Not that hard, I need your note to compare. A uniform rod of mass M and length L

Question

Not that hard, I need your note to compare.

A uniform rod of mass M and length L is sitting vertically with one end resting on a frictionless surface, perfectly balanced. At time t = 0, the top end of the rod is given a tiny push to the right so that it is no longer stable, and the rod falls down. Find the acceleration of the end of the rod just as it hits the ground. Find the angular velocity of rod just as it hits the ground. A uniform rod of mass M and length L is attached to a frictionless hinge on the ground. Initially, the rod is sitting vertically, perfectly balanced. At time t = 0, the top end of the rod is given a tiny push to the right so that it is no longer stable, and the rod falls down. Find the acceleration of the end of the rod just as it hits the ground. Find the angular velocity of rod just as it hits the ground.

Explanation / Answer

a)

Torque on the rod when it touches the ground, T = r*F = (L/2)*m*g


we know,

T = I*alfa

m*g*L/2 = m*L^2/3 * alfa


alfa = 3*g/2*L

relation between tangential acceleration and angular acceleration,

a_tan = L*alfa = (3/2)*g

b)

as the rod falls down its potentail energy decreases and kinetic energy increases.


final kinetic enrgy = initial potential energy

0.5*I*w^2 = m*g*L/2

0.5*m*L^2/3 * w^2 = m*g*L/2

==> w^2 = 3*g/L

==>w = sqrt(3*g/L)

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