Angular momentum and rotational inertia

In summary: I2=...Yeah, I can’t figure out how to connect the two :/ Their masses aren’t the same and neither are their radii. So I’m not sure how to connect I1=... and I2=...Hint: What equation can you write relating the angular momentum of the original wheel to its rotational inertia and angular velocity? And what equation can you write relating the angular momentum of the new wheel to its rotational inertia and angular velocity?Hint: What equation can you write relating the angular momentum of the original wheel to its rotational inertia and angular velocity? And what equation can you write relating the angular momentum of the new wheel to its rotational inertia and angular velocity?In summary, the problem involves designing a bicycle wheel with
  • #1
starstruck_
185
8

Homework Statement


You decide to design a bicycle that will have only 3/4 of the angular momentum of the original wheel when both wheels are traveling along a road at the same velocity. The original wheel had a diameter of d1=37cm and rotational inertia of I1=0.32 m2kg. If your new wheel has a diameter of d2= 34cm, what is its rotational inertia?
2. Relevant equation
Angular momentum denoted by L.

L= wI (where w is the angular velocity, and I is the rotational inertia)
Rotational inertia of a ring = mR2.
w= v/r (where v is the linear velocity).

The Attempt at a Solution


I know that the angular momentum for the new wheel is 3/4 of the first wheel, their linear velocities are the same, I need rotational inertia of the new wheel.

Lnew wheel= 3/4 (v/d1/2) * m*(d2/2)^2

I honestly have no clue where to start or go with this question, or whether I'm even using all of the required equations :(
 
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  • #2
starstruck_ said:
the angular momentum for the new wheel is 3/4 of the first wheel
That was your intention, but you did not necessarily achieve that.
If you really knew it was 3/4 of the old the problem becomes trivial. You are given the MoI of the original, so you could just multiply it by 3/4.

Let the mass, radius and MoI of the original be m0, r0 and I0, and the new wheel be m1, etc.
What equations can you write relating these?
 
  • #3
haruspex said:
That was your intention, but you did not necessarily achieve that.
If you really knew it was 3/4 of the old the problem becomes trivial. You are given the MoI of the original, so you could just multiply it by 3/4.

Let the mass, radius and MoI of the original be m0, r0 and I0, and the new wheel be m1, etc.
What equations can you write relating these?

I=mr^2?
 
  • #4
starstruck_ said:
I=mr^2?
Yes, but there are two masses, two radii, two MoIs. Always use different representations for different variables, like m0, etc.
Can you find another equation relating the masses?
 
  • #5
starstruck_ said:

Homework Statement


You decide to design a bicycle that will have only 3/4 of the angular momentum of the original wheel when both wheels are traveling along a road at the same velocity. The original wheel had a diameter of d1=37cm and rotational inertia of I1=0.32 m2kg. If your new wheel has a diameter of d2= 34cm, what is its rotational inertia?
2. Relevant equation
Angular momentum denoted by L.

L= wI (where w is the angular velocity, and I is the rotational inertia)
Rotational inertia of a ring = mR2.
w= v/r (where v is the linear velocity).

The Attempt at a Solution


I know that the angular momentum for the new wheel is 3/4 of the first wheel, their linear velocities are the same, I need rotational inertia of the new wheel.

Lnew wheel= 3/4 (v/d1/2) * m*(d2/2)^2

I honestly have no clue where to start or go with this question, or whether I'm even using all of the required equations :(
You have i1 given from this you can find m1
 
  • #6
Abhishek kumar said:
You have i1 given from this you can find m1
Even no need to calculate mass you can simply apply formula for angular momentum Iω.
 
  • #7
Abhishek kumar said:
Even no need to calculate mass you can simply apply formula for angular momentum Iω.

So, would I2= 3/4s of I1? Since they both have the same velocity?
 
  • #8
starstruck_ said:
So, would I2= 3/4s of I1? Since they both have the same velocity?
You seem not to have understood my first sentence in post #2. Forget the 3/4; that was your intent but you might not have achieved it.
Work with the radii and the formula I=mr2.
Write out the formula twice, once for the original wheel and once for the new wheel, using subscripts to distinguish variables that might have different values. You should notice something else that you need to work out.
 
  • #9
haruspex said:
You seem not to have understood my first sentence in post #2. Forget the 3/4; that was your intent but you might not have achieved it.
Work with the radii and the formula I=mr2.
Write out the formula twice, once for the original wheel and once for the new wheel, using subscripts to distinguish variables that might have different values. You should notice something else that you need to work out.

I rearranged for the mass 1.

I1 = m1*R1^2
I1/R1^2 = M1
4I1/ d1^2 = M1
 
  • #10
starstruck_ said:
I rearranged for the mass 1.

