Torque and Angular Acceleration for a Rigid Body Problem

[Yosty22]

Homework Statement

A 12.0-kg box resting on a horizontal, frictionless surface is attached to a 5.00-kg weight by a thin, light wire that passes over a frictionless pulley. The pulley has the shape of a uniform solid disk of mass 2.00kg and diameter 0.500m. After the system is released, find (a) The tension in the wire on both sides of the pulley, (b)the acceleartion of the box, and (c)the horizontal and vertical components of the force that the axle exerts on the pulley.

Homework Equations

Torque=R (cross) F = I \alpha
Newton's Laws

The Attempt at a Solution

I started with free body diagrams and writing out equations. For mass 1, the 12 kg box on the table, the only forces acting in the x direction are T₁. On the hanging mass, m₂, the only forces are the weight of the hanging mass and T₂. I tried to figure out the torque about the pulley, using τ=I \alpha. I=.5mr² and \alpha = a/r, so Torque = .5mr²*(a/r). However, I don't really know what to do from there. I figure both of the tensions create a torque, but I'm not sure what I need to do from here.

Any help would be greatly appreciated

 

[tiny-tim]

Hi Yosty22!

I figure both of the tensions create a torque …

That's correct.

Find the torque from each tension …

their difference is the total torque that you've just found.

 

[Yosty22]

Oh, duh. Thanks a lot, tiny-tim. Much appreciated. :)

 

[haruspex]

Fwiw, there are a couple of important details missing from the problem statement:
- That the wire is horizontal from the 12kg block to the pulley.
- That the pulley is only frictionless in relation to its axle; in relation to the wire it has unlimited(?) friction.

 

[Nickie]

.

I would suggest using the equations provided in the problem and applying them to the given scenario. First, you can use Newton's Second Law, F=ma, to find the acceleration of the system. Since the system is released, it will experience a net force in the direction of the weight of the hanging mass, which can be calculated using F=mg. This force will cause the system to accelerate in the same direction.

Next, you can use the equations for torque, τ=Iα and τ=rF, to find the tension in the wire and the force exerted on the pulley by the axle. The torque created by the weight of the hanging mass can be calculated using τ=mgr, and the torque created by the tension in the wire can be calculated using τ=Tr, where T is the tension in the wire.

To find the tension in the wire, you can set the two torque equations equal to each other and solve for T. This will give you the tension on both sides of the pulley. To find the horizontal and vertical components of the force exerted by the axle, you can use trigonometric functions to break down the force into its components.

Overall, it is important to carefully analyze the forces and torques acting on the system and use the appropriate equations to solve for the unknown quantities. It may also be helpful to draw a free body diagram for each object in the system to visualize the forces and torques.