# Linear and angular acceleration of a falling cylinder? [long]

by Bob Loblaw
Tags: acceleration, angular, cylinder, falling, linear
 P: 68 1. The problem statement, all variables and given/known data http://img248.imageshack.us/img248/1639/cylinderjl0.jpg A uniform cylinder with a radius of R and mass M has been attached to two cords and the cords are wound around it and hung from the ceiling. The cylinder is released from rest and the cords unwind as the cylinder descends. (a) draw a proper free body diagram for the cylinder; (b) Apply Newton’s second law to the cylinder; (c) apply Newton’s second law in rotational form to the cylinder; (d) the two equations you have written so far contain three unknowns; what is the relationship between the linear acceleration of the cylinder and its angular acceleration? (e) Solve for the linear acceleration of the cylinder; (f) What is the tension in the cords? The attempt at a solution A) The free body diagram should include the forces of tension pulling the cylinder up and the weight of the cylinder, right? B) Essentially it would be Tension-weight of cylinder=angular acceleration, is that correct? C)I know that Newton's second law in rotational form is: Net external torque = moment of inertia x angular acceleration So I would need to solve for moment of inertia: I=mr^2 X Angular acceleration = net external torque. D) So I have Tension-weight of cylinder=angular acceleration I=mr^2 X Angular acceleration = net external torque. Linear Acceleration and Angular Acceleration are related by: a. The Mass. b. The Radius. c. The Torque. d. The Force. In this problem, the mass, radius, and torque are unknown. The force working on the cylinder I believe is just gravity. E and F) I am not sure how to do these. I know this is a lengthy problem, but any help of guidance would be appreciately greatly! Even if ou don't know - I would settle for just a word of encouragement.
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P: 4,124
 Quote by Bob Loblaw 1. The problem statement, all variables and given/known data http://img248.imageshack.us/img248/1639/cylinderjl0.jpg A uniform cylinder with a radius of R and mass M has been attached to two cords and the cords are wound around it and hung from the ceiling. The cylinder is released from rest and the cords unwind as the cylinder descends. (a) draw a proper free body diagram for the cylinder; (b) Apply Newton’s second law to the cylinder; (c) apply Newton’s second law in rotational form to the cylinder; (d) the two equations you have written so far contain three unknowns; what is the relationship between the linear acceleration of the cylinder and its angular acceleration? (e) Solve for the linear acceleration of the cylinder; (f) What is the tension in the cords? The attempt at a solution A) The free body diagram should include the forces of tension pulling the cylinder up and the weight of the cylinder, right?
yes.

 B) Essentially it would be Tension-weight of cylinder=angular acceleration, is that correct?
not angular acceleration. The linear acceleration of the center of mass...

so write the $$\Sigma F_y = ma$$ equation...

 C)I know that Newton's second law in rotational form is: Net external torque = moment of inertia x angular acceleration So I would need to solve for moment of inertia: I=mr^2 X Angular acceleration = net external torque.
Yes, but the moment of inertia of the cylinder isn't mr^2... look that up... so:

net torque = I*alpha (alpha is angular acceleration)

what goes in the left side of this equation?

 D) So I have Tension-weight of cylinder=angular acceleration I=mr^2 X Angular acceleration = net external torque. Linear Acceleration and Angular Acceleration are related by: a. The Mass. b. The Radius. c. The Torque. d. The Force.
So in this part you need to relate a and alpha... do you know the relationship between velocity and angular velocity?

 In this problem, the mass, radius, and torque are unknown. The force working on the cylinder I believe is just gravity. E and F) I am not sure how to do these. I know this is a lengthy problem, but any help of guidance would be appreciately greatly! Even if ou don't know - I would settle for just a word of encouragement.
Once you write your equations in parts B, C, and D... just solve them to get the answers to E and F.
 P: 68 Thanks. I made some changes and now I have: A) My free body diagram has two tension forces up and weight pointed down. B) Newton's 2L looks like 2T-W=ma C) net torque = I*alpha I=1/2mv^2 for a cylinder so: net torque = 1/2mv^2*alpha. I am not certain about the left side. Is net torque equal to zero? I am not sure of the relationship between linear and angular velocity. My guess is that the radius of the cylinder will roll as it is unwound from the rope in such a way that the rotations per second will relate to the speed of the fall?
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P: 4,124
Linear and angular acceleration of a falling cylinder? [long]

 Quote by Bob Loblaw Thanks. I made some changes and now I have: A) My free body diagram has two tension forces up and weight pointed down. B) Newton's 2L looks like 2T-W=ma
yes, exactly.

 C) net torque = I*alpha I=1/2mv^2 for a cylinder so: net torque = 1/2mv^2*alpha.
you mean:

net torque = 1/2mr^2*alpha. (not v but r).

 I am not certain about the left side. Is net torque equal to zero?
What is the definition of torque? How do the tensions in the cords relate to the torque?

 I am not sure of the relationship between linear and angular velocity. My guess is that the radius of the cylinder will roll as it is unwound from the rope in such a way that the rotations per second will relate to the speed of the fall?
if an object rolls without slipping... like a tire on a road... then:

v = rw

that means that

a = r*alpha
 P: 68 By definition, torque=force x radius. The relationship between tension and torque isn't clear to me right now.
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P: 4,124
 Quote by Bob Loblaw By definition, torque=force x radius. The relationship between tension and torque isn't clear to me right now.
yes, that's the definition. the way I think of it is... torque is

force*(perpendicular distance from the axis to the line of the force)

The force is T. Think of the axis of the cylinder... and the line of the force... draw a line segment going from the axis... to the line of the force(this line segment is just a radius of the cylinder)... the length of that line segment is r...

