Classical mechanics: ball rolling in a hollow sphere

In summary, a uniform ball is rolling without slipping inside a fixed hollow sphere. The problem is to find the energy conservation equation and the period of small oscillations of the ball about its equilibrium position. The solution involves variables such as the ball's radius, mass, angle of position, moment of inertia, rotational velocity, kinetic energy, potential energy, total energy, and acceleration due to gravity. After correcting a mistake in the equation for kinetic energy, the correct solution is obtained.
  • #1
blalien
32
0
[SOLVED] Classical mechanics: ball rolling in a hollow sphere

Homework Statement


This problem is from Gregory:

A uniform ball of radius a and centre G can roll without slipping on the inside surface of a fixed hollow sphere of (inner) radius b and centre O. The ball undergoes planar motion in a vertical plane through O. Find the energy conservation equation for the ball in terms of the variable [tex]\theta[/tex], the angle between the line OG and the downward vertical. Deduce the period of small oscillations of the ball about the equilibrium position.

So in summary, we have:
a: radius of ball
m: mass of ball
[tex]\theta[/tex]: angle of the ball's position, relative to the vertical line connecting the center and bottom of the hollow sphere
I: moment of inertia of ball
[tex]\omega[/tex]: rotational velocity of ball
T: kinetic energy of ball
V: potential energy of ball (V=0 at height [tex]\theta[/tex]=[tex]\pi[/tex]/2, the center of the sphere)
E: total energy of ball
g: acceleration due to gravity

Homework Equations


I = 2/5ma^2

The Attempt at a Solution


First of all, I'm assuming that [tex]\omega[/tex]=[tex]\theta[/tex]'. It sounds intuitive, but I could be wrong there.

I'm given, as a solution, that the period of small oscillation (that is, sin([tex]\theta[/tex])=[tex]\theta[/tex]) is 2[tex]\pi[/tex](7(b-a)/5g)^(1/2), which I'm not getting in my results. I have a very strong hunch that my mistake comes from bad energy equations. So, would you mind taking a look of these?

T = 1/2mv^2 + 1/2I[tex]\omega[/tex]^2
v = [tex]\omega[/tex]*(b-a)
So T = 1/2m([tex]\omega[/tex]*(b-a))^2 + 1/2(2/5ma^2)[tex]\omega[/tex]^2
T = m[tex]\omega[/tex]^2/10(7a^2-10ab+5b^2)
V = -(b-a)mgcos([tex]\theta[/tex])

So E = T + V = that stuff
Am I correct here?
 
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  • #2
blalien said:

The Attempt at a Solution


First of all, I'm assuming that [tex]\omega[/tex]=[tex]\theta[/tex]'. It sounds intuitive, but I could be wrong there.

That is wrong. Draw a simple diagram to figure it out.
 
  • #3
Hah, it's always the little mistakes in the beginning that steal away an hour of my life.

That fixed everything. Thank you so much for catching that.
 

Related to Classical mechanics: ball rolling in a hollow sphere

1. What is classical mechanics?

Classical mechanics is a branch of physics that describes the motion of macroscopic objects, such as balls, planets, and cars, using Newton's laws of motion and the principles of energy and momentum.

2. How does a ball roll in a hollow sphere?

A ball rolling in a hollow sphere follows the laws of classical mechanics, specifically the principle of conservation of angular momentum. As the ball rolls, it creates a torque that causes it to rotate around the center of the sphere. This rotation causes the ball to follow a circular path inside the sphere.

3. What is the role of gravity in a ball rolling in a hollow sphere?

Gravity plays a crucial role in the motion of a ball rolling in a hollow sphere. It provides the force that acts on the ball, causing it to accelerate and start rolling. Additionally, gravity also affects the ball's path and speed as it moves inside the sphere.

4. How does the mass of the ball and the size of the sphere affect the motion?

The mass of the ball and the size of the sphere both impact the motion of the ball rolling inside. A heavier ball will have more momentum and will be harder to stop, while a larger sphere will provide a longer path for the ball to follow. These factors can affect the speed and trajectory of the ball as it rolls.

5. What are the practical applications of studying ball rolling in a hollow sphere in classical mechanics?

Studying ball rolling in a hollow sphere can have several practical applications, such as understanding the motion of objects in a curved path, designing better sports equipment like bowling balls or golf balls, and predicting the behavior of rolling objects in various scenarios, such as on a sloped surface or in a rotating container.

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