Question

A block slides down a frictionless hemi-spherical bowl (shown on the back of this page). It is released from rest at the point (2,0). Calculate the velocity, tangential acceleration, radial acceleration, and angular acceleration when the block goes through angular displacetments of 30°, 60, 90, 120, 150, and 180 Oa the diagram, draw vectors to scale representing each of the linear quantities.

Answer 1

Initial velocity of block = 0 (it is released). Tangential force = mg cos theta a_T = g cos theta < i > at theta = 30 degree and 150 degree velocity = > a_T = g cos 30 g R sin theta = 1/2 mg v^2 = 8.49 m/s^2 v = squareroot 2gR sin theta v = 4.43 m/s a_R = v^2/R = 2g R sin theta/R a_R = 4.43^2/2 = 9.81 m/s^2 alpha = 8.48/2 = a_R = 2 g sin theta 4.24 rad/s^2 - alpha = a_T/R = g cos theta/R < ii > at theta = 60 and 120 degree a_T = g cos 60 = 4.9 m/s^2 v = squareroot 2 times 9.8 times 2 times sin 60 = 5.83 m/s a_R = 17 m/s^2 alpha = 2.45 rad/s^2 at theta = 90 degree a_T = g cos 90 = 0 v = squareroot 2gR