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

[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

- radius of ball=$R$

- 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?

 

[BvU]

Hello Joshua,
My physics intuition tells me that

but they won't be useful

may be off the mark. How about you ?

 

[Joshua Benabou]

@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.

 

[BvU]

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 !

 

[Joshua Benabou]

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

 

[BvU]

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

 

[Joshua Benabou]

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.

 

[haruspex]

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.

 

[Joshua Benabou]

@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.

 

[haruspex]

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.

 

[Joshua Benabou]

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

 

[haruspex]

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

No, you already have N_d. The equation involves only variables you already have in your equations.
To make progress, please post your attempt at that equation.