Physics (Springs) Question?

Question

A 1.4 kg block is released from rest and allowed to slide down a frictionless surface and into a spring. The far end of the spring is attached to a wall, as shown. The initial height of the block is 0.38 m above the lowest part of the slide and the spring constant is 432 N/m.

(a) What is the block's speed when it is at a height of 0.25 m above the base of the slide?
___m/s
(b) How far is the spring compressed?
___ m
(c) The spring sends the block back to the left. How high does the block rise?
__m

Answer 1

I don't see a diagram so I can't answer the question in detail, but from the description, it looks like it is all about conservation energy.

The formula for gravitational energy is:
Eg = M x g x H

where H is the vertical distance and g is the acceleration due to gravity (9.8 m/s on the surface of the Earth)

The formula for kinetic energy is:

Ek = (1/2) x M x V^2

where V is the velocity. (This assumes only linear velocity - things get more complicated when there is rolling, etc.)

The formula for the energy in a simple spring is

Es = (1/2) x K x D^2

where:

K is the spring constant of the spring (assumed constant for simple springs)

D is the displacement of the spring (for a linear spring the change in the length of the spring)

When the block starts from rest at the top of the slide, it has a certain potential energy.

As it slides down, it converts this potential energy to kinetic energy. As the slide is frictionless, the conversion is lossless.

So when the block has gone from 0.38 meters of height to 0.25 m, it has converted the potential energy associated with that 0.12 height to kinetic energy. So:

M x g x 0.12 = (1/2) x M x V^2

and you can compute the V which is the answer to part a.

When the spring is fully compressed, it is because the velocity has gone to 0 so all the kinetic energy has become spring energy. (If the spring is along the slide (i.e. at an angle) then the extra potential energy also goes into the spring energy)

Given the total energy, the mass M, and the spring constant, you can compute the displacement of the spring, for the answer to part b.

Since no energy is lost, if the block is pushed back and up, it ends up at its original height.