What is the Final Angular Velocity of a Cylinder?

[James50]

I've got a synoptic paper coming up and for the life in me I can't remember how to do some basic dynamics!

Homework Statement

A massless rope is wrapped several times around a solid cylinder of radius R = 20 cm, and mass M = 20 kg, which is at rest on a horizontal surface. Someone pulls 1 m or the rope with a constant force of 100 N, setting the cylinder in motion. Assuming that the rope neither stretches nor slips, and that the cylinder rolls without slipping, what is the final angular velocity of the cylinder of mass M and radius R. The moment of inertia of the cylinder is MR²/2.

Homework Equations

T = F.R
L = Iw
T = dL/dt

T is torque, F force, R radius, I moment of inertia, W angular speed, L angular momentum, Y is the angle

The Attempt at a Solution

Torque is 100x0.2 = 20 Nm. Moment of inertia is 0.4 kg m². Now rearranging torque as a function of angular momentum, you get T = I dw/dt.
Some chain rule... dw/dt = w dw/dY. A little bit of integrating... 0.5 w² = (T/I)Y -> using Y=0 as the lower boundary.

Okay, now because 1m is pulled from the rope, and the circumference of the cylinder is 1.256m, it works out that the sphere has rotated by 5 rads. Put that, along with T=20 and I=0.4 into the above equation, you come out with 22.36 rad /s. The actual answer is 18.4 rad /s. Ugh!

 

[tiny-tim]

Welcome to PF!

Hi James50! Welcome to PF!

(have a theta: θ and an omega: ω )

I've got a synoptic paper coming up and for the life in me I can't remember how to do some basic dynamics!

eek!

You clearly need to do a lot of worked examples on rolling motion.

(btw, what's a synoptic paper? )

With rolling, you need two types of equation, one for rotational motion and one for linear motion …

you also need a rolling constraint (usually v = rω).

And in this case, since you're given distance (instead of time), mightn't a work energy equation save you … erm … both work and energy?

 

[James50]

Hi James50! Welcome to PF!

(have a theta: θ and an omega: ω )

Thanks! I saw some symbols hanging around in the latex reference things but then it messed up all my paragraphing!

You clearly need to do a lot of worked examples on rolling motion.

(btw, what's a synoptic paper? )

You're telling me! Nuclear physics and baryon decuplets I can do... But rotating bodies!
A synoptic paper is a general one, it lasts for something like 4 hours and has questions on all the modules in my degree (I've done over 30, ugh!)

And in this case, since you're given distance (instead of time), mightn't a work energy equation save you … erm … both work and energy?

I can't believe I didn't think of doing it that way. Got the right answer straight away, thanks! Although I still can't figure out the normal way of getting there!

 

[tiny-tim]

A synoptic paper is a general one, it lasts for something like 4 hours and has questions on all the modules in my degree (I've done over 30, ugh!)

hmm … I think I'd call that panoptic

Although I still can't figure out the normal way of getting there!

Do F = ma and τ = Iω.

Do a forum search for "rolling" to find some other examples to work through.