A block of mass m is pushed a horizontal distance D from position A to position B, along a horizontal plane with friction coefficient µ. Then the mass is pushed from B to A. If the horizontal force pushing the mass from A to B is P~, and the force pushing the mass from B to A is - P~, what is the total work done by friction?

The total work done by friction is - 2 · μ · m · g · D

Explanation:

Hi there!

The work done by a force is calculated as follows:

W = F · d · cos θ

Where:

W = work.

F = force that does the work.

d = displacement.

θ = angle between the displacement and the force.

If the force is horizontal, as in this case, cos θ = 1

The friction force is calculated as follows:

Ffr = μ · m · g

Where:

μ = friction coefficient.

m = mass of the object.

g = acceleration due to gravity.

Then, in this case, the work done by friction when pushing the block from A to B will be:

W AB = - Ffr · D

W AB = - μ · m · g · D

Notice that the friction force is negative because it is opposite to the pushing force P.

When the block is pushed from B to A, the work done by friction will be:

W BA = Ffr · (-D)

W BA = - μ · m · g · D

Now, the displacement is negative and the friction force is positive (in the opposite direction to - P).

The total work done by friction will be:

W AB + W BA = - μ · m · g · D - μ · m · g · D = - 2 μ · m · g · D