# How Does a Dog Running on a Ball Affect Its Motion on an Inclined Plane?

• Joshua Benabou
In summary, the conversation discusses the motion of a ball on an inclined plane with a small dog sitting on top. The variables and equations related to the ball's motion are provided, but the role of the dog's acceleration and additional equations are questioned. The moderator suggests using all available knowledge, including the fact that acceleration is a vector, to solve for the friction force applied by the dog.
Joshua Benabou
<Moderator's note: Moved from a technical forum and thus no template.>

We place a ball on an an inclined plane (angle ##\theta##). At the top of the ball is a small dog who always stays at the top over the course of the ball's movement.

What is the motion of the ball?
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I apologize for the lack of diagram.

Let's note:

- mass of ball=##M##

- mass of dog=##m##

- point ##Q## =center of ball, point ##P##=position of dog at the top of the ball

- moment of inertia of ball = ##I=kMR^2##

- ##f_d## =static friction force applied horizontally at ##P## to the ball by the dog

- ##f_b##=static friction force applied parallel to the inclined plane to the ball by the plane

- ##N_d## the normal force between the dog and the ball

- ##N_b## the normal force between the ball and the inclined plane

- ##\alpha##=magntitude of angular acceleration of the ball

- ##a##=magntitude of acceleration of center of mass of the ball

We are looking for ##\alpha## and ##a##.

We have the following equations:

1. ##I\alpha=(f_d+f_b)R## (torque applied to the ball)

2. ##Ma=f_d\cos\theta+N_d\sin\theta-f_b+Mg\sin\theta## (projection of forces applied to ball in the direction parallel to the incline)

3. ##mg\sin\theta-f_d\cos\theta-N_d\sin\theta=ma## (since the vector ##QP## is constant, the dog has the same linear acceleration as the ball; this equation comes from projecting the forces applied to the dog in the direction parallel to the incline)

4. Condition for rolling without slipping: ##a=R\alpha##

Note that projecting forces along the direction perpindicular to the incline will give use 2 more equation (one for the forces applied to the ball, one for the forces applied to the dog), but they won't be useful.

The above equations are enough to determine ##a## in terms of the force ##f_d##, but that's the best we can do.

Is it thus necessary to make a choice about how to model the dog? What conditions am I missing?

Last edited by a moderator:
Hello Joshua,
My physics intuition tells me that
Joshua Benabou said:
but they won't be useful
may be off the mark. How about you ?

@BvU : I say this because I wrote them out, and they involve other variables like the normal force ##N_b##. The thing is we don't know anything about these normal forces. Also, do note that we aren't given the friction coefficients so we can't write for example ##f_b=\mu_sN_b##. This is why I say only the projections along the direction of the incline will give us useful relations.

Friction coefficients are unnecessary. You may assume they are big enough.
Do you take the center of the ball as the axis of rotation ?

Oh, and: post in homework and use the template !

@BvU: I still don't see how to get more equations to solve for the friction force $f_d$.

Path of little doggie is what ? So the resultant of the forces on doggie has known direction. What forces work on him/her ?

I already reponded to this question in the OP.the dog has the same acceleration as the ball, and the forces acting on him are f_d, N_d and mg, which gives equation 3. Can you please tell me explicitly what I am missing because as I said the 4 equations I found are underdetermined.

Joshua Benabou said:
I already reponded to this question in the OP.the dog has the same acceleration as the ball, and the forces acting on him are f_d, N_d and mg, which gives equation 3. Can you please tell me explicitly what I am missing because as I said the 4 equations I found are underdetermined.
BvU is hinting that you have not used all your knowledge regarding the acceleration of the dog. Acceleration is a vector.

BvU
@haruspex: See equation $3$. I already used that the acceleration is a vector. As I said, projecting in the direction perpindicular to the incline doesn't give useful equations.

Joshua Benabou said:
projecting in the direction perpindicular to the incline doesn't give useful equations.
Are you sure? What equation do you get? It seems to me that none of the equations you have imply the dog has no acceleration perpendicular to the plane. It follows that such an equation must be extra info.

@haruspex: no, as introducing that equation introduxes new variables like the normal forces!

Joshua Benabou said:
@haruspex: no, as introducing that equation introduxes new variables like the normal forces!

## What is the purpose of a dog running on a rotating ball?

Dogs running on a rotating ball is a form of exercise and mental stimulation. It allows the dog to use their natural instincts and energy in a controlled and safe way, promoting overall physical and mental well-being.

## Is it safe for a dog to run on a rotating ball?

As long as it is done correctly and under supervision, running on a rotating ball can be safe for dogs. It is important to start slow and gradually increase the intensity and duration of the exercise to prevent injuries. It is also recommended to use a ball with a non-slip surface and to always have a spotter present.

## What are the benefits of a dog running on a rotating ball?

Running on a rotating ball has both physical and mental benefits for dogs. It can improve their cardiovascular health, muscle strength, and coordination. It also provides mental stimulation and can help alleviate boredom and destructive behaviors.

## What breeds of dogs are suitable for running on a rotating ball?

Most breeds of dogs can participate in running on a rotating ball, but it is important to consider the size and physical abilities of the individual dog. Smaller breeds may have difficulty balancing on a larger ball, while larger breeds may put too much weight and strain on their joints. It is best to consult with a veterinarian before starting this type of exercise with your dog.

## How often should a dog run on a rotating ball?

The frequency of running on a rotating ball will depend on the individual dog and their physical abilities. It is recommended to start with short sessions and gradually increase the frequency as the dog becomes more comfortable and skilled. It is important to watch for signs of fatigue or discomfort and adjust accordingly.

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