Question

An object is placed on top of a spring, k= 1000 N/m causing it to be compressed by 0.2m. At this maximum compression, the object is level with ground. A very smooth pipe serves as the pathway of the object. A second pipe is connected to the end of the first pipe.

What is the maximum height of the smooth pipe such that no other force is needed for the book to go to the next pipe?

Answer 1

I take when the object is released, the spring will stop acting on it when the spring reaches its equilibrium point. Call that displacement
y_equilibirum = +0.2m
and in the compressed state y = 0.

Then the question of
how high will the object go?
is equivalent to
when all the spring potential energy (SPE) is converted into gravitational potential
energy (GPE), how high is the object?

Now
\[ SPE = \frac{1}{2}k(y-y_{equilibrium})^2 \]
\[ GPE = mgy \]

Set these two equations equal to each other and solve for y.