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A uniform horizontal disk of radius 5.50 m turns without friction at = 2.30 rev/s on a vertical axis through its center, as in the figure below. A feedback mechanism senses the angular speed of the disk, and a drive motor at A ensures that the angular speed remain constant while a m = 1.20 kg block on top of the disk slides outward in a radial slot. The block starts at the center of the disk at time t = 0 and moves outward with constant speed v = 1.25 cm/s relative to the disk until it reaches the edge at t = 465 s. The sliding block experiences no friction. Its motion is constrained to have constant radial speed by a brake at B, producing tension in a light string tied to the block.

A uniform horizontal disk of radius 5.50 m turns without friction at = 2.30 rev/s on a vertical axis through its center, as in the figure below. A feedback mechanism senses the angular speed of the disk, and a drive motor at A ensures that the angular speed remain constant while a m = 1.20 kg block on top of the disk slides outward in a radial slot. The block starts at the center of the disk at time t = 0 and moves outward with constant speed v = 1.25 cm/s relative to the disk until it reaches the edge at t = 465 s. The sliding block experiences no friction. Its motion is constrained to have constant radial speed by a brake at B, producing tension in a light string tied to the block.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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### Rotational Motion of a Sliding Block on a Disk This educational exercise involves analyzing the motion of a block sliding outward on a spinning disk. The disk has a radius of 5.50 meters and rotates without friction at an angular speed of 2.30 revolutions per second. #### Diagram Description - The illustration shows a horizontal disk rotating about a vertical axis through its center. - A feedback mechanism maintains the angular velocity, and a drive motor at point \( A \) controls this speed. - A 1.20 kg block is situated on the disk, initially starting its movement from the center at time \( t = 0 \). - The block slides along a radial slot and moves outward at a constant speed of \( v = 1.25 \, \text{cm/s} \). #### Task Descriptions **(a)** **Torque Function:** Calculate the torque as a function of time that the drive motor must provide. Use the formula \( \tau = 2mr\omega \), where \( m \) is the block’s mass, \( r \) is the radial distance, and \( \omega \) is the angular speed. **(b)** **Torque at \( t = 465 \) s:** Determine the torque value just before the block reaches the edge of the disk. **(c)** **Power Function:** Identify the power output required from the drive motor as a function of time. **(d)** **Power at Slot's End:** Calculate the power when the block is about to reach the end of the slot. **(e)** **String Tension:** Find the tension in the light string as a function of time, which constrains the block's motion through a brake at \( B \). **(f)** **Work Done by Motor:** Compute the work done by the drive motor during the period of \( 465 \) seconds using \( W_{\text{motor}} = mv^2\omega^2t^2 \). **(g)** **Work Done by String Brake:** Analyze the work done by the string brake on the block, given by \( W_{\text{block}} = -\frac{1}{2}mv^2\omega^2t^2 \). **(h)** **Total Work on System:** Evaluate the total work done on the system, including both the disk and the block. This exercise involves concepts of rotational motion, torque, power, and energy
Transcribed Image Text:### Rotational Motion of a Sliding Block on a Disk This educational exercise involves analyzing the motion of a block sliding outward on a spinning disk. The disk has a radius of 5.50 meters and rotates without friction at an angular speed of 2.30 revolutions per second. #### Diagram Description - The illustration shows a horizontal disk rotating about a vertical axis through its center. - A feedback mechanism maintains the angular velocity, and a drive motor at point \( A \) controls this speed. - A 1.20 kg block is situated on the disk, initially starting its movement from the center at time \( t = 0 \). - The block slides along a radial slot and moves outward at a constant speed of \( v = 1.25 \, \text{cm/s} \). #### Task Descriptions **(a)** **Torque Function:** Calculate the torque as a function of time that the drive motor must provide. Use the formula \( \tau = 2mr\omega \), where \( m \) is the block’s mass, \( r \) is the radial distance, and \( \omega \) is the angular speed. **(b)** **Torque at \( t = 465 \) s:** Determine the torque value just before the block reaches the edge of the disk. **(c)** **Power Function:** Identify the power output required from the drive motor as a function of time. **(d)** **Power at Slot's End:** Calculate the power when the block is about to reach the end of the slot. **(e)** **String Tension:** Find the tension in the light string as a function of time, which constrains the block's motion through a brake at \( B \). **(f)** **Work Done by Motor:** Compute the work done by the drive motor during the period of \( 465 \) seconds using \( W_{\text{motor}} = mv^2\omega^2t^2 \). **(g)** **Work Done by String Brake:** Analyze the work done by the string brake on the block, given by \( W_{\text{block}} = -\frac{1}{2}mv^2\omega^2t^2 \). **(h)** **Total Work on System:** Evaluate the total work done on the system, including both the disk and the block. This exercise involves concepts of rotational motion, torque, power, and energy
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