A slotted disk is rotating with a constant angular velocity of w. A ball mass of m slides out from the center of the disk. Kinetic friction force F=mu_k |N| acts on the mass, where N is the normal force between the ball and the channel. Using the body coordinate system i^, j^ fixed to the disk, derive the equation of motion for the mass using r(t) as the generalized coordinate.

Question

A slotted disk is rotating with a constant angular velocity of w. A ball mass of m slides out from the center of the disk. Kinetic friction force F=mu_k |N| acts on the mass, where N is the normal force between the ball and the channel. Using the body coordinate system i^, j^ fixed to the disk, derive the equation of motion for the mass using r(t) as the generalized coordinate.

Answer 1

I DIDN'T POST THE PROBLEM:

Enter the equation of motion in terms of r, dotr,m,w and mu_k. (I have w standing for omega)

ddotr=

Answer 2

To derive the equation of motion for the mass sliding on the slotted disk, we need to consider the forces acting on the mass and apply Newton's second law.

Let's start by considering the forces acting on the mass in the body coordinate system i^, j^ fixed to the disk.

1. Normal force (N): The component of the normal force acting perpendicular to the slotted disk is responsible for providing the centripetal force to keep the mass moving in a circular path. It can be calculated as N = m * w^2 * r, where m is the mass of the ball, w is the angular velocity of the disk, and r is the distance from the center of the disk to the ball.

2. Friction force (F): The kinetic friction force acts tangentially to the slotted disk opposing the motion of the ball. It can be calculated as F = μk * N, where μk is the coefficient of kinetic friction.

Now, let's apply Newton's second law in the i^ (horizontal) and j^ (vertical) directions.

In the i^ direction:
ma_i = -F

Since F = μk * N, we can substitute it into the equation:
ma_i = -μk * N

In the j^ direction:
ma_j = N - mg

Since N = m * w^2 * r, we can substitute it into the equation:
ma_j = m * w^2 * r - mg

Now, let's express the acceleration components, a_i and a_j, in terms of the derivative of the generalized coordinate, r(t), with respect to time, t.

Since we are using r(t) as the generalized coordinate, we know that a_i = d^2r/dt^2 and a_j = d^2r/dt^2.

Substituting these expressions back into the equations, we get:

m(d^2r/dt^2) = -μk * m * w^2 * r

m(d^2r/dt^2) = m * w^2 * r - mg

Simplifying the equations, we have the equation of motion for the mass sliding on the slotted disk:

d^2r/dt^2 = -μk * w^2 * r + w^2 * r - g

This equation describes the motion of the mass along the slotted disk as a function of time, r(t).