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Statement of a problem № 3271

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Consider a point mass m with momentum p rotating at a distance r about an axis. Starting from the definition of the angular momentum L=rxp of this point mass, show that dL/dt = t, where t is the torque. A uniform rod of length l and mass M rests on a frictionless horizontal surface. The rod pivots about a fixed frictionless axis at one end. The rod is initially at rest. A bullet of mass m travelling parallel to the horizontal surface and perpendicular to the rod with speed v strikes the rod at its centre and becomes embedded in it. Using the result above, show that the angular momentum of the rod after the collision is given by |L|= 1/2lv Is L = (l/2)mv also correct? What is the final angular speed of the rod? Assuming M = 5m, what is the ratio of the kinetic energy of the system after the collision to the kinetic energy of the bullet before the collision?

Solution:
Consider a point mass m with momentum p rotating at a distance r about an axis.

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