Analyzing Forces on a Mass in Uniform Circular Motion

Q1. For what speed is the tension the same in both wires for a sphere revolving in a horizontal circle?

Background

Topic: Uniform Circular Motion and Tension Forces

This question tests your understanding of the forces acting on a mass moving in a horizontal circle, specifically how to analyze the tension in supporting wires and relate it to the speed of the mass.

Key Terms and Formulas

• Uniform Circular Motion: Motion of an object in a circle at constant speed.

• Tension ($T$): The force transmitted through a string, rope, or wire when it is pulled tight by forces acting from opposite ends.

• Centripetal Force: The net force causing the inward acceleration of an object moving in a circle, given by $F_c = \frac{mv^2}{r}$.

• Newton's Second Law: $\sum F = ma$

Step-by-Step Guidance

1. Draw a free-body diagram for the sphere, showing all forces acting on it (tensions in both wires and gravity).

   

2. Resolve the forces into vertical and horizontal components. The vertical components of the tensions must balance the weight of the sphere, and the horizontal components provide the centripetal force.

3. Set up equations for the vertical and horizontal force balances. For the vertical direction: $T_1 \sin\theta + T_2 \sin\theta = mg$. For the horizontal direction: $T_1 \cos\theta = T_2 \cos\theta = \frac{mv^2}{r}$ (assuming symmetry and equal angles).

4. Since the question asks for the speed at which the tensions are equal, set $T_1 = T_2 = T$ and use the above equations to solve for $v$ in terms of $m$, $g$, $r$, and $\theta$.

Try solving on your own before revealing the answer!

Final Answer: $v = 5.6\ \mathrm{m/s}$

By equating the vertical and horizontal components and solving for $v$, you find the speed at which the tensions in both wires are equal.