Solving the Frequency of Small Oscillations in a Spherical Dish

In summary, the problem is to find the frequency of small oscillations for a marble of radius b rolling back and forth in a shallow spherical dish of radius R. This can be solved using either conservation of energy or Newton's second law. The answer is w^2 = 5g/7R and this can be shown by setting up PE = KE + Rotational KE. The only difference is that one person used R-b instead of R for the radius, but the results are still close enough.
  • #1
S0C0M988
6
0
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.
 
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  • #2
S0C0M988 said:
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.
 
  • #3


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 =
 

Related to Solving the Frequency of Small Oscillations in a Spherical Dish

1. What is the purpose of solving the frequency of small oscillations in a spherical dish?

The purpose of solving the frequency of small oscillations in a spherical dish is to understand and predict the behavior of objects placed inside the dish when subjected to small perturbations or vibrations. This can have applications in various fields such as physics, engineering, and materials science.

2. How is the frequency of small oscillations in a spherical dish calculated?

The frequency of small oscillations in a spherical dish can be calculated using the formula: f = (1/2π) * √(g/R), where f is the frequency, g is the acceleration due to gravity, and R is the radius of the dish. This formula applies to objects with small amplitudes of oscillation.

3. What factors can affect the frequency of small oscillations in a spherical dish?

The frequency of small oscillations in a spherical dish can be affected by various factors such as the mass of the object, the shape and size of the dish, the elasticity of the material, and the presence of any external forces or damping effects. These factors can alter the natural frequency of the object-dish system.

4. How does the frequency of small oscillations in a spherical dish relate to the energy of the system?

The frequency of small oscillations in a spherical dish is directly proportional to the energy of the system. This means that as the frequency increases, the energy of the system also increases. This relationship is important in understanding the dynamics of the system and how it responds to different inputs and perturbations.

5. Can the frequency of small oscillations in a spherical dish be changed?

Yes, the frequency of small oscillations in a spherical dish can be changed by altering the parameters of the system such as the mass, shape, and size of the object or the dish. External forces or damping effects can also affect the frequency. By adjusting these parameters, the natural frequency of the system can be tuned to a desired value.

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