Question

Is angular momentum constant in uniform circular motion?

Answer 1

Explanation:

1. In a uniform circular motion, an object keeps on revolving around a fixed axis or fixed point at a constant speed.
2. Circular motion becomes a little difficult as we include vectors in them.
3. As we know that the speed of the object remains constant but not the velocity. Velocity is nothing but the speed of the body along with the direction of the motion.
4. Although the speed remains constant throughout the circular motion, the direction of motion keeps on changing every instant. Thus the velocity keeps on changing every instant.
5. A simple direction of a body in uniform circular motion is as follows in the figure given:

1. In a uniform circular motion. Although the velocity of the body changes but the angular velocity of the body remains fixed.
2. We know that the centrifugal force is given by: $\therefore \mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$
3. Where m is the mass of the body, r is the radius of the circular path and v is the velocity of the body. Since we squared the velocity the change in its direction stops playing any role, Hence v² also becomes constant. the centrifugal force on the body also remains the same.
4. We also know that the linear momentum of a body is given by: $\mathrm{p}=\mathrm{mv}$
5. Where, clearly, there is no change in the mass of the body but the velocity of the body keeps on changing every instant. Thus, the linear momentum of the body also changes every moment.
6. We know that acceleration is nothing but the change in velocity of the body. Also, the velocity of the body keeps on changing every moment. Hence the acceleration also keeps on changing every moment.
7. Hence, the angular momentum of the body is given by : $\begin{array}{rcl}\mathbf{\therefore }\mathbf{L}& \mathbf{=}& \mathbf{r}\mathbf{\times }\mathbf{p}\\ & \mathbf{\Rightarrow }& \mathbf{L}\mathbf{=}\left|r\right|\left|p\right|\mathbf{sin}\left(\theta \right)\end{array}$
8. Here, we know that the radius makes a constant angle of 90⁰ with linear momentum. This means that $\sin \left(\theta \right)$ is constant. Also, $\left|p\right|\mathrm{and}\left|\mathrm{r}\right|$ are also constant.

Hence, The angular momentum of the body remains constant in a uniform circular motion.