Acceleration of objects related to moment of inertia

In summary: Yes, that is correct. If the pulleys are not joined, then they will each have their own independent motion and the equations for each mass will not be related to each other.
  • #1
songoku
2,266
319

Homework Statement


A system is given below. Pulley A has radius R and mass 2M and pulley B has radius 1/2 R and mass M. If the pulley is assumed as solid cylinder, find the acceleration of each object
3.jpg


Homework Equations


τ = I.α
F = m.a
a = α.R

The Attempt at a Solution


Equation of motion for block M:
ƩF = m.a
T1 - M.g = M.a1

Equation of motion for second block:
ƩF = m.a
3/2 M.g - T2 = 3/2 M . a2

Equation relating torque:
τ = I.α

I'm not sure how to find the moment of inertia of the system, is it Isystem = I1 + I2 = 1/2 (2M) (R)2 + 1/2 (M) (1/2 R) ?
 
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  • #2
Note - the two pulleys will have the same angular acceleration; torque is force time radius.
[edit] hit the wrong button ... sorry: yes, you add the I's.
You get the I for the pulley from the I for a point mass by integrating - "integrate" is a fancy way of adding things up.
 
Last edited:
  • #3
songoku said:
Equation of motion for block M:
ƩF = m.a
T1 - M.g = M.a1

Equation of motion for second block:
ƩF = m.a
3/2 M.g - T2 = 3/2 M . a2

Equation relating torque:
τ = I.α

I'm not sure how to find the moment of inertia of the system, is it Isystem = I1 + I2 = 1/2 (2M) (R)2 + 1/2 (M) (1/2 R)2 ?

Simon Bridge said:
You get the I for the pulley from the I for a point mass by integrating - "integrate" is a fancy way of adding things up.

Well i guess you can add the moments to get net moment, after all, that's what we do in integration.

But aren't the pulleys given as 2 (or separate?) so will they have same acc. Its not specified anywhere !
 
  • #4
The question is not very clear on that point is it?
When it tells you "the pulley is a solid cylinder", I take that to mean the composite pulley. Otherwise the question does not make a lot of sense.

If you like OP can do it either way - pulleys locked together or pulleys not connected at all. (No mention of a friction coefficient for limited slip between them so it is one or the other.)
 
  • #5
I think there are two concentric pulleys.

Next, after finding the Isystem = I1 + I2 = 1/2 (2M) (R)2 + 1/2 (M) (1/2 R)2, I proceed as follow:

Equation of motion for block M:
ƩF = m.a
T1 - M.g = M.a1
T1 - M.g = M.α.R1...(1)

Equation of motion for second block:
ƩF = m.a
3/2 M.g - T2 = 3/2 M . a2
3/2 M.g - T2 = 3/2 M.α.R2...(2)

τ = I.α
T2.1/2 R - T1.R = 1/2 (2M) (R)2 + 1/2 (M) (1/2 R)2.α...(3)

By using elimination, solve for α. Am I right?

cupid.callin said:
But aren't the pulleys given as 2 (or separate?) so will they have same acc. Its not specified anywhere !

I don't get the meaning. Can there be two cases? Can they have different α?

Thanks
 
  • #6
songoku said:
I think there are two concentric pulleys.

Next, after finding the Isystem = I1 + I2 = 1/2 (2M) (R)2 + 1/2 (M) (1/2 R)2, I proceed as follow:

Equation of motion for block M:
ƩF = m.a
T1 - M.g = M.a1
T1 - M.g = M.α.R1...(1)

Equation of motion for second block:
ƩF = m.a
3/2 M.g - T2 = 3/2 M . a2
3/2 M.g - T2 = 3/2 M.α.R2...(2)

τ = I.α
T2.1/2 R - T1.R = 1/2 (2M) (R)2 + 1/2 (M) (1/2 R)2.α...(3)

By using elimination, solve for α. Am I right?

Yes Your method is correct !

I don't get the meaning. Can there be two cases? Can they have different α?

Thanks

but Well when i read the question again, i guess you can ignore my statement, the method is fine ...
 
