Oscillations of an Exercise Ball

In summary, a physics student is seeking help on finding the equation of motion for an exercise ball with a non-constant mass distribution. The ball oscillates due to the force of gravity acting as the restoring force and the rotational inertia acting as the inertial force. The equation of motion is given by I(d^2/dt^2)(theta)+mgsin(theta)=0, and for easier solving, a small angle approximation and the parallel axis theorem can be used.
  • #1
TacoTycoon
1
0
Hey all,

First I wanted to say hello, as I am new to this forum. I'm a third-year physics student at the University of Toronto.

Anyhow, the question goes like this:

We have an exercise ball (one of those large, inflatable ones) with a diameter of 0.7m. If you've ever seen one of these, you'll know that the material isn't constant throughout the ball, and at the point where you blow the ball up the mass is heavier. Thus, the ball will always try to roll to the equilibrium position where that point is below the absolute centre.

The question is: find the equation of motion for the oscillations of the ball if you displace it slightly from it's equilibrium. The ball will rock back and forth, as shown in this video:

http://www.physics.utoronto.ca/~phy255/OscWave/dloads/Ballroll.m4v

What you're doing by having this 'extra mass' at a point on the ball is shifting the centre of mass away from the centre of the ball.

I believe that it's the force of gravity acting on the centre of mass which is the 'restoring' force of the oscillation, and the rotational momentum of the ball which is the 'inertial' counter-force. I haven't been able, however, to model this mathematically. Does anyone have any ideas? Thanks!

--
Edit: Just read the rules of the homework forum, and saw that I had to post my work:

For the restoring force:

F=ma, this will be gravity. I pictured the centre of mass similarly to a pendulum on a string, and if we take theta to be the angle the ball has rotated from equilibrium, then:

F=-mgsin(theta)

For the inertial force:

F=ma
F=I(d^2/dt^2)(theta) which is the term describing torqe (the d thing means second deriv)

so the overall equation would be: I(d^2/dt^2)(theta)+mgsin(theta)=0

I know this is non-linear, but the question never asks us to solve the equation of motion. Sorry for my notation, I have to get used to writing math online.
 
Last edited:
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  • #2


Hello and welcome to the forum! It's great to have a fellow physicist here.

In terms of your question, you are definitely on the right track. The equation you have written is indeed the correct equation of motion for the oscillations of the ball. As you mentioned, the gravitational force is acting as the restoring force, and the rotational inertia is acting as the inertial force.

To make this equation easier to solve, you can make a small angle approximation, assuming that the amplitude of the oscillation is small. This will simplify the equation and make it easier to solve. Also, you can use the parallel axis theorem to calculate the moment of inertia of the ball with the extra mass at a point away from the center.

I hope this helps and good luck with your studies!
 
  • #3


Hello and welcome to the forum! Your analysis of the oscillations of the exercise ball is correct. The equation of motion for the oscillations can be written as:

I(d^2/dt^2)(theta)+mgsin(theta)=0

where I is the moment of inertia of the ball, m is the mass of the ball, g is the acceleration due to gravity, and theta is the angle of rotation from equilibrium.

This equation is indeed non-linear, and therefore cannot be solved analytically. However, it can be solved numerically using computational methods. You can also make simplifying assumptions to approximate the equation and make it solvable, but this may not accurately represent the real behavior of the ball.

In terms of your notation, it is important to be consistent and clear when writing mathematical equations. You can use symbols such as theta for angles, but make sure to define them clearly beforehand. Also, using units for each variable can make the equation more meaningful and easier to understand.

Overall, you have a good understanding of the physics behind the oscillations of the exercise ball. Keep exploring and learning!
 

1. What causes an exercise ball to oscillate?

The oscillations of an exercise ball are caused by the forces of gravity and elasticity. When the ball is compressed or stretched, it stores potential energy which is then released and converted into kinetic energy, causing the ball to oscillate.

2. How does the size of the exercise ball affect its oscillations?

The size of the exercise ball affects its oscillations by changing the amount of potential energy it can store. A larger ball will have a greater surface area and thus more potential energy, resulting in larger and slower oscillations. A smaller ball will have less potential energy and therefore smaller and faster oscillations.

3. Can the material of the exercise ball affect its oscillations?

Yes, the material of the exercise ball can affect its oscillations. A more elastic material will store more potential energy and result in larger oscillations, while a less elastic material will have smaller oscillations. Additionally, the weight of the material can also impact the oscillations.

4. How does air resistance affect the oscillations of an exercise ball?

Air resistance can dampen the oscillations of an exercise ball. As the ball moves through the air, it experiences friction which converts some of its kinetic energy into heat energy, causing the oscillations to decrease over time. This is why exercise balls will eventually come to a stop if left untouched.

5. How can the oscillations of an exercise ball be used in a workout?

The oscillations of an exercise ball can be used in workouts to improve balance, core strength, and coordination. Exercises such as sitting or standing on the ball and performing various movements can challenge the body to maintain stability and control the oscillations. This can be beneficial for athletes and individuals looking to improve their overall fitness.

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