Finding Work Done by Compressing A Spring

[rk21619]

Homework Statement

I did a lab today in Physics in which we launched ball from a spring loaded cannon directly into a pendulum that captured the ball, held it, and swung upwards with it (representing a totally inelastic collision). One question that confuses me:

> How much work did you do in joules in compressing the spring of the spring gun for the long-range case? Which law of conservation is your answer based upon?

Some additional information: the long-range case is where we fired the ball at it's maximum speed (which I've calculated to be 4.79 m/s^2.

Homework Equations

E = K + U = 0
initial momentum = final momentum
m(v_initial) = (m + M)(v_final)
K = (1/2)(m + M)v^2
U = (1/2)kx^2 [for the spring]

The Attempt at a Solution

I'm guessing the law of conservation to use here is for that of energy, in which K + U = 0, where K is kinetic and U is final energy. The system has all potential energy (no kinetic) when the spring is compressed, which is known as (1/2)kx^2, but I don't know k...

If any other information is needed, let me know. Thanks for the help!

 

[luitzen]

The potential energy stored in the spring is there because you put it into it. That's the work you did.

 

[rk21619]

The potential energy stored in the spring is there because you put it into it. That's the work you did.

Yes, well, this means 2 things to me:

That the answer is (1/2)kx^2, because that's the energy I put into it (which I can't find - I don't know k, and I also don't know how far the spring got compressed [aka x]).

Or, we use the law of conservation of energy so that:
K + U = 0
K_f - K_i + U_f - U_i = 0 [there is no initial kinetic energy or final potential, so...]
U_i = K_f
So, the work I did is equal to K_f, which is (1/2)mv^2.

I did out this calculation and got some ridiculous number .0246 J... I definitely did more work than that, so I'm still so confused.

 

[luitzen]

The method is correct. I don't know how heavy that ball is, but when I run the calculation for a 10g marble with a final speed of 5 m/s I get 0.125J. So unless you are using something lighter and/or moving slower, you'll probably have made a mistake in the calculation.

 

[asteriatic]

Firstly, great job on completing your lab today and coming up with some good questions! In order to find the work done by compressing the spring, we can use the equation W = Fd, where W is work, F is the force applied, and d is the distance over which the force is applied. In this case, the force applied is due to the compression of the spring, and the distance over which it is applied is the distance the spring is compressed.

To find the force, we can use Hooke's Law, which states that the force applied by a spring is directly proportional to the distance it is compressed or stretched. The equation for Hooke's Law is F = kx, where k is the spring constant and x is the distance the spring is compressed. In this case, since we do not know the spring constant, we can use the maximum speed of the ball (4.79 m/s) to find it.

We can use the equation for kinetic energy, K = (1/2)mv^2, to find the initial kinetic energy of the ball before it hits the pendulum. Since we know the mass of the ball, we can solve for the initial kinetic energy. We can then use the equation for conservation of momentum, m(v_initial) = (m + M)(v_final), to find the final velocity of the ball and pendulum system.

Once we have the final velocity, we can use the equation for potential energy of a spring, U = (1/2)kx^2, to find the potential energy of the compressed spring. This potential energy represents the work done by compressing the spring.

Overall, your answer will be based on the law of conservation of energy, as you correctly mentioned. This law states that energy cannot be created or destroyed, only transferred from one form to another. In this case, the kinetic energy of the ball before it hits the pendulum is converted into potential energy of the compressed spring.

I hope this helps clarify the process for finding the work done by compressing the spring. Keep up the good work in your physics studies!