  • #7
cupid.callin said:
Yes Your method is correct !



but Well when i read the question again, i guess you can ignore my statement, the method is fine ...

Thank you and Simon for the help :)
 
  • #8
If the pulleys are not joined together, then there would be two cases to solve for.
You are not explicitly told the two pulleys are joined but it is kinda implied in the question ... interpreting problems scientifically is something you have to learn how to do. A variation would be to have the two pulleys separate but leave the second mass unknown ... you get told that the two masses hit the ground at the same time and asked to calculate the unknown mass. I think this one wants you to notice that the heavier mass does not dominate the motion.

But it's all good fun - enjoy :)
 
  • #9
Simon Bridge said:
If the pulleys are not joined together, then there would be two cases to solve for.
You are not explicitly told the two pulleys are joined but it is kinda implied in the question ... interpreting problems scientifically is something you have to learn how to do. A variation would be to have the two pulleys separate but leave the second mass unknown ... you get told that the two masses hit the ground at the same time and asked to calculate the unknown mass. I think this one wants you to notice that the heavier mass does not dominate the motion.

But it's all good fun - enjoy :)

OK now let say that the pulleys are not joined together. Honestly, I can't imagine how the system works.

If the pulleys are joined, then they will have same angular acceleration and one will move upwards while the other downwards.

I can't even imagine the figure if the pulleys are not joined. In my mind, there are two separate pulleys and they are independent to each other, one will move clockwise and the other moves anti-clockwise. Both masses will move downwards and the equation. We will get equations from each pulley that are not related to each other, unless there is additional information given from the question (such as the masses reach the ground at the same time).

Am I correct this far? Thanks
 
  • #10
songoku said:
I can't even imagine the figure if the pulleys are not joined. In my mind, there are two separate pulleys and they are independent to each other, one will move clockwise and the other moves anti-clockwise. Both masses will move downwards and the equation. We will get equations from each pulley that are not related to each other, unless there is additional information given from the question (such as the masses reach the ground at the same time).

Am I correct this far? Thanks

Yes you are correct. These type of questions are not mostly asked because of their simplicity while solving ...
 
  • #11
one will move clockwise and the other moves anti-clockwise. Both masses will move downwards [...]
That's correct.

I can't even imagine the figure if the pulleys are not joined.
Exploit the third dimension ... imagine the big pulley is completely separate, separate axle and everything, and it is several feet behind the small one, but the axles line up. That would give you the same picture viewed end-on.

If you've seen block and tackle, you get two pulleys mounted side-by-side which can rotate separately.
 
  • #12
cupid.callin said:
Yes you are correct. These type of questions are not mostly asked because of their simplicity while solving ...

Simon Bridge said:
That's correct.


Exploit the third dimension ... imagine the big pulley is completely separate, separate axle and everything, and it is several feet behind the small one, but the axles line up. That would give you the same picture viewed end-on.

If you've seen block and tackle, you get two pulleys mounted side-by-side which can rotate separately.

OK thanks again :)
 

1. What is the moment of inertia of an object?

The moment of inertia of an object is a measure of its resistance to rotational motion. It is similar to mass in linear motion and is influenced by the object's mass, shape, and distribution of mass.

2. How does the moment of inertia affect an object's acceleration?

The moment of inertia directly affects an object's acceleration through rotational motion. Objects with a larger moment of inertia require more torque to achieve the same angular acceleration compared to objects with a smaller moment of inertia.

3. How does the acceleration of an object relate to its moment of inertia?

The acceleration of an object is inversely proportional to its moment of inertia. In other words, as the moment of inertia increases, the acceleration decreases, and vice versa.

4. Can the moment of inertia of an object be changed?

Yes, the moment of inertia of an object can be changed by altering its mass, shape, or distribution of mass. For example, if an object's mass is moved further away from its axis of rotation, its moment of inertia will increase.

5. How is the moment of inertia calculated?

The moment of inertia can be calculated using the formula I = mr², where I is the moment of inertia, m is the mass of the object, and r is the distance between the object's mass and its axis of rotation. This formula can be applied to both point masses and extended objects.

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