I1 = m1*R1^2
I1/R1^2 = M1
4I1/ d1^2 = M1

I2= M2R2^2
I2= M2(d2^2/4)

L2= 3/4 L1 = 3/4 (4I1/d1^2) (d1^2/4) = 3/4I1

So L2= 3/4 I1?
 
  • #11
starstruck_ said:
I1/R1^2 = M1
Ok. in terms of m1, r1 and r2, what do you think m2 will be?
starstruck_ said:
So L2= 3/4 I1?
What part of "forget the 3/4" do you not understand?
 
  • #12
haruspex said:
Ok. in terms of m1, r1 and r2, what do you think m2 will be?

What part of "forget the 3/4" do you not understand?

Okay sorry, I’ll ignore the 3/4. I am having trouble understanding how to connect I1 with I2 .

Am I working with the fact that they have the same velocity? So I rearrange for the velocity and set both equations equal to each other and rearrange for M2?
 
  • #13
starstruck_ said:
Okay sorry, I’ll ignore the 3/4. I am having trouble understanding how to connect I1 with I2 .

Am I working with the fact that they have the same velocity? So I rearrange for the velocity and set both equations equal to each other and rearrange for M2?

Nevermind not sure how I can do that
 
  • #14
starstruck_ said:
Am I working with the fact that they have the same velocity?
No. The question concerns the inherent properties of the wheels. These are independent of how fast they are rotated.
Please try to answer my question in post #11.
 
  • #15
haruspex said:
No. The question concerns the inherent properties of the wheels. These are independent of how fast they are rotated.
Please try to answer my question in post #11.

Yeah, I can’t figure out how to connect the two :/ Their masses aren’t the same and neither are their radii. So I’m not sure how to connect I1= M1R1^2 with I2= M2R2^2

I just know that M2 would equal to 4I2/d2^2 but that doesn’t help much
 
  • #16
starstruck_ said:
Their masses aren’t the same
Right, but how would you expect a change of radius to affect the mass?
 
  • #17
haruspex said:
Right, but how would you expect a change of radius to affect the mass?

The change in radius would reduce the circumference, so the wheel has less mass - the difference in the masses is the amount of mass that fits in their difference of circumference.
 
  • #18
starstruck_ said:
The change in radius would reduce the circumference, so the wheel has less mass - the difference in the masses is the amount of mass that fits in their difference of circumference.
Right, but we need to make that quantitative. If the radius were halved, what do you think the mass ratio would be?
 
  • #19
haruspex said:
Right, but we need to make that quantitative. If the radius were halved, what do you think the mass ratio would be?

If the radius was halved, the circumference would be halved, so you’d have half the mass of the bicycle wheel?
 
  • #20
starstruck_ said:
If the radius was halved, the circumference would be halved, so you’d have half the mass of the bicycle wheel?
right.
In this problem, the radius is reduced from 37cm to 34cm. so what do you expect the new mass to be?
 
  • #21
haruspex said:
right.
In this problem, the radius is reduced from 37cm to 34cm. so what do you expect the new mass to be?

34/37 of the original mass?
 
  • #22
starstruck_ said:
34/37 of the original mass?
Right.
Since I=mr2, what will be the effect on I?
 
  • #23
haruspex said:
Right.
Since I=mr2, what will be the effect on I?

I2= 34/37 of I1?
 
  • #24
starstruck_ said:
I2= 34/37 of I1?
The radius changed too. Look at the whole expression: mr2.
 
  • #25
haruspex said:
The radius changed too. Look at the whole expression: mr2.
Here new wheel's angular momentum is 3/4 of older one that means I1ω1=3/4(I2ω2) and wheels are moving with same velocity.
 
  • #26
Abhishek kumar said:
Here new wheel's angular momentum is 3/4 of older one
No. Please read the question carefully, and post #2.
 

Related to Angular momentum and rotational inertia

1. What is angular momentum?

Angular momentum is a measure of the amount of rotational motion an object has. It is calculated by multiplying the moment of inertia (a measure of an object's resistance to rotational motion) by its angular velocity (the rate at which it rotates).

2. How is angular momentum conserved?

Angular momentum is conserved in a closed system, meaning that it remains constant as long as there are no external forces acting on the system. This is known as the law of conservation of angular momentum.

3. What is rotational inertia?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is dependent on the mass and distribution of the object's mass from its axis of rotation.

4. How does rotational inertia affect an object's motion?

The higher the rotational inertia of an object, the more force is required to change its rotational motion. This means that objects with higher rotational inertia will tend to rotate at a slower rate.

5. How is angular momentum related to rotational inertia?

Angular momentum and rotational inertia are directly proportional to each other. This means that as the rotational inertia of an object increases, so does its angular momentum, assuming the angular velocity remains constant.

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