Marble Path

Marble Path

Angle θ related to φ

Most problems in dynamics can be solved by using energy methods. This method is particularly helpful with complex systems. Energy will be used for the problem to demonstrate how it can be used.

First, the total energy, both potential and kinetic, needs to be determined for the system. The gravitational potential energy V of the marble is

     V = mgh

The kinetic energy of the marble, including both linear and rotational, is

One problem is that there is three variables, h, v, and dφ/dt that describe the marble motion and position. Energy methods can only have one unknown variable, so the three must be related.

The marble's vertical location h is directly related to angle θ by

     h = (R - r) (1 - cosθ)

The marble linear velocity is related to the angular velocity dθ/dt through the arc as

Finally, the angle φ can be expressed in terms of the angle θ through the relationship

Adding potential and kinetic energy (total energy), and using h, v, and dφ/dt relationships give

Solution Results: Initial Angle = -10^o

Solution Results: Initial Angle = -35^o

Solution Results: Initial Angle = -60^o

The moment of inertia for a solid sphere is

     I = 2/5 mr²

Substituting I, and simplifying gives

The total energy, E, is constant regardless of the marble location. Thus, the first derivative with respective to time gives,

Rearranging gives,

If the motion is restricted to small oscillations, sinθ can be approximated by θ and the equation of motion can be simplified to

and the natural frequency ω_n is

The general solution to this equation is

     θ(t) = A sinω_nt + B sinω_nt

where

Results are given for initial angles of -10^o, -35^o and -60^o degrees. Note that for comparison, the results have been non-dimensionalized with respect to the initial conditions.