Q4.16 Non-conservative forces (S)

Explain why the force of friction is non-conservative.

A block of mass m rests on a plane making an angle of θ with the horizontal. The coefficient of kinetic friction between block and plane is μk. What is the work done by the force of friction when:

  1. the block moves a distance d up the plane;
  2. the block moves a distance d down the plane;
  3. the block moves along path (1) then back along (2) returning to the same point?
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Hint

Think carefully about the sign of the work.
 

Solution

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Solution

Friction dissipates mechanical energy, converting it to heat.

Since the block moves parallel to the plane, there is no acceleration perpendicular to the plane; therefore the net force perpendicular to the plane is zero. This implies that N=mgcosθ. The magnitude of the frictional force on the moving block is (see S2.8)

FF=μkFN=μkmgcosθ.

Graphic - No title - noncons
  1. When the block moves a distance d up the plane, the frictional force acts down the plane; thus the work done by the frictional force is negative:

    \[ W_{up}=\int\vec{F} \cdot d\vec{r}=-F_Fd=-\mu_{k}mgd\cos\theta. \]

  2. When the block moves down the plane, FF acts up the plane, and the work done is still negative:
    Wdown=-FFd=-μkmgdcosθ.
  3. On the round trip, the total work done is
    Wround=-2μkmgdcosθ.
    This is clearly not zero, which shows that the frictional force is non-conservative.