Pulley question about a mass on a string

[slain4ever]

Homework Statement

i'm having a lot of trouble with this question.


A pulley, consisting of a 0.40 m diameter wheel mounted on a horizontal frictionless axle, is firmly attached to the ceiling. A light rope wrapped around the pulley supports a 0.50 kg object, as shown in the diagram below. Any motion of the rope causes the pulley to turn, with no slipping between the rope and pulley. When released from rest, the object accelerates downward at 5.0 m/s2.

1) find the tension in the rope.
2) the torque on the pulley from the rope
3) the angular acceleration of the pulley
4) the moment of inertia of the pulley.

Homework Equations

F=T+ma = 0

The Attempt at a Solution

I'm stuck on the first part i keep getting 2.5N since the tension = ma it should be T=5m/s^2 * 0.5 kg but the answers say 2.4N and I can't figure out why. I am assuming it has something to do with the pulley but since it is frictionless i don't see how it adds tension.

 

[PhanthomJay]

Relevant equations

F=T+ma = 0

The Attempt at a Solution

I'm stuck on the first part i keep getting 2.5N since the tension = ma it should be T=5m/s^2 * 0.5 kg but the answers say 2.4N and I can't figure out why. I am assuming it has something to do with the pulley but since it is frictionless i don't see how it adds tension.

Look at your relevant equation again,T does not = ma. Use your relevant equation by first identifying the value of F acting on the object.

 

[slain4ever]

f= ma
0.5 kg * 5 m/s^2
=2.5 N

 

[PhanthomJay]

looking at the forces acting on the object, its weight acts down and the tension acts up. Since the object is falling and accelerating down, the weight force must be greater than the tension force, thus, from Newton 2,
F_net = ma
mg -T =ma
Solve for T.
Note that tension forces always pull away from the objects on which they act, and that acceleration is always in the direction of the net force.

 

[asteriatic]

I would approach this question by first understanding the basic concepts involved. A pulley is a simple machine that can change the direction of a force and can also multiply the force applied. In this case, the pulley is attached to the ceiling and the object is suspended by a rope that is wrapped around the pulley. The force of gravity is acting on the object, causing it to accelerate downward at 5.0 m/s^2.

To find the tension in the rope, we can use Newton's second law, which states that the sum of all the forces acting on an object is equal to its mass times its acceleration. In this case, the object is not accelerating horizontally, so the sum of the forces in that direction is equal to zero. This means that the tension in the rope must be equal to the force of gravity acting on the object. Therefore, the tension in the rope can be calculated as T = mg = (0.50 kg)(9.8 m/s^2) = 4.9 N.

Next, we can calculate the torque on the pulley from the rope. Torque is the product of force and distance from the axis of rotation. In this case, the force acting on the pulley is the tension in the rope, and the distance from the axis of rotation is the radius of the pulley, which is 0.20 m. Therefore, the torque can be calculated as T = Fr = (4.9 N)(0.20 m) = 0.98 Nm.

To find the angular acceleration of the pulley, we can use the equation for rotational motion, which states that the net torque on an object is equal to its moment of inertia times its angular acceleration. In this case, the net torque is the torque from the rope, and the moment of inertia of a disk is equal to 1/2 * mass * radius^2. Therefore, the angular acceleration can be calculated as α = τ/I = (0.98 Nm)/(1/2 * 0.50 kg * (0.20 m)^2) = 19.6 rad/s^2.

Finally, we can calculate the moment of inertia of the pulley. As mentioned before, for a disk, the moment of inertia is equal to 1/2 * mass * radius^2. Therefore, the moment of inertia can be calculated as I = 1/2 * (