B. A vehicle on a monorail has a speed S = 10 m/s and an acceleration S=4m/s2 relative to the ground reference XYZ when it reaches point A. Inside the vehicle, a 3 kg mass slides along a rod which at the time of interest is parallel to the X axis. This rod rotates about a vertical axis with w=1 rad /s and w=2 rad/s2 relative to the vehicle at the time of interest. Also at this time, the radial distance d of the mass is .2 m and its radial velocity v = .4 m/s inward. What is the dynamic force on the mass at this instant?

To find the dynamic force on the mass at this instant, we need to consider the forces acting on the mass.

Let's break down the forces:

1. Centripetal Force:
The mass is moving in a circular path due to the rotation of the rod. The centripetal force is given by the formula Fc = m * (r * ω^2), where m is the mass, r is the radial distance, and ω is the angular velocity. In this case, m = 3 kg, r = 0.2 m, and ω = 1 rad/s. Therefore, the centripetal force is Fc = 3 * (0.2 * 1^2) = 0.6 N.

2. Radial Force:
The mass is also experiencing a radial force due to its radial velocity. The radial force is given by the formula Fr = m * (r * α), where m is the mass, r is the radial distance, and α is the radial acceleration. In this case, m = 3 kg, r = 0.2 m, and α = 2 rad/s^2. Therefore, the radial force is Fr = 3 * (0.2 * 2) = 1.2 N.

3. Net Force:
To find the net force acting on the mass, we need to consider both the centripetal force and the radial force. Since the radial force is inward and the centripetal force is directed towards the center of the circular path, we subtract the radial force from the centripetal force to get the net force.

Net Force = Centripetal Force - Radial Force
Net Force = 0.6 N - 1.2 N = -0.6 N

Therefore, the dynamic force on the mass at this instant is -0.6 N. Note that the negative sign indicates that the force is in the opposite direction of the radial velocity, i.e., outward.