Hoop Rolling Up Ramp: Confirm if Altitude Higher Than in Part A?

[CornDog]

Homework Statement

A large hoop rolls without slipping across a horizontal surface. The hoop has a constant translational speed of 9.2 m/s, a mass m of 16 kg, and a radius r of 1.3 m. The moment of inertia of the hoop about its center of mass is 1mr^2. The hoop approaches a 22° incline of height 4 meters and rolls up the incline without slipping.

All parts of the problem up to this point are shown below under "Relevant Equations"

f) You lift it beyond the maximum angle, so when the hoop hits the ramp, it climbs but it does slip. Does the hoop go to a higher altitude than it did in part a?

Homework Equations

a) calculate the total kinetic energy of the hoop as it rolls along the horizontal surface.
Found to be 1354.24 J
b) i) Calculate the magnitude of the velocity of the hoop just as it leaves the top of the incline.
Found to be 6.74 m/s
ii. Specify the direction of the velocity of the hoop just as it leaves the top of the incline.
Found to be 22 degrees
c) Neglecting air resistance, calculate the horizontal distance from the point where the hoop leaves the incline to the point where the sphere strikes the level surface.
Found to be 7.48 m
d) Calculate the force of friction (while its on the ramp) and its direction (up the ramp being positive, down the ramp being negative)
Found to be 29.38 N
e) You increase the angle of the ramp to see if the hoop will go farther. What is the MAXIMUM angle of the ramp (above horizontal) you can increase it to without the hoop slipping, if the ramp has coefficient of static friction = 0.42.
Found to be 40.03 degrees

The Attempt at a Solution

I believe the answer would be no, it does not reach a higher altitude because once you exceed the angle found in part (e), the static friction is overcome by the rotational velocity (omega) of the hoop, which allows the hoop to spin faster. This would therefor increase the rotational kinetic energy of the hoop, allowing less for translational kinetic energy and thus less translational velocity so the hoop would not get as high.

I'm just looking for a confirmation of whether or not I am correct, and an explanation if I am incorrect as this is a crucial question on my homework assignment.

Thanks,
CornDog

EDIT - This assignment is due about 1 day from right now, so quick responses would be appreciated.

 

[berkeman]

Thread moved from Advanced Physics to Introductory Physics.

CornDog -- you sure about that hoop moment of inertia? That's what you were given? Maybe it's a typo?


EDIT -- No, it's the right moment of inertia for a hoop.

 

[OlderDan]

Homework Statement

A large hoop rolls without slipping across a horizontal surface. The hoop has a constant translational speed of 9.2 m/s, a mass m of 16 kg, and a radius r of 1.3 m. The moment of inertia of the hoop about its center of mass is 1mr^2. The hoop approaches a 22° incline of height 4 meters and rolls up the incline without slipping.

All parts of the problem up to this point are shown below under "Relevant Equations"

f) You lift it beyond the maximum angle, so when the hoop hits the ramp, it climbs but it does slip. Does the hoop go to a higher altitude than it did in part a?

Homework Equations

a) calculate the total kinetic energy of the hoop as it rolls along the horizontal surface.
Found to be 1354.24 J
b) i) Calculate the magnitude of the velocity of the hoop just as it leaves the top of the incline.
Found to be 6.74 m/s
ii. Specify the direction of the velocity of the hoop just as it leaves the top of the incline.
Found to be 22 degrees
c) Neglecting air resistance, calculate the horizontal distance from the point where the hoop leaves the incline to the point where the sphere strikes the level surface.
Found to be 7.48 m
d) Calculate the force of friction (while its on the ramp) and its direction (up the ramp being positive, down the ramp being negative)
Found to be 29.38 N
e) You increase the angle of the ramp to see if the hoop will go farther. What is the MAXIMUM angle of the ramp (above horizontal) you can increase it to without the hoop slipping, if the ramp has coefficient of static friction = 0.42.
Found to be 40.03 degrees

The Attempt at a Solution

I believe the answer would be no, it does not reach a higher altitude because once you exceed the angle found in part (e), the static friction is overcome by the rotational velocity (omega) of the hoop, which allows the hoop to spin faster. This would therefor increase the rotational kinetic energy of the hoop, allowing less for translational kinetic energy and thus less translational velocity so the hoop would not get as high.

I'm just looking for a confirmation of whether or not I am correct, and an explanation if I am incorrect as this is a crucial question on my homework assignment.

Thanks,
CornDog

EDIT - This assignment is due about 1 day from right now, so quick responses would be appreciated.

I assume your part f) is actually asking you to compare the altitude to that attained in part e), since part a) has nothing to do with altitude, but the problem is poorly stated since it does not tell you what to assume about friction. The answer depends on what you assume about the coefficient of kinetic friction. If you assume that coefficient of kinetic friction is the same as that of static friction, then lifting the ramp higher will in fact increase the maximum altitude. You can prove that by finding the maximum atltitude in part e) and then raising the ramp to that height (it is long enough to do that) and then calculate the maximum height that could be achieved with the same coefficient of friction for a ramp of sufficient length for the hoop to keep climbing it until it stops. If that height is greater than the ramp height, you know the hoop will make it to the end of the ramp with some vertical velocity, and it will go higher.

A much simpler calculation is if you assume no kinetic friction at all (not realistic, but it makes the problem much easier). In that case, only the translational part of the kinetic energy can be converted to potential energy, and the hoop cannot even make it to the height attained with the ramp at 22 degrees. If the question really does say to compare to part a), this is probably the expected assumption.

Your answers to all the other parts look OK

 

[CornDog]

I think what the question is asking is if the hoop would get higher than whatever height it would attain in its trajectory in part (c). However, I think this is supposed to be a purely conceptual question, as he's only looking for a Yes or No answer.

Your part about now knowing the kinetic friction struck me as weird also, but I just assumed I was doing something wrong, which was why I posted. Good to know I'm not alone.

I know for a fact all the other answers are correct, as it's an online assignment and I've already entered them. Part (f) won't tell us if we're right until after the due date tomorrow morning, which is why I wanted to make sure I was correct.

Thanks,
CornDog

 

[AlephZero]

Re the friction, the problem says the hoop rolls without slipping, so I would take that to mean the friction force was big enough to prevent slipping.

One the thing that is not obvious to me is what it means by "the hoop approaches a 22° incline..." If the incline is literally a plane, there is an impact when the hoop hits the start of the incline and the horizontal and inclined planes are both tangent to the circular hoop. If the impact was elastic it seems possible the hoop could bounce, but there's nothing about the elasticity (or not) in the question.

 

[CornDog]

That's for the beginning of the question. The part that I need help with is:

f) You lift it beyond the maximum angle, so when the hoop hits the ramp, it climbs but it does slip. Does the hoop go to a higher altitude than it did in part a?

Note that is says it does slip now. Also, our class hasn't done anything with bounce in any of our problems like this, so I'm left to assume that it doesn't bounce on contact.

Thanks,
CornDog