Calculating the average friction force on a sliding book

Key concept: This problem uses work-energy ideas and the definition of average frictional force along a slope. The key is to connect the force to the energy changes while the book moves.

Step by step

What this question is testing

The task asks for the average frictional force acting on a book while it slides down an incline. In incline-motion problems, friction is usually the force that removes mechanical energy from the system. That means the most reliable path is to use the energy balance rather than trying to guess the force directly.

If the book is sliding, the friction force acts opposite the direction of motion. Its average value can be found from the work it does over the distance traveled along the slope.

Method to use

The work done by friction is

$W_f = -F_{ ext{fric, avg}} \cdot d$

where $d$ is the distance moved along the incline. The negative sign shows that friction opposes motion. If the problem gives the energy lost to friction, or enough information to find it, then

$F_{ ext{fric, avg}} = \frac{\text{energy lost to heat}}{d}$

using the magnitude of the energy loss.

So the solution strategy is:

1. Identify the distance the book travels along the incline.
2. Determine the energy converted into heat by friction.
3. Divide energy lost by distance to get the average friction force.

Why this works

Work is force multiplied by displacement in the direction of the force. Since friction is not usually constant at every instant, the question asks for the average frictional force. That average is the single constant force that would produce the same total energy loss over the same distance.

This is a standard mechanics interpretation: if a resistive force removes $E$ joules over $d$ metres, then its average size is $E/d$ newtons.

Common exam point

Make sure you use the distance along the incline, not the vertical drop, unless the question explicitly says otherwise. Also keep the sign convention straight: the magnitude of the average frictional force is usually what is required in an answer line, while the work done by friction is negative.

Final form of the answer

If the energy lost to heat over the slide is $E$ joules and the book travels a distance $d$ metres down the slope, then

$\boxed{F_{\text{fric, avg}} = \frac{E}{d}}$

newtons, acting opposite the motion.

Pitfall alert

A common mistake is to use the vertical height change instead of the distance along the incline. That gives the wrong force because friction acts along the surface, not vertically. Another frequent error is forgetting that the work done by friction is negative. When the question asks for the average frictional force, the numerical answer is usually the positive magnitude, but the direction should still be stated as opposite the motion. If you have already found the energy lost to heat in a previous part, do not re-derive it from scratch unless the question requires that.

Try different conditions

If the incline were rougher or the slide distance were doubled, the same energy-loss method still applies, but the average friction force changes with the ratio $E/d$. For example, if a variation asked: 'A book loses 48 J of mechanical energy while sliding 6 m down the incline. Find the average frictional force,' then the answer would be $48/6 = 8$ N. If the distance became 12 m for the same 48 J loss, the average friction would drop to 4 N. The method is unchanged: use work done by friction divided by distance along the slope.

Further reading

work done by friction, energy dissipation, inclined plane mechanics