# Mass on Pulley problem need to find angular acceleration

1. Oct 14, 2013

### nickb145

1. The problem statement, all variables and given/known data
A string is wrapped around a pulley of radius 3.45cm , and a weight hangs from the other end. The weight falls with a constant acceleration 3.00m/s2 .

What's the angular acceleration of the pulley?
2. Relevant equations
i'm writing out the equations i would think that are involved.

a=Δw/Δt
Centripital force Fr=mv^2/R ( i think)
Newtons second law
ω=ωi+at

3. The attempt at a solution
I am just so stuck...and don't know how to start.
I drew out a free body diagram and Drew out the forces for the mass and pulley

For the mass all i have is the Tension and mg. there is no weight for the pully so all i have is tension

I would need to sum the forces right?

2. Oct 14, 2013

### CAF123

Consider a portion of the string wrapped around the pulley. When the mass falls a distance x, what is the corresponding angular displacement of the pulley?

3. Oct 14, 2013

### nickb145

that would be 2∏

4. Oct 14, 2013

### NihalSh

No, that's not right. $2π$ means whatever distance $x$ hanging mass falls, the pulley would complete one revolution, that doesn't sound right, does it???............ Think again.

$$Δθ=\frac{x}{R}$$

all need to do from here on is to differentiate it twice, then you'll have the relation between angular and linear acceleration.

5. Oct 14, 2013

### CAF123

Yes, correct.

6. Oct 14, 2013

### nickb145

Akk.. I didn't read that correctly

7. Oct 14, 2013

### nickb145

SO if i understand this To get the angular displacement i need to take the constant acceleration and divide it by radius (in meteres .0345) then i can find angular acceleration?

Sorry for the stupid questions :(

8. Oct 15, 2013

### NihalSh

Yes!!!

$$a=αR$$

9. Oct 15, 2013

### nickb145

Ok!

So according to my book Angular acceleration is

α= Δω/ΔT I would think that wouldn't be right because i don't know the time.

but aside from my own thoughts and use a=αR is R the Angular displacement? If so, multiply it by the acceleration. then convert it to rad/s^2

10. Oct 15, 2013

### nickb145

that makes no sense, hold on let me think it out. i need to find the angular velocity i think...but how do i find the time...

Last edited: Oct 15, 2013
11. Oct 15, 2013

### NihalSh

Your book is perfectly right. R is, in this case, radius of pulley.

$$ω=\frac{v}{R}$$
$$α= \frac{Δω}{ΔT}=\frac{1}{R}*\frac{Δv}{ΔT}=lim_{ΔT→0} (\frac{1}{R}*\frac{Δv}{ΔT})=\frac{1}{R}*(lim_{ΔT→0} (\frac{Δv}{ΔT}))=\frac{a}{R}$$

Your question asked you to find angular acceleration, there is no mention of angular velocity!!!!

12. Oct 15, 2013

### nickb145

you're right. just thinking 'out loud' i guess.

sorry I'm kind of confused by this: could you explain it? My mind will probably kick into gear
$$α= \frac{Δω}{ΔT}=\frac{1}{R}*\frac{Δv}{ΔT}=lim_{ΔT→0} (\frac{1}{R}*\frac{Δv}{ΔT})=\frac{1}{R}*(lim_{ΔT→0} (\frac{Δv}{ΔT}))=\frac{a}{R}$$

13. Oct 15, 2013

### NihalSh

look in the original post I mentioned, $θ=\frac{x}{R}$.....this is the definition of angle in radians. $x$ is the length of arc of the given circle. Like I said previously, differentiating it twice would give you the required relation $a=αR$. You could view the above calculations as following:
$$Δθ=\frac{Δx}{R}$$
$$ω_{average}=\frac{Δθ}{Δt}=\frac{1}{R}*\frac{Δx}{Δt}$$
$$ω_{instantaneous}=lim_{Δt→0} (\frac{Δθ}{Δt})=lim_{Δt→0} (\frac{1}{R}*\frac{Δx}{Δt})=\frac{1}{R}*(lim_{Δt→0} (\frac{Δx}{Δt}))=\frac{dθ}{dt}=\frac{1}{R}*\frac{dx}{dt}=\frac{v}{R}$$
$$Δω=\frac{Δv}{R}$$
$$α_{average}= \frac{Δω}{Δt}=\frac{1}{R}*\frac{Δv}{Δt}$$
$$α_{instantaneous}= lim_{Δt→0} (\frac{Δω}{Δt})=lim_{Δt→0} (\frac{1}{R}*\frac{Δv}{Δt})=\frac{1}{R}*(lim_{Δt→0} (\frac{Δv}{Δt}))=\frac{dω}{dt}=\frac{1}{R}*\frac{dv}{dt}=\frac{a}{R}$$

$lim$ are used to define derivatives. That's all there is to it. I have showed it in both notations, first with limits and then slightly on the right with $\frac{d}{dt}$ notation. I am assuming you have studied basic calculus, otherwise I wouldn't be talking about limits and derivatives.

If you could be a little bit more clear maybe I could clear your doubt, what is it that confuses you about these equations???

Last edited: Oct 15, 2013
14. Oct 15, 2013

### nickb145

Ak sorry! But yes i was thinking of the other question that my HW was asking me. Which in this case was asking for the angular velocity of the pully when the Mass was 1.3m above the ground.

But i have the angular acceleration, and thank you for the assistance!
But i'm still trying to figure out the angular velocity

15. Oct 15, 2013

### nickb145

EDIT: I think i found the time 1.3=gt2/2 which is t=.5150787
now i plug it into ω=(angular acceleration) x Time and i get 44.81 but it says that it isn't right.

suggestions?

16. Oct 16, 2013

### NihalSh

how did you find time? and time for what?........I think you haven't mentioned some remaining part of the question. Because according to the information in the original post, there is no way and no need to find time. and even if there is finding time for what????

17. Oct 16, 2013

### nickb145

I needed to find angular velocity. But i did figure it out

Last edited: Oct 16, 2013
18. Oct 16, 2013

### NihalSh

It's always better to mention the whole question for completeness. I can't help any further in solving for angular velocity, without knowing the complete question. But since you have figured it out, its no problem.

19. Oct 16, 2013

### nickb145

No problem, thank you for the help though.

I sort of mentioned the second question as "Which in this case was asking for the angular velocity of the pully when the Mass was 1.3m above the ground."

20. Oct 16, 2013

### NihalSh

There should be initial height given to solve for that.