Question

A ball rolls along a horizontal frictionless surface without slipping toward an incline. The ball moves up the incline a certain distance before reversing direction and moving back down the incline. In which of the following situations does the ball move the greatest distance up the incline before turning around?

A)the incline is frictionless
B)there is sufficient static friction between the ball and the incline to prevent the ball from slipping.
C)there is insufficient static friction to prevent slipping, but there is some kinetic friction.
D) the amount of friction is irrelevant

Answer 1

The correct answer is choice B. but I don't understand why. The answer states that for the first one, only translational kinetic energy converted to gravitational energy(mgh), and in the second, both translational and rotational kinetic energy converted to gravitational energy. Thus mgh in the second choice is bigger than in the first. But in choice b, energy is lost by friction.

Answer 2

The progress so far I made is the following:
Choice A: \[\frac{ 1 }{ 2 }mv ^{2}+\frac{ 1 }{ 2 }I*\alpha ^{2} =mgh _{a}+\frac{ 1 }{ 2 }I*\alpha ^{2} \]
Choice B:\[\frac{ 1 }{ 2 }mv ^{2}+\frac{ 1 }{ 2 }I*\alpha ^{2} = mgh _{b}-friction*distance\]

Answer 3

Sorry the second equation should be:
Choice B: \frac{ 1 }{ 2 }mv ^{2}+\frac{ 1 }{ 2 }I*\alpha ^{2} = mgh _{b}-friction*distance\[\frac{ 1 }{ 2 }mv ^{2}+\frac{ 1 }{ 2 }I*\alpha ^{2}--friction*distance = mgh _{b}\]

Answer 4

It is necessary to show that \[\frac{ 1 }{ 2 }I \alpha^2 >\] friction*distance

Answer 5

Please help

Answer 6

Anyone can help me?

Answer 7

@Vincent-Lyon.Fr @LastDayWork @Mashy

Answer 8

@dan815

Answer 9

Hi! Since we have an rolling object, energy isn't \(\rm taken\) by friction. Friction \(\rm converts\) the ball's translational kinetic energy to rotational. So energy is not lost.
If the ball has that much more KE, it will help.

When I see it, I think about momentum. If you had a ball slip up a hill, it's translational velocity is all it has.

But if it's rolling can help it get up a hill (since it won't just slip), then it angular velocity will help it continue.

I have to go! I hope this helps! Take care?

Answer 10

Helo, but if there exists an acceleration, the friction will take away the energy. Friction does not take away energy only when no net force exist.

Answer 11

And in this problem, gravity pulls the ball down

Answer 12

"A ball rolls along a horizontal frictionless surface without slipping"
that statement itself is wrong .. ball cannot roll without friction :D

Answer 13

but thats irrelevant i guess :-/.

Answer 14

The best way to say it, I think, is that friction allows the transfer of rotational kinetic energy to translational kinetic energy.

Friction will take away energy even if it is the only force in the case of something \(\rm sliding\). But, in the case of a ball, the static friction keeps the part of the ball that touches the ground still. The result is that the ball topples over itself. But its kinetic energy isn't reduced.

In the case of the incline, the rotational kinetic energy is converted, by the friction, to translational kinetic energy. It's weird, but it's okay. Energy is an abstract concept, arrived at mathematically, and we study its application.

But, if you want to think about it in a more tangible sense, you can think about its momentum. When you roll a ball, it does seem to lose momentum. If there is friction, the ball rolls with the ground. But the friction force on the bottom of the ball doesn't slow it down if the ball is perfectly round and the ground perfectly flat.

In the case of the ball rolling without friction, it's rolling is unimportant. When it hits an area with friction, the friction force will cause the ball to change its tangential velocity so that it matches the ground. This is just what friction does - opposes motion. So the ball will, eventually, roll with the ground. Then static friction is in effect.
Now, the translational motion will be slowed up the ramp, and then directed downward, by the gravitational acceleration. Gravitational acceleration doesn't directly affect the rotation, though. It pulls down on \(everything\), which does not create a torque due to the ball's symmetry.
Since there's gravity, though, the ball is pulled to the incline. This gives it a normal force, and so friction.
So we're back the friction force, now.

The ball has rotational momentum. The inertia of the ball implies that this momentum isn't going to change without a force. That force is the friction force, because it is the only unbalanced torque. Gravity, though, is what pulls it down the ramp.

