How Do You Calculate Oscillation Periods and Forces on a Dam?

[blackbody]

Hi, I have a couple general (pretty much abstract) mechanics questions, and I'm not sure I'm going the right way about doing them. Any help would be appreciated.

1)On a flat surface with friction, you have a massless spring with a spring constant (k) attached to a wall on one end, and on the other end to a solid cylinder of radius R, which can roll back and forth, due to oscillation. How can you find the time for one period/oscillation?

Ok, so the total mechanical energy is the sum of the kinetic (translational and rotational) and potential energies:
E = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2+ \frac{1}{2}kx^2

I don't know whether to consider this SHM...how would I go about doing this?

2) You have a dam with a certain height of water against it. The pressure of the water can be given as a function of the height of the water p(h). What is the total force acting on the dam?

I'm thinking you just integrate the pressure function from 0 to the height?

 

[OlderDan]

Hi, I have a couple general (pretty much abstract) mechanics questions, and I'm not sure I'm going the right way about doing them. Any help would be appreciated.

1)On a flat surface with friction, you have a massless spring with a spring constant (k) attached to a wall on one end, and on the other end to a solid cylinder of radius R, which can roll back and forth, due to oscillation. How can you find the time for one period/oscillation?

Ok, so the total mechanical energy is the sum of the kinetic (translational and rotational) and potential energies:
E = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2+ \frac{1}{2}kx^2

I don't know whether to consider this SHM...how would I go about doing this?

2) You have a dam with a certain height of water against it. The pressure of the water can be given as a function of the height of the water p(h). What is the total force acting on the dam?

I'm thinking you just integrate the pressure function from 0 to the height?

As long as you are rolling without slipping, \omega is proportional to v. Combine the first two terms to get something that looks like \frac{1}{2}Mv^2
where M is a constant that is made up of the mass and moment of inertia. You should be able to take it from there.

You have the right idea about the dam

 

[thepatientmental]

For the first question, it seems like you are on the right track. The system can be considered as simple harmonic motion (SHM) since the restoring force (from the spring) is proportional to the displacement (from equilibrium position). To find the time for one period, you can use the equation T = 2π√(m/k) where m is the mass attached to the spring and k is the spring constant.

For the second question, you are correct in thinking that you can integrate the pressure function from 0 to the height to find the total force acting on the dam. This is because the total force is equal to the pressure multiplied by the area (F = pA), and the area in this case is just the height of the water times the width of the dam. So the integral would be ∫p(h)dh from 0 to the height.