vec F$ is the result force, $\vec \tau$ is the resultant torque, $d \vec r$ is an infinitesimal change in displacement of the centre of mass, and $d \vec \theta$ is an infinitesimal rotation about the axis of rotation going through the centre of mass.

Answer 4

I have not seen a good treatment of work from friction in the earlier responses to this question nor in responses to other related questions. For a rigid body, there is no heating effect from friction. Consider the work done from sliding friction. (Rolling friction does no work since there is no sliding of the body; similar to a fluid flowing in a rough pipe with no slip at the pipe surface; the friction force at the pipe surface does no work on the fluid.) For example, numerous past questions and responses address the effect of sliding friction for rotational motion in a plane, but incorrectly address the heating effect from friction. Here are three: [text]https://physics.stackexchange.com/questions/165453/tricky-conceptual-question-ball-sliding-and-rolling-down-incline?r=SearchResults&s=1|99.3755 (1)
[text]https://physics.stackexchange.com/questions/527671/does-friction-do-work-or-dissipate-heat?r=SearchResults&s=2|48.8239 (2) [text]https://physics.stackexchange.com/questions/571004/work-done-by-friction-on-a-sphere-sliding-down-the-inclined-plane?r=SearchResults&s=26|31.0922 (3) The responses evaluate torque as inertia times angular acceleration. Then the responses indicate that friction causes heating. Heating does occur in reality, however the relationship torque = inertia times angular acceleration implicitly assumes a rigid body and for a rigid body there is no internal dissipation of energy (no heating). For a rigid body friction only changes the kinetic energy of the body. Consider question/response #1 above for the case of sliding friction. The response nicely evaluates the dynamics of the motion, but then says “when the sphere starts slipping, you lose energy, … energy is lost in heat.” This is not correct; since the response implicitly assume a rigid body, the effect of friction is only to change the kinetic energy of the object. The initial kinetic energy of the object is zero. The final kinetic energy of the object is its translational kinetic energy plus its rotational kinetic energy. It can be shown (see below) that the total work done by gravity and by friction is equal to the change in kinetic energy. The point is that evaluations that implicitly assume a rigid body and then state friction causes heating are incorrect. To address heating effects, the rigid body constraint should be removed and the first law of thermodynamics applied. For the question response 1 above for sliding friction, the following calculation shows that work from gravity and friction = change in kinetic energy of the object.

Does anyone have a relatively simple approach for evaluating the heating for a solid non-rigid body? For example, in fluid mechanics the mechanical energy equation sometimes includes a friction loss term to approximate the energy loss that would be accurately evaluated using the first law of thermodynamics and conservation of mass and momentum.