A slender rod of length l is pivoted about a point C located...

A slender rod of length $l$ is pivoted about a point $C$ located at a distance $b$ from its center $G .$ It is released from rest in a horizontal position and swings freely. Determine $(a)$ the distance $b$ for which the angular velocity of the rod as it passes through a vertical position is maximum, $(b)$ the corresponding values of its angular velocity and of the reaction at $C .$

Key Concepts

Reaction Forces at a Pivot

When a body rotates about a pivot, forces act at the point of support to provide the necessary centripetal force and to balance gravitational forces. Studying reaction forces involves understanding how dynamic components, such as centripetal and tangential accelerations, along with static gravitational forces, combine at the pivot point. This is key for calculating the magnitude of the reaction force exerted by the pivot during the rod's motion.

Physical Pendulum Dynamics

A physical pendulum is any rigid body that swings about a pivot point under the influence of gravity. Unlike a simple pendulum, its moment of inertia and mass distribution play a significant role in its motion. Analyzing a physical pendulum involves combining torque, rotational inertia, and energy conservation, which is essential for determining properties such as the angular velocity and optimum pivot location.

Conservation of Energy

This concept involves equating the potential energy lost by the system to the kinetic energy gained. In the context of mechanical systems like the swinging rod, the gravitational potential energy drop as the rod moves from the horizontal to the vertical position is converted into rotational kinetic energy. This principle is crucial for determining the angular velocity at different points in the rod's swing.

Moment of Inertia and the Parallel Axis Theorem

Moment of inertia measures how mass is distributed with respect to an axis and affects the rotational motion of a body. For a rod pivoted at a point not passing through its center of mass, the parallel axis theorem is used to compute the moment of inertia about the pivot. This theorem states that the moment of inertia about any axis parallel to one through the center of mass is equal to the moment of inertia about the center plus the mass times the square of the distance between the axes.

Transcript

00:01 Hello friends, a cylinder rod of length l is bio weighted above a point c, located at a distance b from its center g.

00:14 So you can see the figure.

00:18 C g is v and length of the rod a .v is called to l.

00:28 It is released from rest in a horizontal position.

00:33 So b1 is called to zero at horizontal position.

00:50 And since freely we have to calculate first part, the distance to be we have to find for which angular velocity to be maximum at vertical position.

01:31 And we have to calculate angular velocity and reaction at c.

01:52 Let us see in position 1.

01:57 That is horizontal position.

01:58 For horizontal position, v1 is 0, omega 1 is 0, total kinetic energy is 0, height h is equal to, just a moment, leave it.

02:45 Now for vertical position, v2 is equal to v omega 2.

03:10 We can draw the diagram here.

03:12 This is vertical position.

03:19 This is c point.

03:22 This is g.

03:27 This is v2.

03:37 This distance is given b.

03:49 Moment of energy of the rod.