EK1001 Physics #137

[MCAT guy]

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Ball A and B are the same size and of uniform density. Ball A is twice as massive as Ball B. Both balls are rolled down the same incline. Which of the following is true?

A. Ball A will accelerate at a greater rate because it has greater rotational inertia
B. Ball A will accelerate more slowly because it has greater rotational inertia
C. Both balls will accelerate down the plane at the same rate
D. Both balls will roll down the plane at a constant velocity

I thought B was a better answer, but it was wrong. Anyhow the explaination:

Although the more massive ball has twice the inertia, the frictional force is twice as great.

Hmmm. It still seems like Ball A would have a larger downward force and therefore more acceleration regardless of frictional force stealing away some of this downward force. I mean, lets say the coefficient of kinetic friction was 0.000000001, basically a nonfactor, then the more massive ball will accelerate faster correct? This seems like a horrible question or maybe my reasoning is wrong. Doesn't it depend on the coefficient of kinetic friction? I don't like their explaination or I'm confused, one or the other.

Can someone help here?

 

[CardiacArrest]

Anytime the MCAT brings up something about an inclined plane and an accelerating object, remember that mass is ALWAYS negligible. Unless there is an external force acting on the object (frictional forces, air resistance), mass has no effect on the acceleration of an object.

F=ma=mgsin(theta)
a=gsin(theta)

Thus, the acceleration rate will be the same for both balls.
Not sure how rotational inertia plays in though. I'm guessing since inertia is dependent on mass, it shouldn't really play a role, but I might be wrong.

 

[UCB05]

"same size and of uniform density" in the question is meant to tell you that their moments of inertia are the same. If one of the balls were, say, hollow, then it would roll down slower.

 

[unhappytnt]

Does the question neglect air resistance? If there is no air resistance, they will accelerate at the same speed.

 

[Phantastic]

Hmmm. It still seems like Ball A would have a larger downward force and therefore more acceleration regardless of frictional force stealing away some of this downward force. I mean, lets say the coefficient of kinetic friction was 0.000000001, basically a nonfactor, then the more massive ball will accelerate faster correct? This seems like a horrible question or maybe my reasoning is wrong. Doesn't it depend on the coefficient of kinetic friction? I don't like their explaination or I'm confused, one or the other.

Can someone help here?

You're dealing with a sphere here. So its static friction, as its not slipping correct?

Would it matter if this were a block? My final acceleration down the slope as I've solved it is:

Assume down plane of incline is (+)-x-coordinate, perpendicular to plane above is (+)-y-coordinate.

F(net) = mgsin(theta) - mgcos(theta)µ(s)

Therfore, according to Newton's 2nd law (F=ma):

a(net) = g(sin(theta)-cos(theta)µ(s))

If that math is correct (assuming no air resistance), then you can clearly see how the mass never comes into play with these types of problems.

Anytime the MCAT brings up something about an inclined plane and an accelerating object, remember that mass is ALWAYS negligible. Unless there is an external force acting on the object (frictional forces, air resistance), mass has no effect on the acceleration of an object.

Indeed, and I believe we can include friction as well, as this problem aptly informs.

"same size and of uniform density" in the question is meant to tell you that their moments of inertia are the same. If one of the balls were, say, hollow, then it would roll down slower.

Exactly. I think we've extracted every bit of basic science information in this problem. Feel that? That's confidence.