Torque and angular acceleration question

[Spartan301]

I'm having some trouble converting mass to force in this problem. Any help would be appreciated.

Homework Statement

The pulley shown in the illustration has a radius of 2.70 m and a moment of inertia of 39.0 kg*m^2. The hanging mass is 4.20 kg and it exerts a force tangent to the edge of the pulley. What is the angular acceleration of the pulley?

Given:
Radius: 2.70 m
moment of inertia: 39.0 kg*m^2
Mass: 4.20 kg

Homework Equations

Objective: Find angular acceleration.

Battle Plan:
Convert from kg to N.
Multiply force and radius to get net torque.
Divide the Torque by the moment of inertia to find the angular acceleration.

The Attempt at a Solution

Outcome:
4.20 kg x lb rate = 9.25941501 lbs
9.25941501 lbs x 4.448 = 41.185878 N
41.185878 N x 2.70 m = 111.2018706 N*m
111.2018706 kg*m^2/s^2 / 39.0 kg*m^2 = 2.851330015 rad/s^2

 

[Doc Al]

I'm having some trouble converting mass to force in this problem.

What do you mean? Finding the force of gravity on some mass? (W = mg; use Newtons, not pounds!)

The pulley shown in the illustration has a radius of 2.70 m and a moment of inertia of 39.0 kg*m^2. The hanging mass is 4.20 kg and it exerts a force tangent to the edge of the pulley. What is the angular acceleration of the pulley?

Is that hanging mass the only thing attached to the pulley? (Illustration?)

 

[Spartan301]

What do you mean? Finding the force of gravity on some mass? (W = mg; use Newtons, not pounds!)

Is that hanging mass the only thing attached to the pulley? (Illustration?)

[URL]http://img713.imageshack.us/f/photoon20110326at0945.jpg/[/URL]

Can you see it alright? http://img713.imageshack.us/f/photoon20110326at0945.jpg/

 

[Doc Al]

OK, I see the diagram.

So...

What forces act on the pulley? Apply Newton's 2nd law for rotation. (Consider the torque on the pulley.)

What forces act on the hanging mass? Apply Newton's 2nd law.

Combine those two equations to solve for the acceleration.

 

[Spartan301]

OK, I see the diagram.

So...

What forces act on the pulley? Apply Newton's 2nd law for rotation. (Consider the torque on the pulley.)

What forces act on the hanging mass? Apply Newton's 2nd law.

Combine those two equations to solve for the acceleration.

So...

You have to find the force of gravity on the block, then add it to the force the block exerts on the pulley.

4.20 kg x 9.81 m/s^2 = 41.202 N

Then to find the force exerted by the block on the pulley,
4.20 kg x lb rate = 9.25941501 lbs
9.25941501 lbs x 4.448 = 41.185878 N
41.185878 N x 2.70 m = 111.2018706 N*m

Do you add them?

= 152.4038706 N+N*m
152.4038706 / moment of inertia
152.4038706 / 39.0 kg*m^2
3.907791554 m/s^2
3.907791554 m/s^2 / radius
3.907791554 m/s^2 / 2.70 m = 1.447330205 rad/s^2

That's the closest I've come! It's supposed to be 1.60 rad/s^2

How can I make this more precise?

You said combine the two laws. Do I do this:
mass x acceleration = moment of inertia x angular acceleration ?

because then I'd need at lease the linear or tangential acceleration, and in order to find that I'd need the angular acceleration. I'd be the one going in a circle instead of the pulley. ;)

 

[Doc Al]

So...

You have to find the force of gravity on the block,

That's one of the forces acting on the block.

then add it to the force the block exerts on the pulley.

No, don't add them.

Again, what force acts on the pulley? (I want a label, not a number.)

What forces act on the hanging mass? (Hint: Two forces act on it.)

4.20 kg x 9.81 m/s^2 = 41.202 N

OK, that's the weight of the hanging mass.

You said combine the two laws. Do I do this:
mass x acceleration = moment of inertia x angular acceleration ?

No. I want you to write two separate equations,

One for the hanging mass:
ΣF = ma

And one for the pulley:
Torque = I*alpha

Then you'll solve these equations together to find the acceleration.

 

[Doc Al]

And I don't want to see pounds anywhere!

 

[Spartan301]

Heh. Okay, no lbs. Just Newtons. Newtons might add pounds, but that's only if you stuff them with figs.

Torque acts on the pulley.

Gravity and the moment of inertia act on the block.

ΣF = ma
Torque = I*alpha

We found the force ΣF = ma = 41.202 N of the block. That is the force that pulls on the pulley along with gravity.

Now we have the moment of inertia. 39.0 N. That's the resistance against the pull of the block.

Torque = I*alpha, but all I have is the moment of inertia. I think what you're trying to say is that torque needs to be found from the first equation we used. Then we can divide the torque by the moment of inertia to find angular acceleration. Right?

 

[Doc Al]

Torque acts on the pulley.

What force produces that torque? (What's directly touching the pulley?)

Gravity and the moment of inertia act on the block.

Moment of inertia is irrelevant for the block, since it's not rotating. Also, moment of inertia is not a force!

