Conservation of Energy with Rotation Practice Problems

A paper cutter is made of a uniform bar attached by its lower end to a horizontal base by a frictionless axle. The bar of mass m_b and length l_b is initially at rest and perpendicular to the base. When the bar is barely moved from its initial position it rotates around the axle. Determine the expression of the angular velocity at the instant the bar strikes the base.

A rigid, uniform bar is attached by its lowest end to a frictionless horizontal hinge. The bar is initially perpendicular to the horizontal plane. The bar of length 1.0 m and mass 0.250 kg is released from rest and rotates in a vertical plane. Calculate the speed of the highest end of the bar when it reaches the horizontal plane.

A pole-like structure measuring 1.80 m in length stands upright and begins to fall, with its bottom end fixed in place. Applying the principle of conservation of energy determine the speed that the upper end of the structure attains just before it reaches the ground.

Two spherical balls of masses m₁ = 32.0 kg and m₂ = 45 kg are linked by a thread which is passed over a cylindrical pulley of radius r = 0.422 m and mass 2.5 kg. The system is initially held still in such a state that m₁ is on the floor while m₂ is at a height of 4.2 m above the floor. Using the principle of conservation of energy, calculate the speed of m₂ just when it hits the floor after the system is released. Assume that there is no friction.