Section 3.1: Uniform Circular Motion
Uniform circular motion occurs when an object moves along a circular path with a constant speed. Although the speed is constant, the velocity changes continuously because the direction changes.
Key Concepts:
- Velocity is always tangent to the circle.
- Centripetal acceleration points toward the center of the circle: \( a_c = \frac{v^2}{r} \)
- Centripetal force required: \( F_c = m a_c = \frac{m v^2}{r} \)
Example 1
A 2 kg mass moves in a circle of radius 3 m at 4 m/s. Find the centripetal acceleration and force.
Centripetal acceleration: \( a_c = \frac{v^2}{r} = \frac{4^2}{3} = 5.33\text{ m/s²} \)
Centripetal force: \( F_c = m a_c = 2 \times 5.33 = 10.66\text{ N} \)
Practice Problems
- A 1.5 kg ball rotates in a circle of radius 2 m at 3 m/s. Find centripetal acceleration and force.
- A car of mass 1200 kg moves around a curve of radius 50 m at 20 m/s. Determine centripetal force.
- A satellite orbits Earth at 7000 km from Earth's center. Find acceleration if speed is 7.5 km/s.
- Object in horizontal circle with radius 4 m, tension 50 N, mass 2 kg. Find speed.
- A 0.5 kg mass on a 0.2 m string rotates at 10 m/s. Find the tension.