Bead on a Rotating Hoop
1. IntroductionConsider 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
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:
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