3 Questions about Rotational Motion; Test Tomorrow

[snuffy]

3 Questions about Rotational Motion; Test Tomorrow! Please Help!

Problem 1

Homework Statement

A uniform disk of radius .12m and mass 5kg is pivoted so that it rotates freely about its axis. A string wrapped around the disk is pulled with a force of 20N.
a. What is the torque exerted on the disk?
b. What is the angular acceleration of the disk?
c. If the disk starts from rest, what is the angular velocity after 3s?

Homework Equations

τ = rFsinθ
I=1/2MR^2 for a uniform disk about its center
T=I(alpha)

The Attempt at a Solution

a. τ = rFsinθ
= (.12)(20)(sin 90)
= 2.4 NM

b. T=I(alpha)
T=.5MR^2(alpha)
2.4=.5(5)(.12^2)(alpha)
alpha=66.66 rad/s^2

c. wf-wi=(alpha)(t)
wf=(66.66)(3)
=199.98

Is that right??


Problem 2

Homework Statement

A pulley of radius R1 and rotational inertia I1 is mounted on an axle with negligible friction. A light cord passing over the pulley has two blocks of mass m attached to either end, as shown above. Assume that the cord does not slip on the pulley. Determine the answers to parts (a) and (b) in terms of m, R1, I1, and fundamental constants.
a. Determine the tension T in the cord.
b. One block is now removed from the right and hung on the left. When the system is released from rest, the three blocks on the left accelerate downward with an acceleration g/3 . Determine the following.
i. The tension T3 in the section of cord supporting the three blocks on the left
ii. The tension Tl in the section of cord supporting the single block on the right
iii. The rotational inertia I1 of the pulley

Here's the picture:
http://courses.ncssm.edu/aphys/problems/AP%20Problems/ap160809.doc 1st problem.

Homework Equations

I am very confused. I do not know how to answer this problem. I assume I must relate the tension to the torque so:
τ = rFsinθ
=(Tension)(radius)

But how would I know that the tension is the force I should use? What about the force of gravity on the two masses? How do you do this?


Problem 3

Homework Statement

A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The rotational inertia of the pole and the rod are negligible. A) Determine the rotational inertia of the rod-and-block apparatus attached to the top of the pole. B) Determine the downward acceleration of the large block. C) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare with the value 4mgD? (greater, equal, less)
Here's the picture:
http://www.physics4students.com/pages/apc/apcaday/apcaday_040.doc
1st problem

Homework Equations

W-T=4ma
Once again, I just don't know how to approach this problem. I'm sorry.

Could someone just walk me through the initial steps? I need to get the answer myself.

 

[Doc Al]

Problem 1

Homework Statement

A uniform disk of radius .12m and mass 5kg is pivoted so that it rotates freely about its axis. A string wrapped around the disk is pulled with a force of 20N.
a. What is the torque exerted on the disk?
b. What is the angular acceleration of the disk?
c. If the disk starts from rest, what is the angular velocity after 3s?

Homework Equations

τ = rFsinθ
I=1/2MR^2 for a uniform disk about its center
T=I(alpha)

The Attempt at a Solution

a. τ = rFsinθ
= (.12)(20)(sin 90)
= 2.4 NM

b. T=I(alpha)
T=.5MR^2(alpha)
2.4=.5(5)(.12^2)(alpha)
alpha=66.66 rad/s^2

c. wf-wi=(alpha)(t)
wf=(66.66)(3)
=199.98

Is that right??

I didn't check your arithmetic, but your method is good.

Problem 2

Homework Statement

A pulley of radius R1 and rotational inertia I1 is mounted on an axle with negligible friction. A light cord passing over the pulley has two blocks of mass m attached to either end, as shown above. Assume that the cord does not slip on the pulley. Determine the answers to parts (a) and (b) in terms of m, R1, I1, and fundamental constants.
a. Determine the tension T in the cord.
b. One block is now removed from the right and hung on the left. When the system is released from rest, the three blocks on the left accelerate downward with an acceleration g/3 . Determine the following.
i. The tension T3 in the section of cord supporting the three blocks on the left
ii. The tension Tl in the section of cord supporting the single block on the right
iii. The rotational inertia I1 of the pulley

Here's the picture:
http://courses.ncssm.edu/aphys/problems/AP%20Problems/ap160809.doc 1st problem.

Homework Equations

I am very confused. I do not know how to answer this problem. I assume I must relate the tension to the torque so:
τ = rFsinθ
=(Tension)(radius)

But how would I know that the tension is the force I should use? What about the force of gravity on the two masses? How do you do this?

