Question

yoyo is their central axes as shown. The two outer disks are identical, each with mass M, radius R, and moment of inertia 1/2MR. The central, smaller disk has mass M/2 and radius R/2. A light, flexible string of negligible mass is wrapped counterclockwise around the central disk of the yoyo. The yoyo is then placed on a horizontal tabletop and the string is gently pulled with a constant force F. The coefficient of friction between the yoyo and the tabletop is p. The tension in the string is not sufficient to cause the yoyo to leave the tabletop 1푤 Otheespertta.com 14% Part (a) Develop an expression for the moment of meta ICM about the central axis of the yoyo in temns of the given quantities and fundamental constants ICM-17/16 (MR) Correet ude e yo-yo with the table r-FR2 Correct p 1400 Part (c) 1=57/16 ( M R2 ) Develop an expression for the moment of inertia of the yo-yo about the contact point with the table. Correct! > 14% Part (d) Determine the magnitude of the linear acceleration of the center of mass of the yoyo the moment the string becomes taught. Grade Summary 8 F/57 M Potentsial 96%

Answer 1

To find the acceleration of a falling yo-yo, calculate its moment of inertia, use Newton’s second law for rotation to find the angular acceleration, and relate this to the linear acceleration using the radius.

Step-by-step explanation:

Calculating the Acceleration of a Falling Cylinder (Yo-Yo)

For a solid cylinder (yo-yo) with mass m and radius r that falls as the string unwinds without slipping, we need to calculate the linear acceleration of the cylinder. The moment of inertia of a solid cylinder is I = 1/2 MR², where M is the mass and R is the radius. The torque (τ) exerted by the tension T in the string is equal to T × r. This torque is related to the angular acceleration α by τ = Iα. Since the cylinder does not slip, its angular acceleration is related to the linear acceleration a by a = rα. By combining these equations, we can solve for a.

Here are the steps:

1. Calculate the moment of inertia (I) using I = 1/2 Mr².
2. Apply Newton’s second law for rotation, net τ = Iα, to find α.
3. Use the relationship a = rα to find the linear acceleration a.