Gravity acts on the block. What else does? (What's directly touching the block?)

ΣF = ma
Torque = I*alpha

We found the force ΣF = ma = 41.202 N of the block. That is the force that pulls on the pulley along with gravity.

You need to set the net force on the block (ΣF) equal to ma.

Now we have the moment of inertia. 39.0 N. That's the resistance against the pull of the block.

Torque = I*alpha, but all I have is the moment of inertia. I think what you're trying to say is that torque needs to be found from the first equation we used. Then we can divide the torque by the moment of inertia to find angular acceleration. Right?

You'll find alpha by solving those two equations together. (How does alpha of the pulley relate to the acceleration 'a' of the block?)

One step at a time. Let's work on the block first. Name the two forces--and their directions--that act on the block. Then write ΣF = ma. (Take down as positive, since you know the mass will accelerate down.)

 

[Spartan301]

Okay there's the gravity pulling the block down, and there's the rope that's pulling upwards. (it's going down, but I mean the rope is resisting) Is it called tension?

ΣF = ma
ΣF = (4.20 kg)(9.81 m/s^2)
ΣF = 41.202 N

Now the rope pulls back up on the block again. I guess it would pull back up on the block with the same resistance as the pulley has to rotating, correct? That's 39.0 kg*m^2

Sounds like we're shifting focus away from the acceleration of the block and concentrating on the acceleration of the rope.

 

[Spartan301]

Hey Doc? Is linear acceleration the same as tangential acceleration? Because if we used all this to find the linear acceleration of the rope, maybe we could find the tangential acceleration of the pulley!

 

[Doc Al]

Okay there's the gravity pulling the block down, and there's the rope that's pulling upwards. (it's going down, but I mean the rope is resisting) Is it called tension?

Yes! There are two forces on the block:
(1) Gravity, equal to mg, acting down.
(2) The tension in the rope, acting up. Call that force T.

ΣF = ma
ΣF = (4.20 kg)(9.81 m/s^2)
ΣF = 41.202 N

No. mg is just one of the forces acting on the block. The net force on the block (taking down as positive) is ΣF = mg - T. Applying Newton's 2nd law, we get:
ΣF = ma
mg - T = ma

That's our first equation. Note that there are two unknowns, the acceleration and the tension, so we need a second equation. That's where the pulley comes in.

(Tip: Don't be so quick to plug in numbers. Solve the problem symbolically, as much as you can, then reach for the calculator at the end.)

Now the rope pulls back up on the block again. I guess it would pull back up on the block with the same resistance as the pulley has to rotating, correct? That's 39.0 kg*m^2

No guessing. The rope pulls up on the block with force T; the rope also pulls down on the pulley with force T. What's the torque on the pulley?

Sounds like we're shifting focus away from the acceleration of the block and concentrating on the acceleration of the rope.

Since block and rope are attached, they have the same acceleration. How does the angular acceleration of the pulley relate to the linear acceleration of the block?

 

[Doc Al]

Hey Doc? Is linear acceleration the same as tangential acceleration? Because if we used all this to find the linear acceleration of the rope, maybe we could find the tangential acceleration of the pulley!

Absolutely! The tangential acceleration of the pulley is the same as the linear acceleration of the rope which is the same as the linear acceleration of the block. Good thinking.

How does tangential acceleration relate to angular acceleration?

 

[Spartan301]

Tangential acceleration is the radius times the angular acceleration, which we're trying to get to.

...you said the rope pulls up of the block with a force T and down on the pulley with T. Does that make Torque = 0?

So this rope exerts a force on the pulley. We're going to use the forces around the block and the rope to find a linear acceleration, then jump to the pulley to find the torque by treating it as tangential acceleration, divide by the radius of the pulley, and then we'll have angular acceleration.

Objective: Find Angular Acceleration.

Given:
Radius: 2.70m
moment of inertia: 39.0 kg*m^2
Mass of block: 4.20 kg.

Guest Equation:
mg - T = ma
g - T/m = a

Battle Plan:
find the tension of the rope by examining the givens in the pulley.
Subtract the tension of the rope acting upward, divided by the mass, from the force of gravity to find the acceleration. a = g -T/m
Mass x acceleration = linear acceleration
Linear acceleration = tangential acceleration.
Divide the tangential acceleration by the radius to find angular acceleration

So I need to have instructions on how to find the acceleration of the block.
The block has a mass, and there's a force of gravity, but the rope is causing it to fall more slowly. So I need T!

So how do you find T?

 

[Doc Al]

Tangential acceleration is the radius times the angular acceleration, which we're trying to get to.

Good.

...you said the rope pulls up of the block with a force T and down on the pulley with T.

Right.

Does that make Torque = 0?

No. The rope exerts a tension on the pulley and thus a torque on it. What is that torque?

So I need to have instructions on how to find the acceleration of the block.
The block has a mass, and there's a force of gravity, but the rope is causing it to fall more slowly. So I need T!

So how do you find T?

Again, you need to write the equation for torque:
Torque = I*alpha

That will become your second equation. Then you can solve for the unknowns.