# Homework Help: Energy, Force, and Inertia Problem. Ball down ramp to loop to loop

1. Nov 10, 2012

### rschaefer2

1. The problem statement, all variables and given/known data
A solid sphere of mass m and radius r rolls without slipping along the track shown below. It starts from rest with the lowest point of the sphere at height h above the bottom of the loop of radius R, much larger than r. (Consider up and to the right to be the positive directions for y and x respectively)

What are the force components on the sphere at the point P if h = 3R? (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)

2. Relevant equations
Isphere=(2/5)mr2
K=.5mv2
K=.5Iω2
PE=mgh
ac=v2/R

3. The attempt at a solution
Fx
First, solving for v2
mg(3R)=.5mv2+(.5)(2/5)(mr2)(v2/r2)
mg2R=.7mv2+mgR
v2=20gR/7

Centripetal acceleration is the acceleration in the x direction
a=v2/R=20gR/7R
Fx=ma=-20mg/7 <=== Correct!

Fy
Force in y direction is only the force from gravity
F=ma
Fy=-mg <=== Wrong!

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Last edited: Nov 10, 2012
2. Nov 10, 2012

### TSny

Hello, and welcome!

Note that you are measuring heights from the "floor". Does point P have a height?

EDIT:
OK, I saw that you left out the potential energy at P in the first equation, but it looks like you took care of it in the second equation.

3. Nov 10, 2012

### TSny

I don't see anything wrong with your answer for Fy. Wait! Is the force of gravity the only force with a y component acting on the ball at P?

Last edited: Nov 11, 2012
4. Nov 10, 2012

### rschaefer2

Well, i guess there is friction to? But there wasn't any coefficients given for kinetic friction

5. Nov 10, 2012

### TSny

Right. Is the friction kinetic or static?

6. Nov 10, 2012

### rschaefer2

Actually, static correct? Because the object is rolling.

7. Nov 10, 2012

### TSny

That's right- rolling without slipping. So, even if they did give you a coefficient of static friction it wouldn't be of much help. The static coefficient only let's you find the maximum force of static friction, and there is no reason why the static friction would need to be at its maximum value at P.

So, you'll need to find the friction force some other way. Will the ball have any angular acceleration at P?

8. Nov 10, 2012

### rschaefer2

The ball itself? Well angular acceleration will be the derivative of ω with respect to s? EDIT: s as in seconds, from g

9. Nov 10, 2012

### TSny

That's a correct statement if s denotes time. [Edit: Ah, I see that you are using s for time!] But it's not of much help.

Do you think the ball will be speeding up or slowing down at P?

10. Nov 10, 2012

### rschaefer2

Definitely slowing down, because v will be decreasing

11. Nov 10, 2012

### TSny

Will ω also be decreasing?

12. Nov 10, 2012

### rschaefer2

Yes, because v=ωr

13. Nov 10, 2012

### TSny

Good. So, looks like you have an unknown y-component of acceleration, an unknown angular acceleration, and an unknown friction force. You'll need three independent equations that relate these unknowns. How many equations can you think of?

14. Nov 11, 2012

### rschaefer2

1) $\sumτ$=I(alpha)
2)v=ωr
3)fs=μFn
4) alpha=aT/r
5)aT=(-mg-fs)m
Fn= -20mg/7
I=2/5mv2/r2

I can't seem to relate it using these, i'm probably missing one of the equations.

Last edited: Nov 11, 2012
15. Nov 11, 2012

### TSny

Oooo. Shotgun approach. Need to narrow the field. Eq. #1 looks like a keeper. #4 is also good except the subscipt T is suspicious. What does aT stand for?

Finally, you'll need another equation that relates at least two of the three unknowns: ay, $\alpha$, and the friction force. (Hint: We don't want to upset Newton - which is easy to do.)
[EDIT: your #5 equation looks like it might be the third equation. Not sure the signs are correct though and there appears to be another mistake in it also.]

Last edited: Nov 11, 2012
16. Nov 11, 2012

### rschaefer2

at=ay

Okay, thanks! I'll try to work them out.

17. Nov 11, 2012

### TSny

Ok, good. (I was going to have to quit for now anyway - it's late and I need my beauty rest.)

18. Nov 11, 2012

### rschaefer2

My attempt:

Equation 1 expands to:
mg+fs=(2/5)mr2(ay/r)

plugging the right side of this equation into equation 3:

ay=(2/5)mray/m

and this is where i get stuck, because the acceleration terms cancel out.

19. Nov 11, 2012

### TSny

What you have written on left side doesn't represent the type of quantity you wrote earlier on the left side of equation 1. (The dimensions of what you have on the left don't match the dimensions you have on the right.)

Also, it might help to think about the direction of the friction force. If the angular speed of the ball is slowing down at P, what must be the direction of the net torque on the ball (clockwise or counterclockwise)? What direction does the friction force need to act in order to produce that direction of torque about the center of the ball?

20. Nov 11, 2012

### rschaefer2

EDIT: From previous, i've changed torque to only static friction in the positive direction, as mg acts through the axis. Also, the friction force needs to counteract the clockwise movement, therefore needing to be positive (counterclockwise).

So i'm pretty sure i've worked this out correctly. I just want to double check, as this is my last attempt for points. Thanks!

Equations:
1) $\sum$τ=Iα
fsr=Iα
α=fsr/I

2)α=ay/r
fsr/I=ay/r
ay=fsr2/I

3)may=(-mg+fs)
mfsr2/((2/5)mr2)=-mg+fs
5/2fs-fs=-mg
3/2fs=-mg
fs=-2/3mg

Plug back into 1)
α=(-2/3)mgr/I
α=(-10g/6r)

Plug into 2)
(-10g/6r)=ayr
ay=(-10g/6)

Fy=may
Fy=(-10mg/6)

21. Nov 11, 2012

### TSny

Your equation (1) indicates that you are choosing the positive direction for rotation to be the direction of the torque produced by fs. Is that clockwise or counterclockwise?

Your equation (3) indicates that you are choosing positive y in the upward direction.

Now, if ay is positive, is the angular acceleration α positive or negative according to your conventions adopted in (1)?
Equation (2) implies that α is positive when ay is positive. Is that correct?

Dang signs

22. Nov 11, 2012

### rschaefer2

Rolling up the ramp, the ball is rotating clockwise. Since ω is decreasing, the torque acting on the sphere must be in the opposite direction, counter clockwise.

So the positive direction of rotation is counterclockwise

If ay is positive, then the angular speed should also increase, therefore it should rotate clockwise quicker. So equation two should be ay=-αr?

23. Nov 11, 2012

### TSny

Yes, I believe so. Note that you got a negative value of fs which would indicate that the friction force is downward. But you know that it is upward. So, see how it works out with ay=-αr

24. Nov 11, 2012

### rschaefer2

Awesome, thank you! my new answer is (-10/14)mg, with the correct signs. Any confirmation?

25. Nov 12, 2012

### TSny

That's what I got, too. You might want to reduce the fraction. Good!