# Rotational Motion problem

1. Mar 20, 2007

### xaer04

1. The problem statement, all variables and given/known data

From the text: "A $50 kg$ mass is tied to a massless rope wrapped around a solid cylindrical drum. The drum is mounted on a frictionless horizontal axle. When the mass is released, it falls with acceleration $a = 3.7 \frac{m}{s^2}$. Find (a) the tension in the rope and (b) the mass of the drum.

Mass of the falling object = $50 kg$
Acceleration of gravity = $9.8 \frac{m}{s^2}$
Net acceleration of falling object = $3.7 \frac{m}{s^2}$

2. Relevant equations

Newton's 2nd Law
$$\sum \vec{F} = m \vec{a}$$

Definiton of torque in terms of radius and Force applied
$$\tau = r \vec{F} \sin (\theta)$$

Definition of torque in terms of rotational intertia and rotational acceleration
$$\tau = I \alpha$$

Definition of rotational acceleration in terms of radius and tangential acceleration
$$\alpha = \frac{a}{r}$$

Rotational intertia for a cylinder
$$I = \frac{1}{2} MR^2$$

3. The attempt at a solution

I know that $a_\textit{gravity} - a_\textit{rope} = a_\textit{final}$ for the falling block. this gave me an upward acceleration of $6.1 \frac{m}{s^2}$, which gave me my first answer:
$$T = -305N$$

For my second answer, i set the definitons of torque in terms of quantities that i knew and the one quantity that i didn't know - mass of the falling object, mass of the drum, acceleration of the falling object, force of the falling object.

My equation looked like this before i started cancelling out things:

$$(\frac{1}{2}Mr^2) (\frac {a}{r}) = (rma)$$

The radius values and accelerations cancelled out, leaving me with this:

$$M = 2m$$

This can't be right because the mass of the larger object can't be entirely dependant on the mass of the smaller object. Could someone help me out?

Last edited: Mar 20, 2007
2. Mar 20, 2007

### e(ho0n3

I have not checked the numbers. However, I don't understand how you conclude that the difference between the acceleration due to gravity and the rope is equal to the final acceleration. I suggest drawing a force-body diagram. You should quickly determine what the tension is from it.

Why can't that be? It is perfectly possible for the mass of the drum to be twice that of the falling mass.

3. Mar 20, 2007

### xaer04

sorry about that. i attached a free body diagram. also, i posted the formula incorrectly, but the posted answer i obtained like this.

$$9.8 + a_\textit{rope} = 3.7$$
$$a_\textit{rope} = -6.1$$
$$T = -305N$$

as for my question - I predicted that the mass of the drum would be larger, but i suppose I was looking for a more complicated equation. Maybe I'm having an issue grasping that the acceleration and radius cancels out completely in my formula, because it doesn't seem plausible.

#### Attached Files:

• ###### Free Body Diagram1.bmp
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Last edited: Mar 20, 2007
4. Mar 20, 2007

### Staff: Mentor

Where does this come from? Instead, write the force equation for the hanging mass per Newton's 2nd law.

5. Mar 20, 2007

### xaer04

ok, ok, lol... i'll write the force equation.

$$\sum \vec{F} = m \vec{a}$$
$$\sum \vec{F} = \vec{F}_\textit{gravity} + \vec{F}_\textit{rope}$$
$$\sum \vec{F} - \vec{F}_\textit{gravity} = \vec{F}_\textit{rope}$$
$$185N - 490N = -305N$$

I opted to leave mass out and work with acceleration in the beginning for easier calculating... that didn't account for my sign error in my equation though, i just copied it wrong... unless i'm calculating for the tension incorrectly????:(

6. Mar 20, 2007

### Staff: Mentor

Much better. (That acceleration equation made no sense on its own.)

You are messing up the directions of the forces (and thus their signs), that's why you get a negative answer. The forces are gravity and rope tension: which way do they act? What's the direction of the acceleration? Hint: Use a consistent sign convention.

Your expression for torque--the right hand side of the equation--is incorrect.

7. Mar 20, 2007

### xaer04

alright... i'll have upward be positive, giving me -185 + 490 = 305 N. I don't know why i had it backwards, to be honest.

really? Are you talking about this: $\tau = r \vec{F} \sin(\theta)$? I changed F to ma, and i see the point that the rope begins to hang freely as making a horizontal line with the radius, making the Force of the torque being perpendicular: so
$$\sin(\frac{\pi}{2}) = 1$$
leaving me with r*m*a.

8. Mar 20, 2007

### Staff: Mentor

Why? The F you need is the force exerting the torque on the drum--what's that? "ma" would be the net force on the mass--not relevant.

9. Mar 20, 2007

### e(ho0n3

Would the tension on the rope be the force exerting the torque on the drum?

10. Mar 21, 2007

### Staff: Mentor

Of course. It's the rope that pulls on the drum.

11. Mar 21, 2007

### xaer04

ok... so the F on the right side of my equation is 305N, and then a on the left side is 3.7m/s^2, the net acceleration of the closed system. therefore, my final equation should be like this:

$$M = \frac{2F_\textit{rope}}{a_\textit{net}}$$

*calculating*

about 165kg. that makes a LOT more sense, heh:) So my equation was almost right, i just misunderstood what variables meant what. thank you very much, Doc Al and e(ho0n3 :) Thanks to you... i have all week to do other homework.