Section 3.2: Centripetal Acceleration & Force
In uniform circular motion, the velocity vector changes direction constantly. This change produces a centripetal acceleration directed toward the center of the circle.
Centripetal Acceleration:
- Magnitude: \( a_c = \frac{v^2}{r} \)
- Direction: toward the center of the circle
Centripetal Force:
- Net force required to maintain circular motion: \( F_c = m a_c = \frac{m v^2}{r} \)
- Acts toward the center (radial direction)
Example 1
A 1.5 kg object moves in a circle of radius 2 m at 3 m/s. Find the centripetal acceleration and force.
Centripetal acceleration: \( a_c = \frac{v^2}{r} = \frac{3^2}{2} = 4.5 \text{ m/s²} \)
Centripetal force: \( F_c = m a_c = 1.5 \times 4.5 = 6.75 \text{ N} \)
Practice Problems
- A 2 kg mass moves in a circle of radius 1.5 m at 5 m/s. Find acceleration and force.
- A car of 1000 kg goes around a curve of radius 30 m at 15 m/s. Determine centripetal force.
- A satellite orbits at 10,000 km from Earth's center with speed 6 km/s. Find acceleration.
- An object rotates on a string of 0.5 m at 8 m/s. Find the tension.
- A roller coaster car of 500 kg moves through a loop of radius 10 m at 20 m/s. Find the centripetal force at the top.