So torque due to one rope is T*r. Torque due to both ropes is 2T*r.
 P: 68 Thanks, I see the problem a little better now. So: Newtons 2L: 2T-W=ma Rotational form: 2T*r=1/2mr^2*alpha The relationship between the two forms: a = r*alpha So we want to solve for linear acceleration. Would we take r as (1/2mr^2*alpha)/2T? If so, that would still leave me to deal with another unknown, alpha. Thanks for your help so far. I feel fortunate that you were able to assist me.
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P: 4,124
 Quote by Bob Loblaw Thanks, I see the problem a little better now. So: Newtons 2L: 2T-W=ma Rotational form: 2T*r=1/2mr^2*alpha The relationship between the two forms: a = r*alpha So we want to solve for linear acceleration. Would we take r as (1/2mr^2*alpha)/2T? If so, that would still leave me to deal with another unknown, alpha. Thanks for your help so far. I feel fortunate that you were able to assist me.
You can cancel 1 r in:

2T*r=1/2mr^2*alpha

leaving

2T=1/2mr*alpha

so you have the 3 equations:

2T-mg=ma
2T=1/2mr*alpha
a = r*alpha

trying playing with these 3 equations... you should be able to solve for a, alpha and T in terms of m, g and r... try to go from 3 equations in 3 unknowns to 2 equations in 2 unknowns...
 P: 68 I came up with 2r*alpha=a and mr*alpha=T Is it even possible to have non-symbolic answers for this problem?
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P: 4,124
 Quote by Bob Loblaw I came up with 2r*alpha=a and mr*alpha=T Is it even possible to have non-symbolic answers for this problem?
No. you'll have to give your answer in terms of r, m and g. But your equations don't look right...

I blundered... for the first equation we should use:

mg - 2T = ma (1)... since the object is going downwards...
2T=1/2mr*alpha (2)
a = r*alpha (3)

plug in a from (3) into (1). that gives:

mg - 2T = mr*alpha ... solve this along with (2):
2T=1/2mr*alpha

solve the above 2 equations... you should be able to get T and alpha in terms of m, r and g...
 P: 68 Solving for T in equation 2 gives me: T=mr*alpha Putting that into the equation derived from plugging (3) into (1) mg-2mr*alpha=mr*alpha that leaves me with mg=3 or m9.81=3 or .31kg Am I on the right track? The problem asks for linear acceleration of the cylinder as well as tension though I am not sure if I am any closer to finding a real solution. EDIT: Fixed bad equation and got new results.
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 Quote by Bob Loblaw Solving for T in equation 2 gives me: T=mr*alpha Putting that into the equation derived from plugging (3) into (1) 2mr*alpha-mg=mr*alpha that leaves me with 2-mg=1 which is mg=1. Since g is 9.81, m is 0.10kg. Am I on the right track?
I messed up on the first equation... I modified my previous post now...

equation 2 is:

2T = 1/2mr*alpha
gives
T = 1/4 mr*alpha

plugging (3) into (1) (this is the equation I changed)

gives:

mg - 2T = mr*alpha

then plugging in the T gives:

mg - 2(1/4 mr*alpha) = mr*alpha

we can cancel m's

g - (1/2)r*alpha = r*alpha

gives (3/2)r*alpha = g

solving for alpha gives alpha = 2g/(3r)

plugging that into T = 1/4 mr*alpha

gives T = (1/4)mr*(2g)/(3r)
so T= (1/6)mg

and a = r*alpha = r*2g/(3r)
so a= 2g/3

check over this yourself... I could have blundered... look over my last post with the 3 equations again... try going through solving it and see if you get the same answers...
 P: 68 I blundered on my algebra as well. I should have: mg-2T=mr*alpha T=1/4mr*alpha mg-2(1/4)mr*alpha=mr*alpha cancelling the mr*alpha leaves me: mg=1/2 9.81m=1/2 m=.05kg
 P: 68 I came out with the same answers (once I knew what to do) so in real numbers a=2(9.81)/3=6.54 T=(1/6)mg=(1/6)(1/2)=1/12N Is this beast finally slain?
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 Quote by Bob Loblaw I blundered on my algebra as well. I should have: mg-2T=mr*alpha T=1/4mr*alpha mg-2(1/4)mr*alpha=mr*alpha cancelling the mr*alpha leaves me: mg=1/2 9.81m=1/2 m=.05kg
you can't cancel mr*alpha (because of the mg)... we have to divide all the terms by the same thing:

mg-2(1/4)mr*alpha=mr*alpha

if you divide each term by mr*alpha, you get

mg/(mr*alpha) -1/2 = 1
g/(r*alpha) - 1/2 = 1
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P: 4,124
 Quote by Bob Loblaw I came out with the same answers (once I knew what to do) so in real numbers a=2(9.81)/3=6.54 T=(1/6)mg=(1/6)(1/2)=1/12N Is this beast finally slain?
the mg = 1/2 isn't right.
 P: 68 You're right about my algebra being bad. So now I cannot see how mass would be solvable. If that is the case I won't get a real numerical answer. Is that normal for these types of problems?
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 Quote by Bob Loblaw You're right about my algebra being bad. So now I cannot see how mass would be solvable. If that is the case I won't get a real numerical answer. Is that normal for these types of problems?
Yeah, it is definitely normal. The idea is to solve in terms of the variables given... you're given a mass of M and radius R (we used small letters but no matter)... so your answers have to be in terms of those variables... and using constants like g is fine.

so:

alpha = 2g/(3R)
T= (1/6)Mg
so a= 2g/3