A mass attached to a cord around a disk

[disque]

Homework Statement

Imagine a frictionless pulley (a solid cylinder) of unknown mass M and radius r = 0.200 m which is used to draw water from a well. A bucket of mass m = 1.50 kg is attached to a massless cord wrapped around the pulley. The bucket starts from rest at the top of the well and falls for t = 3.00 s before hitting the water h = 9.99 m below the top of the well.

What is the tension in the cord?
What is the torque that is applied to the pulley due to the cord?

Homework Equations

Torque=force*radius
Tension=mass*g?

The Attempt at a Solution

 

[tiny-tim]

Hi disque!

Torque=force*radius

Yes

Tension=mass*g?

No … use conservation of energy to find the speed, differentiate for the acceleration, then use good ol' Newton's second law to find the tension

 

[nlightner]

To calculate the tension in the cord, we can use the equation T = ma, where T is the tension, m is the mass of the bucket, and a is the acceleration of the bucket. Using Newton's second law, we know that the net force on the bucket is equal to its mass multiplied by its acceleration. In this case, the net force is equal to the weight of the bucket (mg) minus the tension in the cord, since the tension is acting in the opposite direction of motion. Therefore, we can write the equation as mg - T = ma. Solving for T, we get T = mg - ma = m(g-a). Plugging in the given values, we get T = (1.50 kg)(9.8 m/s^2 - 9.8 m/s^2) = 14.7 N. This is the tension in the cord.

To calculate the torque applied to the pulley due to the cord, we can use the equation torque = force * radius. In this case, the force acting on the pulley is the tension in the cord, and the radius is the radius of the pulley. Therefore, the torque can be written as T = (14.7 N)(0.200 m) = 2.94 Nm. This is the torque applied to the pulley due to the cord.

It is important to note that this solution assumes a few things, such as the pulley being frictionless and the mass of the cord being negligible. In reality, there may be some friction present and the mass of the cord may affect the calculations slightly. However, these assumptions do not significantly impact the overall solution.