Bead on a Rotating Hoop

1. Introduction

                  Consider a bead of mass m sliding freely on a hoop of radius R rotating with angular velocity ω0 in a constant gravitational field with acceleration g as shown in Figure 1. 

   Since the bead is on rotating hoop, it moves on the surface of the sphere of             
   radius R.  We use the spherical coordinates to describe the position of the               
   bead. The position of the bead is given as follows



    Figure 1: Bead on a rotating hoop

   The velocity components of the bead are as follows:


2. Lagrange’s Equations of motion for bead  

The kinetic and potential energies of the bead are given as follows




Where θ = 0 is the zero potential energy point.  Therefore the Lagrangian   L = T-V is


The Lagrange’s equation for θ is 



Thus one gets the equations of motion for bead on rotating hoop:



In the equation (2.5) frictional forces are not considered. We can easily include the velocity dependent frictional forces in the equation. Let v be the frictional coefficient of the bead on the hoop, on adding the frictional forces the equation of motion becomes



3. Numerical Integration of Equation

                The equation of motion (2.5)  can be time integrated to know the trajectory/ position of the bead at every time step using methods like Euler method, Runge-Kutta method etc,. Runge-Kutta method is better and more accurate. To apply these techniques the equation of motion is transferred to two first order differential equations as follows:



Equation (3.1) defines the state of the bead. Its first order time derivative is given by



To solve the above first order differential equations initial values are required i.e.   θ and  θdot are required.

4.  Code for Bead on a Rotating Hoop

            A code is written in MATLAB for the bead on rotating hoop. The equation of motion is solved using ODE45 of the MATLAB. Animation of the bead motion is plotted. The following are the values used in the code and can be changed accordingly.

Acceleration due to gravity, g = 9.81 m/s2
Mass of the bob, M = 10 N
Radius of the rotating hoop, R = 2 m
Frictional coefficient of the bead, V = 0.5
Angular Velocity of the rotating hoop, ω0 = 20 rad/s

The following are the initial values used in the code


Initial values can be changed accordingly to the requirement. The following figure shows the plots obtained for the bead with the above values run for 20 seconds.

The values of the bead in the code can be changed and the motion of bead can be studied. One can study the interesting behavior in the motion of the bead by playing around with the code changing the values.  

             A Code is written in MATLAB for the Bead on Rotating Hoop. The position of the bead as the hoop rotates is animated. Below video shows the animation done with the CODE written in MATLAB. The boundary conditions in the CODE can be changed and the interesting motion of the bead on the hoop can be studied. 

Bead on a Rotating Hoop

CODE can be obtained from the following link: 

Your suggestions and comments are appreciated if you use the CODE. 

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