# A box slides over friction and hits a spring

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I've been struggling with these problems for days.

A block is released from point A on a track
ABCD as shown in the ﬁgure. Point A is
higher than points B, C, and D. The track is
frictionless except for a portion BC which has
a coeﬃcient of friction µ. The block travels
down the track and hits the spring with spring
constant k.
The acceleration of gravity is 9.8 m/s^2.

Here's the picture:

1. If the spring compresses 6 cm, ﬁnd the
coeﬃcient of friction µ.

2.For this value of µ, to what height (h′) will the
block rise when it returns to point E ?

3.Suppose µ is changed. After releasing the
mass m from point A, the mass travels over
the rough portion of the track BC, bounces
oﬀ the spring, and travels a distance 2.9 m
part way back across the rough portion of the
track to stop at point F .
Under these conditions, what is the value
of µ?

I know I'm supposed to use (initial potential energy + initial kinetic energy + work done by friction = final potential energy + final kinetic energy) to solve them, but I'm not sure how to manipulate the equation. Thank you!

We do want to compare initial and final energies as you suggested above.

To do so, we can note that in each of the situations mentioned (1, 2, and 3) the initial and final kinetic energies are zero $U_i - W_{friction} = U_f$. Also, the initial potential energy is the same in each case: $U_i = mgh$ and the final total energy will be all potential energy. The work done by friction will be $W_{friction} = \mu N d$

1. $mgh - \mu N d = \frac{1}{2}m \Delta x^2$
Here, we are looking for $\mu$ being given all of the other variables (m, g, h, d, and x).
2. $mgh - 2 \mu N d = mgh'$
Here we are looking for h' being given all of the other variables.
3. $mgh - \mu N (d+x) = 0$
Here we are looking for x.

answered Mar 10, 2014 by (2,140 points)