Want an equation to describe a sphere rolling ACROSS (not down) an inclined plane, given an initial force across and taking friction and gravity into account

Question

An example would be found in putting on a golf course. In this case assume that the elevation from start to endpoint is of equal elevation and that the initial force is applied only once.. I can find many example of equations for a sphere rolling down an inclined plane, but have had no success with across.

Answer 1

Hi Mike R.,

Agree with Mark H. above that the question was posed carelessly. Apparently an initial lateral velocity was meant, and the trajectory across-and-down the plane thereafter is a parabola, of course (with a little fudging for the change of axis of rotation of the ball). If there is an initial and continuous lateral force (but no initial lateral velocity), the effective force vector does not change direction, and the path is a straight line diagonally down the slope, but gaining speed continuously. If there is both an initial velocity and a continuous lateral force, it's still a parabola.

It gets more complicated if you need to solve completely involving initial angular momentum of the ball, since the axis of rotation would change with the effective torque provided by gravity and the frictional force, and the moment of inertia of the ball on its axis/(axes). I suppose that'll be a hyperbolic function, though close to a parabola.

-- Cheers, -- Mr. d.

Answer 2

A problem like this is done in pieces.....you have 2 separable actions:

--The motion across the surface in response to an initial impulse.

--The motion down the "fall line" in response to the force of gravity.

Start with a flat, level surface. With no applied impulse, the sphere will not move. If you tip the surface, it will accelerate down the fall line. If the surface stays flat, the sphere will move at a velocity determined by the initial impulse (force acting through a distance. The trick is to convert the initial impulse into angular and linear momentum.

Referring to your problem statement, the elevation cannot be the same between start and finish.....with no applied force, the sphere will go downhill in response to the gravitational force.

Also, note the distinction between applied force and applied impulse. To change the momentum and energy state of an object requires a force acting through a distance (or acting for the equivalent time)