Angular momentum and rotational inertia

[starstruck_]

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.

L_{new 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 :(

 

[haruspex]

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?

 

[starstruck_]

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?

 

[haruspex]

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?

 

[Abhishek kumar]

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.

L_{new 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

 

[Abhishek kumar]

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.

 

[starstruck_]

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?

 

[haruspex]

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.

 

[starstruck_]

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

 

[starstruck_]

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?

 

[haruspex]

I1/R1^2 = M1

Ok. in terms of m1, r1 and r2, what do you think m2 will be?

So L2= 3/4 I1?

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

 

[starstruck_]

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?

 

[starstruck_]

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

 

[haruspex]

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.

 

[starstruck_]

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

 

[haruspex]

Their masses aren’t the same

Right, but how would you expect a change of radius to affect the mass?

 

[starstruck_]

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.

 

[haruspex]

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?

 

[starstruck_]

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?

 

[haruspex]

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?

 

[starstruck_]

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?

 

[haruspex]

34/37 of the original mass?

Right.
Since I=mr², what will be the effect on I?

 

[starstruck_]

Right.
Since I=mr², what will be the effect on I?

I2= 34/37 of I1?

 

[haruspex]

I2= 34/37 of I1?

The radius changed too. Look at the whole expression: mr².

 

[Abhishek kumar]

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.

 

[haruspex]

Here new wheel's angular momentum is 3/4 of older one

No. Please read the question carefully, and post #2.