Question

As shown in the figure, a 1.5-kg box is held at rest against a spring with a force constant k = 705 N/m that is compressed a distance d. When the box is released, it slides across a surface that is frictionless, except for a rough patch that has a coefficient of kinetic friction mu*k = 0.40 and is 6.0 cm in length. If the speed of the box is 2.2 m/s after sliding across the rough patch, determine the initial compression d of the spring.

Answer 1

The first thing to realize is that the part of the question that talks about the block sliding across a frictionless surface DOESN'T MATTER! The only variable that changes during this time is distance and time itself, and the only reason we'd care about either is if we're interested in knowing the TOTAL time the box was in motion or the TOTAL distance the box traveled. But we aren't - we only care about things that changed the speed of the box, and the only thing that did was the friction from the rough patch.

So really we can pretend this question is asking about a box that is released from a spring and then IMMEDIATELY travels across the rough patch. Then solving for compression of the spring is easy, because we can use conservation of energy. All the elastic potential energy that you start with, PE, is converted into either kinetic energy, KE (because the block is in motion), or heat energy, H (due to the WORK done by friction to slow the box down). In other words:

\[PE=KE+H\]
\[{1 \over 2}kd^2={1 \over 2}mv^2+FD\]

d is the compression of the spring and D is the distance over which the force (friction) acted (remember that W=FD). We aren't given the force of friction (F) directly, but we can find it using mu(k) * F(N) = mu(k) *mg. So the equation above becomes:

\[{1 \over 2}kd^2={1 \over 2}mv^2+\mu_kmg D\]
Now it's just a matter of plugging in your values (making sure you have the proper units for each), then solving for d to get your answer!

These paragraphs always look scary when you first read through them. Focus on the specific question being asked and think about the situation. Come up with a general way of solving the problem (e.g. conservation of energy in this case), then sub in more specific variables appropriate to your specific situation.

If you have any questions let me know!