Section 3.3: Circular Motion and Tension
When an object moves in a vertical circle attached to a string or rope, the tension varies depending on position. The centripetal force is provided partly or fully by the tension in the string.
Tension in Circular Motion:
- At the top: \( T_{\text{top}} = m \frac{v^2}{r} - mg \)
- At the bottom: \( T_{\text{bottom}} = m \frac{v^2}{r} + mg \)
- At any angle \( \theta \): \( T = m \frac{v^2}{r} + mg \cos\theta \)
Key Concepts:
- Tension provides centripetal force.
- Weight affects tension, especially at vertical extremes.
- Minimum speed at the top ensures string stays taut: \( v_{\min} = \sqrt{gr} \)
Example 1
A 2 kg mass rotates in a vertical circle of radius 1.5 m with speed 4 m/s at the top. Find the tension at the top.
Tension: \( T_{\text{top}} = m \frac{v^2}{r} - mg = 2 \frac{4^2}{1.5} - 2 \cdot 9.8 = 10.67 - 19.6 \approx -8.93 \text{ N} \)
Negative indicates the string would go slack; minimum speed required is higher to maintain tension.
Practice Problems
- A 1.5 kg ball swings in a vertical circle of radius 2 m at 5 m/s at the bottom. Find tension.
- Determine the minimum speed at the top for a 3 kg mass on a 2 m radius vertical circle.
- A 0.5 kg mass moves in a vertical circle of radius 0.8 m. Speed at top is 2 m/s. Find tension at top and bottom.
- A roller coaster car of mass 400 kg moves through a vertical loop of radius 10 m. Find tension at top if speed is 15 m/s.
- A pendulum bob of 2 kg is at 30° from vertical, moving at 3 m/s. Determine tension in string.