Question

A long, uniform rod of mass M and length l is supported at the left end by a horizontal axis into the page and perpendicular to the rod, as shown. The right end is connected to the ceiling by a Horizontal Axis thin vertical thread so that the rod is horizontal. Express the answers to all parts of this question in terms of some or all of the quantities M, l and g, and any necessary constants (a) Determine the magnitude and direction of the force exerted on the rod by the axis. The thread is then burned by a match. For the time immediately after the thread breaks, determine: (b) the angular acceleration of the rod about the axis shown (c) the translational acceleration of the center of mass of the rod (d) the force exerted on the end of the rod by the axis The rod rotates about the axis and swings down from the horizontal position. (e) Determine the angular momentum of the rod as a function of e, the arbitrary angle through which the rod has swung. (Hint: Acceleration is not constant. Thread

Start from basic physics principles, and include explanations such that another student would be able to follow and understand your reasoning.

Answer 1

a) There are three forces acting on the rod. Out of this weight of the rod tries to rotate clockwise and force exerted by the string and horizontal axis tries to rotate counter clockwise At equilibrium, torque due to both the forces should balance. cicckwiseconter ciockwise T, (considering right end as the pivoting point) Mg Hence magnitude of the force is and the direction of force is upwards

b) We have the relation between torque and angular acceleration as, Mgl MM2 3g 21 -g 21 Hence,angular acceleration is given by

c) acceleration and rotational acceleration We have the following relation between translational 1 3g 2 21 3g 4 3g Hence,translational acceleration of the centre of mass of the rodis

d) Sum of the force exerted by the axis and weight of the rod should give Ma. Mg Mg Hence.force exerted on the end of the rod by the axis is