Question

The bent rod has a mass rho per unit length and rotates about the z-axis with an angular velocity ω. Determine the angular momentum L of the rod about the fixed origin of the axes, which are attached to the rod. Also, find the kinetic energy T of the rod.

Options:
a. Angular momentum is given by: L = Iω, and kinetic energy is T = (1/2)Iω².
b. Angular momentum is given by: L = I/2 ω, and kinetic energy is T = (1/4)Iω².
c. Angular momentum is given by: L = Iω², and kinetic energy is T = (1/2)Iω.
d. Angular momentum is given by: L = Iω, and kinetic energy is T = (1/2)Iω.

Answer 1

The angular momentum of a uniformly rotating rod about a fixed axis is given by L = Iω, and its kinetic energy is T = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity. Option A is correct.

Step-by-step explanation:

Calculation of Angular Momentum and Kinetic Energy of a Rotating Rod

The angular momentum (L) of an object that rotates about a fixed axis, such as a rod, can be determined using the formula L = Iω, where I represents the moment of inertia and ω is the angular velocity. For kinetic energy (T) of the rotating object, the formula used is T = (1/2)Iω².

To calculate the rod's moment of inertia (I), we must integrate the mass distribution along the length of the rod taking into account its rotational axis. If the rod is uniform and rotates about its center, the moment of inertia is I = mL² / 12. If it's rotating around one end, it's I = mL² / 3.

For example, a rod with a mass m and a length L rotating about its center would have an angular momentum L = (mL² / 12)ω and kinetic energy T = (1/2)(mL² / 12)ω².