Download Homework 20 Elastic potential energy. If we will deform an object

Homework 20
Elastic potential energy.
If we will deform an object (say, coil spring) elastically its potential energy will be changed. We
can easily prove that: if we will let the elastically deformed object go, it returns to its original
shape and some parts of the object start moving (are accelerated), so kinetic energy increases.
How to calculate the elastic potential energy? If we stretch or compress a coil spring and
its length will be changed by x, the coil spring will apply the force to the ball of mass m (see the
Figure below):
If we let the coil spring go, the ball, initially at rest, will be accelerated and will get kinetic
energy. A simple way to calculate the kinetic energy of the ball after the coil spring is completely
stretched is to find the work W done by the elastic force of the coil spring on the ball:
(1)
Just to remind, we calculate the work as a product of the force and the displacement along the
force. The problem is that force itself depends on the displacement. The force is maximal when
the coil spring is maximally compressed and decreases to zero as the coil spring is stretching to
its original length. Luckily, in this case we can use the average force which can be calculated as
the sum of the initial and final values of the elastic force divided by 2:
(2)
Then the work is:
(3)
The work, done by the elastic force is equal to the potential energy of the compressed coil spring.
Since the total energy of the ball and coil spring conserves, as the coil spring is completely
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stretched, all potential work of the deformed weightless coil spring becomes the kinetic energy
of the ball:
(4)
So the potential energy of the deformed (compressed or stretched) coil spring depends on the
force constant (elastic constant) of the spring k and the deformation x.
(5)
Problems:
Today I will give you just one problem. It is not easy. If you will not be able to solve it, please,
do not be discouraged.
A 9gram bullet moving at a speed 800m/s hits a 5kg box with sand and gets trapped in the box
(see Figure below). Find the compression length of the coil spring if the elastic constant of the
coil spring is 1000N/m?
Hint: Consider the first moment after the bullet is trapped and use the momentum conservation
law.
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