Conservation of Angular Momentum for a Block on a Table

Q10. A block with mass M, on the end of a string, moves in a circle on a horizontal frictionless table as shown. As the string is slowly pulled through a small hole in the table:

Background

Topic: Conservation of Angular Momentum

This question tests your understanding of angular momentum and kinetic energy in a system where a mass moves in a circle and the radius is changed by pulling the string through a hole.

Key Terms and Formulas

• Angular momentum ($L$): $L = mvr$

• Kinetic energy ($K$): $K = \frac{1}{2}mv^2$

• Torque ($\tau$): $\tau = r \times F$

• Conservation of angular momentum: If no external torque acts, $L$ is constant.

Step-by-Step Guidance

1. Identify the forces acting on the block: gravity, normal force, and the pull on the string. Only the pull changes the radius, but it acts along the radius, so its torque is zero.

2. Recall that angular momentum is conserved when there is no external torque. Since the pull is along the radius, it does not exert a torque on the block.

3. Write the expression for angular momentum: $L = mvr$. As the radius $r$ decreases, the velocity $v$ must increase to keep $L$ constant.

4. Consider the kinetic energy: $K = \frac{1}{2}mv^2$. As $v$ increases, $K$ increases, so kinetic energy does not remain constant.

Try solving on your own before revealing the answer!

Final Answer: A) the angular momentum of M remains constant

Since the force pulling the string acts along the radius, it does not exert a torque, so angular momentum is conserved. The kinetic energy increases as the radius decreases.