A long, thin rod of mass M and length L is standing straight up on a table. Its lower end rotates on a frictionless pivot. A very slight push causes the rod to fall over. As it hits the table, what are (a) the angular velocity

Verified step by step guidance

1. Identify the moment of inertia (I) of the rod about the pivot point at its end. For a uniform rod pivoted at one end, the moment of inertia is given by the formula: I = \frac{1}{3}ML^2.
2. Apply the conservation of mechanical energy principle. Initially, when the rod is vertical, all the energy is potential energy (PE), and as it falls, this energy converts into rotational kinetic energy (KE).
3. Calculate the initial potential energy when the rod is vertical. The center of mass of the rod is at L/2 from the pivot, so the potential energy is PE = Mg\frac{L}{2}, where g is the acceleration due to gravity.
4. Set up the energy conservation equation: initial potential energy equals final kinetic energy. The kinetic energy of a rotating object is given by KE = \frac{1}{2}I\omega^2, where \omega is the angular velocity.
5. Solve the energy conservation equation for \omega (angular velocity) to find the angular velocity of the rod as it hits the table. Rearrange the equation to isolate \omega and solve: \omega = \sqrt{\frac{2Mg\frac{L}{2}}{\frac{1}{3}ML^2}}.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a pivot point, expressed in radians per second. In this scenario, as the rod falls, its angular velocity increases due to the gravitational force acting on its center of mass. Understanding angular velocity is crucial for analyzing the motion of the rod as it transitions from a vertical to a horizontal position.

Moment of Inertia

The moment of inertia quantifies an object's resistance to rotational motion about an axis. For a rod pivoting at one end, the moment of inertia depends on its mass and length, calculated using the formula I = (1/3)ML². This concept is essential for determining how the rod's mass distribution affects its angular acceleration as it falls.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In the context of the falling rod, potential energy is converted into kinetic energy as it falls. This concept helps in calculating the angular velocity at the moment of impact by equating the initial potential energy to the kinetic energy just before the rod hits the table.