The box slides down the helical ramp such that r = 0.5 m, θ = (0.5t3)rad, and z = (2-0.2t2) m, where t is in seconds. Determine the magnitudes of the velocity and acceleration of the box at the instant θ = 2π rad.
Given:
By differentiating, we have
By differentiating, we have
By differentiating, we have
By differentiating, we have
By differentiating, we have
When we have
At:
Component of the box’s velocity is expressed as
The magnitude of the box’s velocity is expressed as
The magnitude of the box’s velocity is
Components of acceleration of the box is expressed as
Accordingly, we have
Accordingly, we have
Magnitude of the acceleration of the box is expressed as
By substituting the values of the parameters into the above relation, we have
Magnitude of the acceleration of the box is