Marble rolling on ramp harmonic motion

In summary: I'm still not sure why my approach didn't work though. Do you mind explaining why?Thank you! I'm still not sure why my approach didn't work though. Do you mind explaining why?In summary, the conversation discusses the calculation of the time it takes for a perfectly solid marble to go down one side and up the other of a spherical bowl. The correct answer is given as option b) and the incorrect answer is given as option d). The conversation also includes a discussion of the incorrect approach taken in the attempt at solving the problem. The correct approach involves considering the velocity of the center of the ball and writing it in terms of both the angular velocity and the angle of rotation of the ball.
  • #1
ln(
42
0

Homework Statement


A perfectly solid marble of radius R rolls without slipping at the bottom of a spherical bowl of a radius 6R. The marble rolls back and forth in the vertical plane executing simple harmonic motion close to the lowest point. How long does it take the marble to go down one side and up the other from its maximum height to its maximum height?

a) Cannot be determined without knowing amplitude
b) ##\pi \sqrt{\frac{7R}{g}}##
c) ##\pi \sqrt{\frac{7R}{2g}}##
d) ##\pi \sqrt{\frac{7R}{5g}}##
e) ##\pi \sqrt{\frac{5R}{7g}}##

Homework Equations


##\tau_{net} = I\alpha##
##I = 2/5 mR^2## (for sphere)
##F = ma##
##\tau = rFsin\theta##

The Attempt at a Solution


I got answer choice d, but the key says b is correct.
I set up ##\tau_{net} = I\alpha = -fR## where ##f## is the static friction force. I believe gravity does not contribute to the torque, since it acts at the center-of-mass; neither does the normal force since it is always antiparallel to the radius vector from the center of the marble to the point of application. That only leaves the static friction force.
Then, I know that ##F = ma = -mgsin\theta + f \approx -mg\theta + f## tangent to the bowl, using the sine approximation for small theta.
Plugging back into torque net, I get that ##(2/5 mR^2)\alpha = -(ma + mg\theta)R##. Divide by an ##mR## and simplify to get ##7/5 R\alpha = -g\theta##. This means that ##\omega = \sqrt{\frac{5g}{7R}}##. Finally I realize that the desired time is half the period and ##T = 2\pi / \omega## so the desired time is ##\pi \sqrt{\frac{7R}{5g}}##, or at least that's what I got.

Where did I go wrong?
 
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  • #2
ln( said:
##7/5 R\alpha = -g\theta##.
OK up to here!

Hint: ##\alpha \neq \ddot \theta##.
##\dot \theta## is not the rate of spin of the ball about its center.
Consider the kinematics more carefully. What is the meaning of the angle ##\theta##?
 
  • #3
TSny said:
OK up to here!

Hint: ##\alpha \neq \ddot \theta##.
##\dot \theta## is not the rate of spin of the ball about its center.
Consider the kinematics more carefully. What is the meaning of the angle ##\theta##?
Hmm.. I think what you're trying to get at is that ##\alpha## isn't directly related to the change in the angle wrt the center of the bowl; rather, it refers to the rotation of the ball. So that means ##\alpha = 5\ddot \theta##?
 
  • #4
ln( said:
Hmm.. I think what you're trying to get at is that ##\alpha## isn't directly related to the change in the angle wrt the center of the bowl; rather, it refers to the rotation of the ball. So that means ##\alpha = 5\ddot \theta##?
Yes. Can you derive the relation ##\alpha = 5\ddot \theta##?
 
  • #5
TSny said:
Yes. Can you derive the relation ##\alpha = 5\ddot \theta##?
I made a guess there by thinking about how if the ball rolled through an angle ##\phi## wrt its center of mass, it would trace out an arc ##R\phi##. But then, wrt the center of the bowl, if the angle changed by that same amount the center-of-mass of the marble would trace out an arc ##(6R - R) \phi##. And then I assume this relation holds for the angular accelerations. I'm not sure how to do it rigorously.

EDIT: I think since ##\alpha = a/r## in general, then wrt to the ball's COM ##\alpha = a/R## but wrt to the center of the bowl ##\alpha' = \ddot \theta = a/(5R)##. So ##\alpha = 5\ddot \theta##.
 
Last edited:
  • #6
ln( said:
I made a guess there by thinking about how if the ball rolled through an angle ##\phi## wrt its center of mass, it would trace out an arc ##R\phi##. But then, wrt the center of the bowl, if the angle changed by that same amount the center-of-mass of the marble would trace out an arc ##(6R - R) \phi##. And then I assume this relation holds for the angular accelerations. I'm not sure how to do it rigorously.
Yes, that's good. Another way is to consider the velocity of the center of the ball ##v_c##. Let ##\phi## represent the angle of rotation of the ball about its center. Write ##v_c## in two different ways: (1) in terms of ##\dot \theta## and (2) in terms of ##\dot \phi##.
upload_2018-1-21_16-24-10.png
 

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  • #7
TSny said:
Yes, that's good. Another way is to consider the velocity of the center of the ball ##v_c##. Let ##\phi## represent the angle of rotation of the ball about its center. Write ##v_c## in two different ways: (1) in terms of ##\dot \theta## and (2) in terms of ##\dot \phi##.
View attachment 218891
##v_c = 5R\dot \theta = R\dot \phi##.
 
  • #8
ln( said:
##v_c = 5R\dot \theta = R\dot \phi##.
Yes. I guess that's pretty much the same as your first derivation in terms of arc lengths.
 
  • #9
ln( said:
EDIT: I think since ##\alpha = a/r## in general, then wrt to the ball's COM ##\alpha = a/R## but wrt to the center of the bowl ##\alpha' = \ddot \theta = a/(5R)##. So ##\alpha = 5\ddot \theta##.
I think you nailed it! :oldsmile:
 
  • #10
TSny said:
I think you nailed it! :oldsmile:
Thank you!
 

1. What is harmonic motion?

Harmonic motion refers to the repetitive back-and-forth motion of an object around a central equilibrium point.

2. How does a marble rolling on a ramp exhibit harmonic motion?

A marble rolling on a ramp exhibits harmonic motion because its motion is controlled by the force of gravity, which acts as a restoring force, pulling the marble back towards the center of the ramp.

3. What factors affect the harmonic motion of a marble rolling on a ramp?

The factors that affect the harmonic motion of a marble rolling on a ramp include the angle of the ramp, the mass of the marble, and the initial velocity of the marble.

4. How can the period of a marble's motion on a ramp be calculated?

The period of a marble's motion on a ramp can be calculated using the equation T = 2π√(mgh)/mg, where T is the period, m is the mass of the marble, g is the acceleration due to gravity, and h is the height of the ramp.

5. What real-life applications use the concept of harmonic motion?

Harmonic motion is used in a variety of real-life applications, including pendulum clocks, musical instruments such as guitars and violins, and even in the movement of atoms and molecules. It is also used in engineering and architecture to design stable structures.

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