To mathematically illustrate the concept of work for a particle, suppose we have a constant force

The work

In this case the work done is simply the force (

Now suppose we have a constant force

The work done on the particle by the force is given by:

This is the same result as before, except that we are now taking the component of

Let’s now consider the most general case where the force

Using Calculus, we can set up a general expression for the work done by the force vector

Let the distance

At the point where the particle is located, let

To evaluate the total work done on the particle between

Now,

where

Substituting equation (2) into equation (1) we get

Now, the velocity of the particle is always tangent to the curve. If we set

Using the chain rule of Calculus we can write

Since,

we can write

This gives us

Substituting equation (4) into equation (3) we get

As a result,

This scalar equation tells us that the work done on the particle between

Once again, the term

is equal to the work done by the forces acting on the particle. These forces can either be conservative or non conservative. The force

If gravity is acting on the particle, the work done by gravity is given by

Where:

Δ

If an elastic spring (obeying Hooke's Law) is acting on the particle, the work done by the spring is

Where:

Thus, if gravity and an elastic spring are the only forces acting on the particle then

This tells us that you do not need to go through the effort of evaluating the integral on the left side of the above equation if the forces acting on the particle are gravity and/or an elastic spring. The work done by these two forces is simply given by the above equations for

To determine the work done by a force on a rigid body we must first treat the rigid body as a collection of particles and then sum the work done on each of those particles, in order to find the total work done on the rigid body.

For each particle in the rigid body we can apply equation (5), without loss of generality. Thus, for each particle in the rigid body (denoted by subscript

Summing over the entire rigid body we have:

By Newton’s Third Law all internal forces cancel out. Therefore the term

is equal to the work done only by the external forces (

From the page on kinetic energy, the general kinetic energy equation for a rigid body at any instant is given by

where the variables are defined on the page on kinetic energy.

Set

Therefore from equation (7),

Set

Therefore from equation (7),

As a result equation (6) becomes

This scalar equation tells us that the work done on the rigid body by external forces (between

Once again, the term

is equal to the work done by the external forces acting on the rigid body. These forces can either be conservative or non conservative.

If gravity (an external force) is acting on the rigid body then the work done by gravity is given by

Where:

Δ

If an elastic spring (obeying Hooke's Law) is acting on the rigid body, the work done by the spring is

Where:

Thus, if gravity and an elastic spring are the only forces acting on the rigid body then

This tells us that you do not need to go through the effort of evaluating the integral on the left side of the above equation if the (external) forces acting on the rigid body are gravity and/or an elastic spring. The work done by these two forces is simply given by the above equations for

Suppose there are two parallel forces

No matter where this force couple is located relative to an arbitrary point

The direction of the moment due to this couple is given by the right hand rule and it is perpendicular to the plane containing the force pair. The magnitude of the moment

If the body rotates in the plane between two angles

where

If

This formula can be useful for, say, determining the work done by a pure moment (torque) acting on a flywheel that rotates between angles

Make sure you use the right hand rule to assign the correct sign for

If a force acts perpendicular to the direction of travel it does no work. This is easy to see if one looks at the equation for work done by a force

The work is given by:

The force

Another example of a force that does no work is the friction force acting on a wheel (or ball) that rolls without slipping, as shown below. The friction force acts at the contact point

However, whether or not there is slipping, the normal force

Power is defined as the rate at which work is performed (with respect to time).

Power acting on a particle

If we know the instantaneous force

Power acting on a rigid body rotating about a fixed point

If we know the instantaneous torque

A flywheel is an example of a rotating rigid body for which we might want to know the power acting on it.

The usefulness of solving dynamics problems using work and energy calculations is that it can greatly simplify the analysis of some problems. You don’t have to draw a free body diagram and analyze the forces involved. Furthermore, there is only a single scalar equation to apply (for each particle, or for each rigid body), as opposed to applying force and moment equations for each of the three directions (

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