Calculating the Period Time of a Ball Inside a Ball

[asi123]

Homework Statement

As you can see, I have a ball inside a ball.
One with a density p1 and the other p2.
the ball stand on a surface with friction so it can move without sliding.
they move the ball from its rest point and release it.
I need to figure out the period time, any idea?

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

As you can see, I have a ball inside a ball.
One with a density p1 and the other p2.
the ball stand on a surface with friction so it can move without sliding.
they move the ball from its rest point and release it.
I need to figure out the period time, any idea?

Hi asi123!

(is there zero friction between the balls?)

The only way to approach this is to work out the equations of its motion first.

Hint: lots of things are being conserved.

 

[baba89]

To calculate the period time of the ball inside a ball, we can use the equation T = 2π√(m/k), where T is the period time, m is the mass of the ball, and k is the spring constant of the surface the ball is resting on. In this case, we can consider the ball inside the ball as a simple harmonic oscillator, with the spring constant being a combination of the densities of the two balls and the force of gravity acting on them.

We can also consider the rotational motion of the ball inside the ball, using the equation T = 2π√(I/mgd), where I is the moment of inertia of the inner ball, mg is the force of gravity acting on it, and d is the distance between the center of mass of the inner ball and the point of rotation.

Alternatively, we can use the conservation of energy principle to calculate the period time. The total energy of the system (ball inside a ball) is equal to the sum of its kinetic and potential energies. By setting the initial potential energy to zero and equating it to the final kinetic energy at any point in the motion, we can solve for the period time.

It is also important to consider any external factors that may affect the motion of the ball, such as air resistance or the elasticity of the materials used. By taking these into account, we can obtain a more accurate calculation for the period time of the ball inside a ball system.