Sphere rolling down an incline plane pulling a rope off a cylinder

In summary: So you can use the linear acceleration of the rope to find the rotational acceleration of the sphere.
  • #1
leeone
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0
The tension in the rope is actually being provided by a solid sphere with radius 23.5 cm that rolls down an incline as shown in the figure. The incline makes an angle of 32° with the vertical. The end of the rope is attached to a yoke that runs through the center of the sphere, parallel to the slope. The friction between the incline and the ball is sufficient that the ball rolls without slipping. The mass of the sphere is closest to

I solved for the Tension in a previous problem to be T= 12.7 N

also angular acceleration of the cylinder is 12.5 rad/s^2 and its mass is 7.25 kg and its radius is 28cm.

I drew some free body diagrams and got

M=2T/((gsin(theta)-(7/5)R(angular acceleration of sphere)) and theta = 58 degrees...I just don't know how to get the angular acceleration of the sphere.

I tried to get it by equating the torques of the cylinder and sphere but then I needed the mass which is what I am solving for.
 

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  • #2
How can you relate the angular and linear acceleration?
 
  • #3
linear acceleration = angular acceleration(R)
 
  • #4
okay my equations I started with we're

Mgsin(theta)- friction-T=M(linear acceleration)

friction(R)-T(R)= (I of sphere)(angular acceleration)
 
  • #5
but here I have three unknowns and two equations even if I know the above relation linear acceleration = (angular acceleration(R)
 
  • #6
leeone said:
okay my equations I started with we're

Mgsin(theta)- friction-T=M(linear acceleration)
This one makes sense.

friction(R)-T(R)= (I of sphere)(angular acceleration)
The tension acts at the center of the sphere, so it does not produce a torque.
 
  • #7
leeone said:
I solved for the Tension in a previous problem to be T= 12.7 N
How did you solve for the tension?
 
  • #8
Using that T(R of cylinder) = I (angular acceleration of cylinder) = (1/2)(M of cylinder)(R of cylinder)^2(angular acceleration of cylinder)
 
  • #9
It was a three question problem and the angular acceleration of the cylinder
was caused by the tension force...at this point they did not inform you the sphere was what was causing it.
 
  • #10
leeone said:
Using that T(R of cylinder) = I (angular acceleration of cylinder) = (1/2)(M of cylinder)(R of cylinder)^2(angular acceleration of cylinder)
That's fine.
 
  • #11
leeone said:
It was a three question problem and the angular acceleration of the cylinder
was caused by the tension force...at this point they did not inform you the sphere was what was causing it.
But they must have given you additional information.

In any case, once you have the tension, you should be able to determine the accelerations.
 
  • #12
Assume that the pulley is at rest at time t0, which is the time that the tension in the rope is applied. The tension remains constant for 2.0 s at which point the rope goes slack, and the pulley continues to spin with no further force applied from the rope for an additional 3.0 s (for a total of 5.0 s of rotation). What, in radians, is the total angular displacement the pulley has undergone since t0?
a) 52 rad b) 83 rad c) 100 rad d) 160 rad e) 25 rad f) 180 rad

This is the only other information...but I was pretty positive this was unrelated to the questionIf they must have given me additional information then why do you say I should be able to determine the accelerations?

Also I am trying to solve for the mass
 
  • #13
leeone said:
Assume that the pulley is at rest at time t0, which is the time that the tension in the rope is applied. The tension remains constant for 2.0 s at which point the rope goes slack, and the pulley continues to spin with no further force applied from the rope for an additional 3.0 s (for a total of 5.0 s of rotation). What, in radians, is the total angular displacement the pulley has undergone since t0?
a) 52 rad b) 83 rad c) 100 rad d) 160 rad e) 25 rad f) 180 rad

This is the only other information...but I was pretty positive this was unrelated to the question


If they must have given me additional information then why do you say I should be able to determine the accelerations?
Did they give you the tension? As stated, there is not enough information to answer that question. (They must have given you something.)
 
  • #14
The tension is 12.7 N? which I solved for from the rotational acceleration of the cylinder...could I equat the torques of the two and solve for the angular acceleration of the sphere in terms of the angular acceleration of the cylinder?
 
  • #15
leeone said:
The tension is 12.7 N? which I solved for from the rotational acceleration of the cylinder...could I equat the torques of the two and solve for the angular acceleration of the sphere in terms of the angular acceleration of the cylinder?
So they gave you the rotational acceleration of the cylinder?

If so, you can use it to find the rotational acceleration of the sphere. Hint: The linear acceleration of the rope is the same as the linear acceleration of the sphere.
 
  • #16
yes they did. could I not use (I of sphere)*(angular acceleration of sphere)=(I of cylinder)*(angular acceleration of cylinder)?
 
  • #17
leeone said:
could I not use (I of sphere)*(angular acceleration of sphere)=(I of cylinder)*(angular acceleration of cylinder)?
Not I, but R.
 
  • #18
Thank You. I tried that previously but got the wrong answer b/c my calculator was in radians and not degrees -_____-
 
  • #19
leeone said:
Thank You. I tried that previously but got the wrong answer b/c my calculator was in radians and not degrees -_____-
D'oh! :smile:
 

1. How does the mass of the sphere affect its acceleration while rolling down the incline plane?

The mass of the sphere does not affect its acceleration while rolling down the incline plane. According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Since the force of gravity acting on the sphere is the same regardless of its mass, the acceleration will also be the same.

2. What is the relationship between the angle of the incline plane and the acceleration of the sphere?

The angle of the incline plane affects the acceleration of the sphere. As the angle increases, the acceleration of the sphere also increases. This is because a steeper incline plane exerts a greater component of the force of gravity in the direction of motion, resulting in a larger net force and therefore a greater acceleration.

3. How does the length of the rope affect the motion of the sphere?

The length of the rope does not affect the motion of the sphere. As long as the rope is long enough to reach the bottom of the incline plane, the sphere will have the same acceleration regardless of its length. The rope only serves to pull the cylinder and keep it in motion, while the sphere's motion is solely determined by the incline plane and the force of gravity.

4. Does the friction between the sphere and the incline plane affect its acceleration?

Yes, the friction between the sphere and the incline plane can affect its acceleration. Friction is a force that opposes motion and can slow down the acceleration of the sphere. The amount of friction depends on the coefficient of friction between the two surfaces and the normal force exerted on the sphere by the incline plane.

5. How does the mass of the cylinder being pulled by the rope affect the motion of the sphere?

The mass of the cylinder being pulled by the rope does not affect the motion of the sphere. The cylinder is being pulled by the rope, and its mass has no direct influence on the motion of the sphere. However, the mass of the cylinder can indirectly affect the motion by adding more weight to the rope, which can increase the tension and exert a larger force on the sphere.

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