# Rotational Dynamics and pulley

• J89
In summary: TM - 0.25 :wink: = 75a and Tm - 0.25 :wink: = -125athen use the last two equations to eliminate Tm, and solve for TM … and from that, find the force on the hook.In summary, to find the force that the ceiling exerts on the hook, you can use Newton's second law and the equations Tm + TM = 250, TM - 0.25 = 75a, and Tm - 0.25 = -125a, where Tm and TM represent the tension in the two sections of rope and a represents the linear acceleration. By solving for TM, you can then
J89

## Homework Statement

Two weights are connected by a very light flexible cord that passes over a 50 N frictionless pulley of radius .300 m. Pulley is a solid uniform disk and is supported by a hook connected to a ceiling. What force does the ceiling exert on the hook? There are two weights around each side of the disk, one is 75 N and the other is 125 N. Answer is 249 N.

I = 1/2 MR^2
F=ma

## The Attempt at a Solution

#### Attachments

• mg2.bmp
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J89 said:
Two weights are connected by a very light flexible cord that passes over a 50 N frictionless pulley of radius .300 m. Pulley is a solid uniform disk and is supported by a hook connected to a ceiling. What force does the ceiling exert on the hook? There are two weights around each side of the disk, one is 75 N and the other is 125 N. Answer is 249 N.

Hi J89!

(I assume that "frictionless pulley" means that there is friction between the pulley and the rope, with no slipping)

The tension in the two sections of rope will be different … call them Tm and TM, and call the linear acceleration a …

then apply good ol' Newton's second law three times, ie to each of the weights and the pulley (separately), to get Tm + TM

Based on the given information, we can use the equation F=ma to determine the acceleration of the system. Since the pulley is frictionless, the net force acting on the system is equal to the sum of the tensions in the cord on either side of the pulley. We can set up two equations, one for each weight, as follows:

For the 75 N weight:
F - T = ma

For the 125 N weight:
T - F = ma

Where F is the force exerted by the ceiling, T is the tension in the cord, m is the mass of each weight (assuming they are equal), and a is the acceleration of the system.

Since the weights are connected by the same cord, the tension on either side of the pulley must be equal. Therefore, we can set the two equations equal to each other and solve for F:

F - T = T - F
2F = 2T
F = T

Now, we can substitute this value of T into either of the original equations to solve for F:

F - T = ma
F - F = ma
0 = ma

Since the acceleration of the system is 0 (due to the weights being balanced by the tension in the cord), the force exerted by the ceiling, F, must also be 0. This means that the ceiling is not exerting any force on the hook.

In order for the system to have an acceleration of 0, the sum of the torques on the pulley must also be 0. The torque on the pulley is equal to the force applied (F) multiplied by the radius of the pulley (0.300 m). Since the force is 0, the torque must also be 0. This means that the force exerted by the ceiling is not necessary for the system to remain in equilibrium, and the pulley can support the weights on its own.

Therefore, the answer of 249 N is not correct. The ceiling does not exert any force on the hook, and the pulley can support the weights on its own due to its rotational dynamics.

## 1. What is rotational dynamics?

Rotational dynamics is the branch of physics that studies the motion of objects that rotate around an axis. It involves the study of torque, angular velocity, and angular acceleration.

## 2. How do pulleys work in rotational dynamics?

Pulleys are used in rotational dynamics to change the direction of a force and to increase or decrease the magnitude of a force. They also help to distribute the weight of an object evenly, making it easier to lift or move.

## 3. What is the difference between linear and rotational motion?

Linear motion is the movement of an object in a straight line, while rotational motion involves the movement of an object around an axis. In rotational motion, the distance from the axis to a point on the object determines the distance traveled, rather than the displacement in a straight line.

## 4. How is torque related to rotational dynamics?

Torque is a measure of the force that causes an object to rotate. It is directly related to the object's mass, the distance from the axis of rotation, and the angle between the force and the lever arm. In rotational dynamics, torque is used to calculate the angular acceleration of an object.

## 5. What are some real-world applications of rotational dynamics and pulleys?

Rotational dynamics and pulleys have many practical applications in everyday life. Some examples include the use of pulleys in elevators, cranes, and exercise equipment. Rotational dynamics is also essential in understanding the movement of objects such as satellites, planets, and spinning tops. It is also used in designing and optimizing machinery and mechanical systems.

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