In this case, gravity reminds me of the force you put on a still object. I don't know how much it applies, but if you "exert a force" on an object with much static friction, it doesn't move. In this case, the ball is not slipping. But the twist is that it is still free to roll as directed by gravitation. Gravitation will make the ball slow and go back down the incline, in time.

Since the ball is in contact with the incline, the ball has to stop rolling before it can head back down. So the rotational momentum must be taken away. This happens slowly, and the ball will continue to roll until all of its rotational momentum is gone.

Answer 15

"The ball has rotational momentum. The inertia of the ball implies that this momentum isn't going to change without a force"
u mean without a torque (cause gravity can't put torque)

Answer 16

Haha, good point, @Mashy !
But it can be rolling on a frictionless surface. A helpful comparison is a bowling ball! The floor it lands on is slippery. It can have hardly any effect on the ball's spin. So, a frictionless surface couldn't make it spin.

But if a ball hit the floor rolling, it would continue to roll. It will probably slip still (people make balls rotate backwards or to the side). So the tangential velocity at the edge doesn't match the ground speed.

There is some significance to the problem's wording, however. Since it is rolling, it will have rotation when it meets the incline.

Answer 17

But its mentioned WITHOUT slipping..
when bowling ball rolls , then it is slipping :P

Answer 18

Right, @Mashy , I mean a torque! Sorry. Torque is a better word for that sentence :) My bad!

Answer 19

Haha, "without slipping" on the frictionless surface before the ramp... That does seem unimportant.

Answer 20

That'd be hard to do...

Answer 21

That is IMPOSSIBLE to do :P

Answer 22

I guess I should show what your answer says in equations, like you showed.
I'll use "i" to indicate "initial" and "f" to indicate "final."
\(KE_i+PE_i=KE_f+PE_f\) for the conservation of mechanical energy.
Since KE is both translational and rotational, I'll write them in that order. So I'll use:
\(\large KE_{i,\rm trans}+KE_{i,\rm rot}+PE_i\\\qquad\qquad=KE_{f,\rm trans}+KE_{f,\rm rot}+PE_f\)
For A:
\(\frac 1 2 mv^2+\frac 1 2 I\omega ^2 +0= 0+\frac 1 2 I\omega^2+mgh_A\)
For B:
\(\frac 1 2mv^2 + \frac 1 2 I\omega^2 + 0 =0+0+mgh_B\)
The final energy, you see, is the same. But for B, it is all potential. Since \(m\) and \(g\) are constant, you know that \(h\) is what varies to decide the potential energy. Since the final energy in B is just potential, it's height is greater. None of that energy had to be shared.

Answer 23

Haha, definitely impossible to exact precision, I think. "Frictionless" is idealized, even. But you can get a ball rotating and then send it out at a velocity such that the tangential velocity at the ground nearly matches ground speed.

So it'd look like it'd be rolling with the ground, even if it really wasn't. And it wouldn't "slide" if it was matching the ground speed exactly.

Answer 24

@theEric
HAHA.. that is cheating bro :P..
isn't it much easier to just have a nice friction surface and do it :D.. but i like ur idea very much :D

Answer 25

The answer is B.I understand this. But the point is the friction exists and its direction is to the up-right. This friction causes a torque counterclockwise which slowing the ball down. And Friction must do something in this problem since it is accelerating. Textbook teaches us that if there is a friction you cannot apply energy conservation.

Answer 26

Don't confuse with the rolling without accelerating. In rolling without accelerating, no friction exists because it does not need a torque to keep it rolling.

Answer 27

A torque is only needed while accelerating. And the torque is caused by friction. And according to Newton's law in this problem. Fnet = F(component of gravity on the axis parallel to the incline) - friction.

Answer 28

so a = (F(component of gravity on the axis parallel to the incline) - F(friction))/mass

Answer 29

And if the friction is toward up right, the work done by friction became positive. than using the energy equation KEi+W(nonconservative)=mgh
mgh is even bigger

Answer 30

This is actually really confusing. Hard to conceptualize it.

Answer 31

The static friction doesn't just take away energy like kinetic friction does. So, the static friction is a way to convert the kinetic energy from rotational to translational, and translational to rotational.
Conservation of energy always applies, but conservation of mechanical energy doesn't always. You can still apply conservation of energy by introducing a variable with the simple meaning of "energy that's not mechanical anymore." Weird, eh?

