Pulley Problem: Tension on one side, Weight on another

[frostedpoptar]

Homework Statement

problem #62

Homework Equations

Torque = F * radius
Tangential accel = radius * alpha
w = mg
Kinematic equations for linear motion

The Attempt at a Solution

I tried to find the net torque and go from there.
For clockwise torque I got T = 10 * .25 = 2.5 Nm
For ccw torque I got T = 1.5g * .25 = 3.675 Nm
For net torque I then get ccwT - cwT = 1.175 Nm

I used that with Torque = I * alpha to find alpha.
1.175 = .5 * m * r^2 * alpha
1.175 = .0625 * alpha
18.8 = alpha

Then to get tangential accel I did

Accel = .25 * 18.8
accel = 4.7 m/s^2

Plug that into kinematics

.3 = .5 * 4.7 * t^2
.357 s = t

This is wrong however, the answer is .56 s.

Can anyone tell me where I went wrong? Thanks.

 

[frostedpoptar]

I also tried to do F = ma and in turn, 4.7 = 1.5a

I then substituted that a into kinematics and got something like .43 s. Still wrong.

 

[PeterO]

I also tried to do F = ma and in turn, 4.7 = 1.5a

I then substituted that a into kinematics and got something like .43 s. Still wrong.

You are also accelerating the pully - we commonly use mass-less, frictionless pulleys in this type of problem - so it will take longer.

 

[frostedpoptar]

Then how do I go about doing this?

 

[Houdini176]

The counterclockwise tension equals T*r, where T=tension in the string and r is the radius of the pulley. Because the mass is accelerating downward, you cannot say that tension on the string on the "left side" of the pulley is equal to the mass. If that were true, the mass wouldn't accelerate. This is where you went wrong. Instead, you have to write a system of equations and solve for the acceleration and, subsequently, time.

The equations you write should be for the net torque (in terms of tensions) on the pulley and the net force on the block. The tension on the right side is 10N, so you only have two unknowns, A and T1 (If T1 is the tension of the left string) and two equations. Remember that the linear acceleration of the pulley equals the linear acceleration of the weight. After you use the dynamics equations for acceleration, use the kinematics. I got 0.56 seconds for my answer this way.

 

[frostedpoptar]

Houdini, thanks so much for the reply.

I see what you're getting at.
It's probably just me but I can't seem to apply it.

Torque cw = T * .25m
Torque ccw = mg - T * .25m?

I'm probably making the same mistake you mentioned in your first paragraph though.

Can you please elaborate?

It's greatly appreciated. Physics Final tomorrow :)

 

[Houdini176]

Houdini, thanks so much for the reply.

I see what you're getting at.
It's probably just me but I can't seem to apply it.

Torque cw = T * .25m
Torque ccw = mg - T * .25m?

I'm probably making the same mistake you mentioned in your first paragraph though.

Can you please elaborate?

It's greatly appreciated. Physics Final tomorrow :)

Okay, well, remember to just take it one step at a time. It looks like you've make the force and the torque the same thing, but you should leave them apart and find one equation for torque and one for force.

Torque_net=T1*r-T2*r=T1*r-10*r

So all I've done here is taken the two torques and subtracted them. You don't know T1 yet, but you do know it's relationship to the net torque, and the net torque's relationship to the acceleration. And T2 is just the 10 Newtons

T1*r-10*r=Ia_angular=Ia/r

Divide by r

T1-10=Ia/r^2=(1/2Mr^2*a)/r^2=(ma/2)

So now you have a pretty easy equation for A and T1 (T1-10=ma/2; this is the "torques equation", but after our manipulation it's more of a "tensions equation" because T1 is just the tension on the string. I'll still refer to it as the "torques equation". Remember that the torque=tension*radius, or t=T*r)

Now you need the pretty simple equation for the forces

Force_net=-m_blocka=T1-m_blockg

This follows from the tension pulling up on the block (Which you have never known), and the block's weight pulling down. The result of adding them is the sought after acceleration of the block times it's mass, -m_blocka.

You'll notice that A in the force equation and the A in the torques equation should be the same, now just solve for T1 (Easier from the torque equation), substitute it into the other equation, and go to town. And no problem, I hope that helps.

You should get a=(m_block*g-10)/(m_block+I/r^2),

 

[frostedpoptar]

Just one thing I'm confused about in your explanation now

In "T1-10=Ia/r^2=(1/2Mr^2*a)/r^2=(ma/2)" how did you get ma/2?

When I simplify this I get

T1-10 = ((1/2)(M)(r^2)(a))/(r^2)
T1 - 10 = ((1/2)(2)(r^2)(a))/(r^2)
1/2 cancels the 2
T1 - 10 = ((r^2)(a))/(r^2)
r^2's cancel
T1-10 = a

Where did I go wrong?

 

[frostedpoptar]

Oh nevermind I see what you did. You just didnt substitute the 2 for m.

 

[Houdini176]

Heh okay cool, I hope I answered your question. Good luck on your final!

 

[frostedpoptar]

OK, I went through your entire method and the answer I got was .43 s.

The correct answer however is .56 s. Did I do something wrong, or do you get the same answer as well?

Thanks for the continued support.

 

[Houdini176]

No problem, I'll do it all out right here

M=Mass of wheel m=mass of block/weight

Tension/Torque Equation

T1-T2=Ia_angular/r
T1-T2=Ia_linear/r^2
T1-T2=Ma/2
T1=Ma/2+T2 (T2=10N, I'll leave it as a variable)

Force Equation

T1-mg=-ma
T1+ma=mg

Here you combine the two, substitute in T1=Ma/2-T2

Ma/2+T2+ma=mg
Ma/2+ma=mg-T2
a(M/2+m)=mg-T2
a=(mg-T2)/(M/2+m)
a=(1.5*9.8-10)/(2/2+1.5)=4.7/2.5=1.88 m/s^2, or 188 cm/s^2

(188/2)t^2=30
t=sqrt(60/188)=0.56

 

[frostedpoptar]

Thanks so much! After looking at what you did I worked it out on my own and it all came out perfectly!

I think this was the only problem I had a problem on in preparation for the final so if I get a 100 it goes out to you!

Thank you a million times! :)

 

[Houdini176]

Sweet, good to hear. Good luck on your final, and it was my pleasure.