The length of a pendulum that is equivalent to a rocking hemisphere

In summary, a solid sphere cut in half results in a hemisphere of radius r and mass M being placed on a rough table. The hemisphere rocks back and forth with small amplitude movements. The length of an equivalent simple pendulum is 1.73r and approximations need to be justified. The center of mass of a hemisphere is located at a distance 3r/8 below the center of the sphere. However, further calculations and derivations are needed to confirm the solution.
  • #1
chriskh

Homework Statement


A solid sphere is cut in half and a homogeneous hemisphere of radius r and mass M is set upon a table(with its flat side up). The surface of the table is perfectly rough. The hemisphere rocks back and forth with small amplitude excursions from equilibrium. What is the length of an equivalent simple pendulum? Justify approximations. Note that the center of mass of a hemisphere is at a distance 3r/8 below the center of the sphere.

Homework Equations

The Attempt at a Solution

 
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  • #2
Can you give it your best shot?
 
  • #3
upload_2017-12-15_3-3-17.png

The answer should be 1.73r

I can't find what's wrong with my solution.
 

Attachments

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  • #4
I'm not sure about your derivation of either ##T## or ##V##. Can you justify those?

You may need to do some serious geometry!
 
  • #5
I think you need 3/8 in the denominator instead of 5/8.
 
  • #6
J Hann said:
I think you need 3/8 in the denominator instead of 5/8.
There's more than that is not right.
 

1. What is a pendulum?

A pendulum is a weight suspended from a pivot so that it can swing freely back and forth under the influence of gravity.

2. What is a rocking hemisphere?

A rocking hemisphere is a half-sphere shaped object that can move back and forth in a rocking motion.

3. How is the length of a pendulum related to a rocking hemisphere?

The length of a pendulum is directly related to the period of a rocking hemisphere. As the length of the pendulum increases, the period of the rocking hemisphere also increases.

4. What is the significance of finding the equivalent length of a pendulum to a rocking hemisphere?

Finding the equivalent length of a pendulum to a rocking hemisphere allows us to understand the relationship between the two objects and make accurate measurements and predictions about their movements.

5. How can the equivalent length of a pendulum to a rocking hemisphere be calculated?

The equivalent length can be calculated by using the equation T=2π√(L/g), where T is the period of the rocking hemisphere, L is the length of the pendulum, and g is the acceleration due to gravity.

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