Hint for a: Is it moving? What's the acceleration?
Hints for b: You are told the acceleration. Apply Newton's 2nd law to the group of three blocks on the left. Then apply it to the single block on the right. (Draw free body diagrams of the three "systems": masses on the left; mass on the right; pulley.)

Problem 3

Homework Statement

A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The rotational inertia of the pole and the rod are negligible. A) Determine the rotational inertia of the rod-and-block apparatus attached to the top of the pole. B) Determine the downward acceleration of the large block. C) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare with the value 4mgD? (greater, equal, less)
Here's the picture:
http://www.physics4students.com/pages/apc/apcaday/apcaday_040.doc
1st problem

Homework Equations

W-T=4ma
Once again, I just don't know how to approach this problem. I'm sorry.

Could someone just walk me through the initial steps? I need to get the answer myself.

Hint for a: What's the rotational inertia of a point mass a given distance from an axis?
Hints for b: Draw free body diagrams for the pole/masses thing and for the block. Apply Newton's 2nd law.

 

[snuffy]

Okay thanks for the confirmation on problem 1.

Problem 2:
a. Would the Tension=2mg? That seems to make sense to me. I applied Newton's Second, it's at rest so T-2mg=0.
b. Still working on it...another hint appreciated...


Problem 3:
a. Would I=(m)(L/2)²+ (m)(L/2)²+ (m)(L/2)²? One for each point mass and the horizontal pole?

b. T=I(alpha) since the torque is equal to Tr and angular acceleration=a/r
(Tension)(radius)=I(a/r)
TR=(3/4mL²)(a/r)
a=4TR²/3ML²

and since Tension-4mg=4ma then T=4ma+4mg. Therefore I could sub it back into the equation for acceleration.

C. Don't know how to do.

 

[Doc Al]

Problem 2:
a. Would the Tension=2mg? That seems to make sense to me. I applied Newton's Second, it's at rest so T-2mg=0.

Good.

b. Still working on it...another hint appreciated...

Apply Newton's 2nd law.

Problem 3:
a. Would I=(m)(L/2)²+ (m)(L/2)²+ (m)(L/2)²? One for each point mass and the horizontal pole?

Not exactly:
- You're told that the horizontal pole can be ignored.
- What's the distance of each point mass from the axis?

b. T=I(alpha) since the torque is equal to Tr and angular acceleration=a/r
(Tension)(radius)=I(a/r)
TR=(3/4mL²)(a/r)
a=4TR²/3ML²

Redo with the correct I.

and since Tension-4mg=4ma then T=4ma+4mg. Therefore I could sub it back into the equation for acceleration.

Careful with signs. The acceleration is downward.

C. Don't know how to do.

Consider how the gravitational PE is transformed into KE.

 

[snuffy]

Okay so...

Problem 2b:
i. T₃-3mg=-3m(g/3)
T₃-3mg=-mg
T₃=-mg+3mg
T₃=2mg

ii. T₁-mg=(mg)/3
T₁=(4mg)/3


iii. T=I(alpha)
I=Torque/alpha
alpha=(g)/(3R)
Torque=T₃-T₁
I=(2mg-(4mg/3))/(g/3R)
I=2mR

Problem 3:
a. So 1/2ml²=I? And if the horizontal rod was not to be ignored it would be 3/4ml²?
b. Another question about this: how is the downward acceleration of the block related to the spinning motion of the apparatus? Is it just the acceleration of the block is equal to the angular acceleration times the radius of the spinning apparatus?

 

[Doc Al]

Okay so...

Problem 2b:
i. T₃-3mg=-3m(g/3)
T₃-3mg=-mg
T₃=-mg+3mg
T₃=2mg

ii. T₁-mg=(mg)/3
T₁=(4mg)/3

OK.

iii. T=I(alpha)
I=Torque/alpha
alpha=(g)/(3R)
Torque=T₃-T₁
I=(2mg-(4mg/3))/(g/3R)
I=2mR

You made a mistake in calculating the torque. (What happened to the radius?)

Problem 3:
a. So 1/2ml²=I?

No.

And if the horizontal rod was not to be ignored it would be 3/4ml²?

The rod would contribute 1/12M(2L)^2 to the total I, where M is the rod's mass.

b. Another question about this: how is the downward acceleration of the block related to the spinning motion of the apparatus? Is it just the acceleration of the block is equal to the angular acceleration times the radius of the spinning apparatus?

That's right, since they are connected by the string.