Gliding mass stopped by spring and friction

Question

(12 points possible)

A small block of mass m=1 kg glides down (without friction) a circular track of radius
R=2 m, starting
from rest at height R. At the bottom of the track it hits a massless relaxed spring with spring constant
k= 7 N/m, which starts to be compressed as the block continues to move horizontally. Note that we
assume no energy loss during this “collision". There is friction between the block and the horizontal surface, and it is not uniform. As a function of distance, the friction coefficient varies like μ(x)=αx, with
α=
0.7 m
−1. Assume for simplicity that static and dynamic friction coefficients are the same, and use
g=10
m/s
2. (See figure)

(a) What is the maximal distance x
1 that the block moves horizontally away from the track at
x=0? (in
meters)

x
1=
unanswered

(b) What time t
1 does it take for the block to travel between
x=0 (relaxed spring) and
x=x
1 (block at
first stop)? (in seconds)

t
1=
unanswered

(c) What will happen after the block reaches point x
1?

The block will stay put forever at x=x1.
The block will move back and reach a second stop exactly at x=0.
The block will move back and reach a second stop somewhere between x=0 and x=x1.
The block will move back and get catapulted up the circular track.