I don't understand:
"A torque is only needed while accelerating. And the torque is caused by friction. And according to Newton's law in this problem. Fnet = F(component of gravity on the axis parallel to the incline) - friction."

Gravity acts as a field force, affecting all of the ball. And it is directed parallel to the ramp, as you say, just because normal force counters the other component of gravity.

Answer 32

Because the ball is rolling up the incline and the friction is supposed to oppose the direction of its motion. which will be toward down left. But it is actually up right. This is very confusing. Would you mind help me explain this please.

Answer 33

Physics is very heavy, conceptually! Hehe, I really agree with you there.

Answer 34

What do you think about the direction of the friction in this problem

Answer 35

Oh, I see what you mean. Friction actually does not show up in the energy conservation equation right?

Answer 36

I think the static friction is always directed up the ramp, to oppose the gravitational force that tries to make it slip.

Answer 37

Right, @kzhou ! Friction doesn't show up in the energy equation. You just know it's there because it allows the rotational kinetic energy to be converted to be converted to translational kinetic energy.

Answer 38

And for the reason why static friction is always directed up the ramp is that I mean you know the ball's angular velocity is decreasing. So the angular acceleration have to be counterclockwise. Thus the friction is directed up the ramp.

Answer 39

Do you think this reasoning is good?

Answer 40

Thank you so much. I am so impressed by your help.

Answer 41

I'll explain that thing you just asked about.
Friction opposes motion at the point of contact! This is static friction because there is no motion.
But, just like lightly pushing a brick so it's static friction opposes its force (that would cause motion), the friction opposes the force of gravity that would cause motion.

If you set a ball on a frictionless hill, it would slide down. Friction, though, wouldn't let it just slide, because that's relative motion at the point of contact.

Also, I like your reasoning! That seems like a very good way to look at it, as far as I know!

Glad I can help! It's an interesting problem that I haven't seen before! But that's the cool thing about physics - you can apply all the well-known principles to new problems, and solve them :)

Answer 42

I'm very glad you put a lot of thought into this as well! I like to see that. It's the only way anyone can learn :)

Answer 43

I've never seen your reasoning applied before, but it seems perfectly reasonable to me :)

Answer 44

Very, very reasonable..

Answer 45

Pretty interesting question, and here is a view:
Firstly, we note that: Total Kinetic Energy [KE] of the Ball = Kinetic Energy due to Translation [TKE] + Kinetic Energy due to Rotation [RKE] Thus: KE=TKE+RKE.

Secondly: Height the ball rises to is proportional to the amount of kinetic energy LOST by the ball at its highest point on the incline since this lost KE = the gain in Potential Energy = Gain in Height [assuming constant mass and gravitational field strength and no frictional losses].

In case [B], when there is enough static friction to make the ball roll up the incline without slipping, the amount of Kinetic Energy Lost at the highest point = TKE+RKE. This is since the ball will come to a complete stop at the highest point.

In case [A], with a frictionless incline, the amount of Kinetic Energy Lost at the highest point = TKE.
Since the incline is frictionless, there is no Torque acting on the ball and hence RKE is unchanged and it will be spinning even at the highest point.

In case [C], since their is some kinetic friction, and some slipping, the amount Kinetic Energy Lost at the highest point = TKE+[some, but not all, RKE]-[Loss due to Kinetic Friction]


So the amount of energy lost at the highest point [which converts to potential energy] is highest in case [B]. Thus the ball will go highest under the situation of Case [B].

Case [D] of course rules itself out since we see that the Cases A, B, C give different heights, and hence friction is relevant.

Answer 46

There are some finer points to consider, which I did not raise in my previous post for the sake of lucidity:
1. The question says that: "A ball rolls along a horizontal frictionless surface without slipping toward an incline". If the horizontal is frictionless, the ball cannot roll on it !!! So we have to interpret it as that the horizontal has zero kinetic friction.
2. When the ball hits the incline, its velocity changes INSTANTLY in Direction. The Impact of the incline on the ball causes this change of direction. It has to be assumed that this Impact does not do any WORK on the ball. This is largely true as long as the ball does not Bounce on impact.

Answer 47

"I think the static friction is always directed up the ramp" <-- True

Reason -
Gravitational force cannot affect angular velocity, but it will certainly reduce the translational velocity. Hence, at the contact point, tendency of slipping would be towards rω


Hence, to prevent slipping, static friction would be directed up the ramp.