Torque on shaft of pulley question

[pjb73]

Hi

Here is my question:
A solid disc pulley with a 5kg mass, radius of 0.5m has one layer of thin rope coiled around it. On the other end of the rope is a metal block with a mass of 10kg.
Calculate the following (without take rope friction into account. also the rope does not slip):
1. The torque on the pulleys shaft.

To solve this I did the following:
Torque = Force x Radius
Torque = 10kg x 0.5m
Torque on pulleys shaft is = 5Nm

Would you please let me know if this is correct?

Thanks.

 

[Doc Al]

(1) 10 km is a mass, not a force.
(2) What is the tension in the rope?

 

[pjb73]

The tension in the rope is mass x gravity = 10 x 9.8 = 98N

So my answer should be:
Torque = Force x Radius
Torque = (10kg x 9.8) x 0.5m
Torque on pulleys shaft is = 49Nm

Is this correct?

Thanks

 

[Doc Al]

The tension in the rope is mass x gravity = 10 x 9.8 = 98N

That's the weight of the hanging mass, not necessarily the tension in the rope.

Assuming the pulley is free to rotate, there will be acceleration. You'll have to figure out the tension by applying Newton's 2nd law to both the pulley and the hanging mass.

 

[pjb73]

To calculate acceleration 1st calculate pulley inertia:

pulley inertia(I) = 1/2mr2 (2 represents squared)
I = 1/2 x 5 x 0.25
I = 0.625

acceleration(a) = hanging mass weight divided by mass of hanging mass plus pulleys inertia divided by radius squared.
a = 98 / (10 + (0.625 / 0.25))
a = 98 / (10 + 2.5)
a = 7.84 m/s

I think the above is the correct acceleration. I don't know how to calculate the tension on the rope, please would you give me another hint?

thanks

 

[haruspex]

I think the above is the correct acceleration. I don't know how to calculate the tension on the rope, please would you give me another hint?

The standard way to solve a problem like this is to introduce an unknown, T, for the tension, another, a, for the linear acceleration, then analyse each of the masses wrt forces/torques/accelerations separately. That would give you a pair of equations from which you could deduce a and T.
You've finessed that by treating the rotational inertia / radius-squared as equivalent to a mass. That works. Having done that, it's easy to get the tension by considering just the hanging mass. There are two forces on it, you know one of them, you know the mass, and you know the acceleration.

 

[pjb73]

Thanks for your reply haruspex

So f = ma
f = 10 x 7.84
f = 78.4N

So the Torque on the pulley shaft - T = fr
T = 78.4N x 0.5m
T = 39.2Nm

Is this correct?

Thanks.

 

[haruspex]

So f = ma
f = 10 x 7.84
f = 78.4N

What exactly does f mean here?

 

[pjb73]

f = force

Is that not correct?

 

[haruspex]

f = force

Is that not correct?

Yes, but which force?

 

[pjb73]

I was thinking the for I was calculating was the tension force on the rope.

 

[Doc Al]

I was thinking the for I was calculating was the tension force on the rope.

When you apply Newton's 2nd law, ƩF = ma, you must decide what system or object you are analyzing, in this case the hanging mass. You then must identify all the forces acting on that mass. The rope tension is just one of the forces acting on the mass.

To really learn how to solve this and similar problems, do not rely on "canned" formulas, like the one you used for acceleration in post #5. Instead, derive your own formulas by applying Newton's 2nd law to both the pulley and the hanging mass. (Just like haruspex has advised.)

Also, do not create multiple threads on the same problem! https://www.physicsforums.com/showthread.php?t=702575