Work Done by a Spring Force — Question 9

Question 9

A frictional approach limits the maximum compression

A 1 kg block moving at 4 m/s first contacts an unstretched horizontal spring of $k=100$ N/m. A constant 6 N friction force opposes the block throughout compression. The spring is ideal.

1. Write the stopping condition for compression magnitude c.

2. Find the maximum compression.

3. Can the block remain at that position if the maximum static friction is also 6 N?

Question 9 - Solution

A frictional approach limits the maximum compression

(a) Include both negative works.

The governing relation is $$\boxed{K(c)=\tfrac12mv_i^2-fc-\tfrac12kc^2}$$ Initial $$K=8\ \mathrm J.$$ At a stop, $$0=8-6c-50c^2,$$ so $$50c^2+6c-8=0.$$

(b) Select a nonnegative compression.

$$c=(-6+\sqrt{1636})/100=(\sqrt{409}-3)/50=0.3445\ \mathrm m.$$ The other algebraic root is negative and is outside the defined compression coordinate.

(c) Test force balance at the stop.

The spring force is $100c\approx34.45\ \mathrm N$, greater than the 6 N available static friction. The block must begin moving back. Zero instantaneous speed alone does not establish equilibrium.