What is the effect of conveyor belt inertia on the movement of a ball?

[marked1000]

This is a possible problem that I can have on my quiz.

There is ball on a conveyor belt. The belt is moving to the right, and a circular motor is running the belt. The belt starts of straight then starts to bend where the circular motor is. The circular motor is just at where the bend occurs and is of course under the belt. The same ball after passing the motor is now on the belt at an angle.

Now we have been told there will be six equations.

Two force equations for when the ball was sitting normally on the belt as the belt motored on. Fx and Fy

Two force equations after it passes the belt. Fx and Fy

Two equations for motor : Iα and a=rα

 

[K^2]

You might need to draw a picture. I can't understand from explanation alone what this setup looks like.

 

[rcgldr]

So the motor is providing a torque Iα and the belt is initially accelerating at rα with the ball accelerating at some fraction of the belts rate depending on it's angular inertia while on the level surface. The force on the ball equals the mass of the ball times it's linear rate of acceleration. Once the ball is on the inclined (or declined) portion of the belt, gravity also exerts a force on the ball.

The unknowns are the angular inertia of the motor (since it's torque was specified as Iα, but I is unknown), the mass of the ball, and the mass of the belt (I'm not sure how to deal with the angular inertia component for the belt). You could probably assume the ball is a sphere of uniform density to determine it's angular inertia.

If Fx and Fy are the forces applied by the belt to the ball, then for the horizontal (level belt) case, you only need to know the rate of acceleration of the belt, and the angular inertia and mass of the ball. What isn't known is how the motor responds once the ball is on a decline or incline, since either the torque and/or the rate of acceleration of the belt will change.