Discusssion
As you watch the sphere move through its path, focus on a few things:
- Direction of the velocity vector. The velocity of the ball is always tangent to the path, as expected. Note that the unit tangent vector e_t points in the same direction as the velocity.
- Direction of the acceleration vector. The acceleration of the ball always points “inward” on the path, again, as expected. When P is slowing down, you see that the acceleration vector points in the negative e_t direction. Similarly, when P is increasing in speed, the acceleration vector points in the positive e_t direction.
- Directions of the polar unit vectors. The e_r unit vector always points outward from point O toward the sphere. The e_θ unit vector is perpendicular to e_r, and in the direction of increasing θ, as seen in the animation above.
Your task is to calculate R, R_dot and R_ddot using the given equation relating R and θ. This will require derivatives with respect to TIME; however, since R is given in terms of θ, you will need to use the chain rule of differentiation. For example: R_dot = dR/dt = (dR/dθ)(dθ/dt).
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