Question

Chapter 10 [ Essential University of Physics( 3^rd ed.)]

Given the following R_disk= 2.50 cm, R_axle= 0.250 cm, M_disk= 25.0 g, M_axle= 0.750 g,  L_axle= 1.00 cm, and state any assumption used. a) Determine the rotational inertia of yo-yo about its center. (b) Derive using force and torque the expression for the linear acceleration a_cm of a yo-yo rolling down a string using an approach similar to that of the object rolling down a ramp. (c) Calculate linear acceleration of the yo-yo for the given values.

Answer 1

A yo-yo is made of two identical disks attached at their centers to a cylindrical axle, which has a smaller radius. A string is attached to and wrapped around this axle. If the yo-yo is released, it displays both downward linear motion and circular motion about the axle.

In the below figure T= tension in the string , r= radius of disk =250 cm,

b= radius of axle= 0.250 cm , M= mass of disk + mass of axle= (25.0+0.75)=25.75 gm

  

(a) Rotational inertia can be given as the sum of the moments of inertia of each of the end disks plus the moment of inertia of the axle:

(b) The net force gives us gravity minus tension if down is the positive direction. This gives the equation

  

  For the angular motion, the torque will dependent on the tension along the string and gravity doesn’t contribute. The equation for angular acceleration is therefore

  

from both equations

  

this is the required equation of the acceleration

(c) according to the question acceleration of the yo yo will be