Rotation, torque, angular momentum?

[kjn11994]

Homework Statement

A cloth tape is wound around the outside of a uniform solid cylinder (mass M, radius R) and fastened to the ceiling as shown in the diagram below. They cylinder is held with the tape vertical and then released from rest. as the cylinder descends, it unwinds from the tape without slipping. The moment of inertia of a uniform solid cylinder about its center is 1/2MR^2.

diagram: http://imgur.com/h0v07

A. draw and label a diagram of the cylinder showing all the forces acting on the cylinder after it is released. label each force clearly.

B. in terms of g, find the downward acceleration of the center of the cylinder as it unrolls from the tape

C. what is the angular velocity of the cylinder after it has descended 3 meters?

D. while descending, does the center of the cylinder move toward the left, toward the right, or straight down? Explain.

Homework Equations

KEr=1/2MR^2

The Attempt at a Solution

i have no clue how to solve this

 

[vela]

You need to show us you made some effort before you can get help here. Did you at least do part A?

 

[kjn11994]

i honestly have no idea where to start...

 

[vela]

Then I would suggest you go back and read your text, particularly the examples. Identifying the forces is pretty basic. You should have covered it in class weeks ago. You can open a new thread after you've made some sort of attempt at solving the problem on your own.

 

[asteriatic]

problem



I can help guide you through this problem by explaining the concepts of rotation, torque, and angular momentum.

Rotation refers to the motion of an object around an axis. In this problem, the cylinder is rotating around its center as it descends.

Torque is a measure of the force that causes an object to rotate. In this case, the force of gravity is causing the cylinder to rotate as it descends.

Angular momentum is a measure of an object's tendency to continue rotating. In this problem, the angular momentum of the cylinder remains constant as it descends, since there are no external forces acting on it to change its rotation.

Now, let's move on to solving the problem. First, we need to draw a free body diagram of the forces acting on the cylinder after it is released. These forces include the weight of the cylinder (mg), the tension in the tape (T), and the normal force from the ceiling (N).

A. The diagram should show the weight acting downwards at the center of the cylinder, the tension acting upwards at the point where the tape is attached to the cylinder, and the normal force acting upwards at the contact point between the ceiling and the cylinder.

B. To find the downward acceleration of the center of the cylinder, we can use the equation F=ma, where F is the net force acting on the cylinder and a is the acceleration. The net force in this case is the weight of the cylinder minus the tension in the tape, since they are acting in opposite directions. So we have:

mg - T = ma

Solving for a, we get a = (g - T/m). Since we know the moment of inertia (1/2MR^2) and the radius (R), we can also express the tension in terms of the angular acceleration (α) using the equation τ = Iα, where τ is the torque and I is the moment of inertia. So we have:

T = 1/2MR^2 * α

Substituting this into our previous equation, we get:

a = (g - 1/2MR^2 * α)/m

C. To find the angular velocity of the cylinder after it has descended 3 meters, we can use the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity (which is 0 in this case), a