Pendulum A small glue ball is fired along the horizontal into the lower end of a

Question

Pendulum A small glue ball is fired along the horizontal into the lower end of a uniform rod, pivoted at its upper end. After the impact, the ball is stuck to the rod and the rod swings up to an angle beta. The mass of the glue ball is m. The rod has mass M and length L. Ignore friction and air resistance. With the glue ball stuck to the lower end of the rod. calculate the moment of inertia of the combined object for an axis through the pivotal point and perpendicular to the plane of rotation. Find the position of the center of mass for the combined object. Find the speed of the ball before the impact. Derive the pendulum's oscillator equation for the combined object if the angle of deflection is very small. Find the frequency of the small oscillations of the combined object.

Explanation / Answer

partA :

MOment of inertia of rod I = ML^2/3

I combined = (ML^2/3) + ML^2

MOI combined = L^2 *(M/2 + m)

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Centre of Mass CoM = ML/2 + M(0)/(m +M) = mL/(m+M)

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from conservation of momentum

mv = (M+m) V1

from conservation of ENErgy

0.5 * (m+M) V'^2 = (m+M) gL *(1 + 1/2 *(m/(M+m))

V' = (c) v=(m+1/3M)/m*sqrt((m+M/2)/(m+M/3)*2gL(1-cos))

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e. f= 1/(2pi) *sqrt((m+M/2)/(m+M/3)*g/L)

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