# Dynamics of rigid bodies

## Homework Statement

Three objects of equal mass, A, B and C, are released from rest at the same instant from the same height on identical ramps. Objects A and B are both blocks, and they slide down their respective ramps without rotating. Object C rolls down the ramp without slipping. Its moment of inertia is unknown.

Objects A, B and C are made of different materials, thus the coefficients of friction between the objects and their corresponding ramps are not necessarily the same.

Object A reaches the bottom of its ramp first, followed by objects B and C, which reach the bottom at the same instant.

Rank the center of mass accelerations, the net forces and the frictional forces exerted on objects A-C according to magnitude, from largest to smallest.

conceptual

## The Attempt at a Solution

Because B and C arrive at the bottom at the same time, then its center of mass accelerations are the same, and ranking the center of mass accelerations from largest to smallest, it would be A>B=C, right?

The net forces would be the same for B and C because the accelerations are the same, and greater for A, as the acceleration is greater.

But what about frictional force? Wouldn't we need to know the coefficients of friction for the ramps to know this?

haruspex
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But what about frictional force? Wouldn't we need to know the coefficients of friction for the ramps to know this?
You can rank the frictional forces for A and B, right? (Just draw the FBDs.)
Let's compare B and C. What do we know about the frictional force in C? What equations can you write?

You can rank the frictional forces for A and B, right? (Just draw the FBDs.)
Let's compare B and C. What do we know about the frictional force in C? What equations can you write?

Hey, sorry to bother you, and I know this was a long time ago, but I am working on this problem too. Regarding your question-- the frictional force would be greater for object B than for object A, as it reaches the bottom first, right? And because object C is rolling without slipping, it would have 0 kinetic friction, correct?

Sorry about bringing back up a 4-year-old question, haha! Just wanted to see if my reasoning is correct. Any help is appreciated!

haruspex
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the frictional force would be greater for object B than for object A
Yes.
because object C is rolling without slipping, it would have 0 kinetic friction, correct?
Right... can you say any more?

• Kevodaboss
Yes.

Right... can you say any more?

What do you mean? Are you talking about static friction, too?

haruspex
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What do you mean? Are you talking about static friction, too?
Yes.

• Kevodaboss
So the static friction would be nonzero, but after the ball starts rolling, it has zero kinetic friction, right? So would the static friction of object C equal the kinetic friction of object B, since they reach the bottom at the same instant? Or does the static friction have no effect on this?

haruspex
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would the static friction of object C equal the kinetic friction of object B,
A good thought... can you justify that with some equations?

• Kevodaboss
Force of static friction on C is μsN, force of kinetic friction on B is μkN. Normal forces are the same because forces due to gravity are the same. However we don't know the coefficients of friction, so I don't see how I can show that these forces of friction are the same...

haruspex
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Force of static friction on C is μsN, force of kinetic friction on B is μkN. Normal forces are the same because forces due to gravity are the same. However we don't know the coefficients of friction, so I don't see how I can show that these forces of friction are the same...
What equations can you write relating these to the linear accelerations? What do you know about those accelerations?

• Kevodaboss
I think Fnet=ma can be applied here... Masses are the same, and the center-of-mass accelerations of B and C are the same as B and C reach the bottom at the same time. It follows that the net force on both are the same, and since normal force and force of gravity are the same on both objects, frictional force must be the same as well. Correct?

haruspex