Solving the Frequency of Small Oscillations in a Spherical Dish

[S0C0M988]

A marble of radius b rolls back and forth in a shallow spherical dish of radius R. Find the frequency of small oscillations. You can solve this problem using conservation of energy or using Newton’s second law. Solve it both ways and show that you get the same answer.

I kind of get the concept and I'm using conservation of energy but he numbers don't work out. I know the answer is supposed to be w^2 = 5g/7R

I set up PE = KE + Rotational KE but it doesn't work.

 

[OlderDan]

A marble of radius b rolls back and forth in a shallow spherical dish of radius R. Find the frequency of small oscillations. You can solve this problem using conservation of energy or using Newton’s second law. Solve it both ways and show that you get the same answer.

I kind of get the concept and I'm using conservation of energy but he numbers don't work out. I know the answer is supposed to be w^2 = 5g/7R

I set up PE = KE + Rotational KE but it doesn't work.

It worked for me except I have R - b instead of R. Assuming a small marble. R is close enough.

 

[kyle8921]

I understand your frustration when the numbers don't seem to work out. However, it is important to carefully review your calculations and assumptions to find where the error may be occurring. Let's take a closer look at the problem and see if we can identify the issue.

First, let's review the concept of small oscillations. In this case, we can assume that the marble is only moving back and forth within a small range, which means we can use the small angle approximation. This means that we can approximate the sine of the angle as the angle itself, which simplifies our calculations.

Next, let's outline the steps for using conservation of energy to solve this problem:

1. Identify the initial and final states: In this case, the marble starts at the top of the dish and ends at the bottom of the dish.

2. Write out the conservation of energy equation: We can write the equation as follows: PE(initial) + KE(initial) + Rotational KE(initial) = PE(final) + KE(final) + Rotational KE(final)

3. Substitute in the appropriate equations: For potential energy, we can use the equation PE = mgh, where m is the mass of the marble, g is the acceleration due to gravity, and h is the height of the marble above the bottom of the dish. For kinetic energy, we can use the equation KE = 1/2mv^2, where m is the mass of the marble and v is its velocity. For rotational kinetic energy, we can use the equation Rotational KE = 1/2Iw^2, where I is the moment of inertia of the marble and w is its angular velocity.

4. Simplify the equation: Since we are using the small angle approximation, we can approximate the height of the marble as b - R, where b is the radius of the marble and R is the radius of the dish. We can also assume that the marble is rolling without slipping, so we can relate the linear and angular velocities as v = rw, where r is the radius of the marble. Substituting these values into the conservation of energy equation and simplifying, we get the following equation:

mgb - 1/2Iw^2 = 1/2mv^2 + 1/2Iw^2

5. Solve for the frequency: We can use the equation v = rw to solve for the velocity, which gives us v =