Final answer:
The motion of a particle moving in a rotating vertical plane is described by two independent one-dimensional kinematic equations. Horizontally, the motion is with constant velocity, while vertically, the particle experiences downward acceleration due to gravity resulting in oscillatory motion.
Step-by-step explanation:
To find the equations of motion for a particle of mass m moving without friction in a vertical plane and rotating with constant angular velocity Ω about the y-axis, we must consider the forces acting on the particle. In this case, the horizontal and vertical motions of the particle are independent of each other.
Horizontal Motion
Since there is no horizontal acceleration, we use the kinematic equation for constant velocity motion:
x = x0 + Vxt
Here, Vx is a constant velocity in the horizontal direction, and x0 is the initial position.
Vertical Motion
The vertical motion is affected by gravity, so the particle experiences a constant downward acceleration -g. The kinematic equations for vertical motion are:
- y = y0 + Vyt - 1/2gt^2
- Vy = Voy - gt
The possible motions of the particle include circular motion in the horizontal plane and oscillatory motion similar to a simple pendulum in the vertical plane.