Section 2.6: Circular Motion and Centripetal Force

Circular motion occurs when an object moves along a circular path. The force that keeps it moving in a circle is called the centripetal force. Understanding centripetal acceleration is key to analyzing circular motion.

Centripetal Acceleration and Force:
  • Acceleration: \( a_c = \frac{v^2}{r} \)
  • Centripetal Force: \( F_c = m a_c = \frac{m v^2}{r} \)
  • Direction: Always towards the center of the circle.
Angular Speed:

\( \omega = \frac{v}{r} \), where \( \omega \) is the angular velocity in rad/s.

Example 1

A 2 kg mass rotates in a horizontal circle of radius 0.5 m at a speed of 4 m/s. Find the centripetal force.

\( F_c = \frac{m v^2}{r} = \frac{2 * 4^2}{0.5} = \frac{32}{0.5} = 64 \text{ N} \)

Example 2

A car of mass 1000 kg takes a circular turn of radius 50 m at a speed of 20 m/s. Find the centripetal acceleration and centripetal force.

Centripetal acceleration: \( a_c = v^2/r = 20^2 / 50 = 400/50 = 8 \text{ m/s²} \)

Centripetal force: \( F_c = m a_c = 1000 * 8 = 8000 \text{ N} \)

Practice Problems

  1. A 5 kg ball moves in a horizontal circle of radius 2 m at 3 m/s. Find the centripetal force.
  2. A car goes around a curve of radius 100 m at 20 m/s. Determine its centripetal acceleration.
  3. A satellite of mass 500 kg orbits a planet in a circle of radius 10,000 km at speed 2000 m/s. Find the centripetal force.
  4. A stone tied to a string of length 1.5 m is swung in a horizontal circle at 6 m/s. Find the tension in the string.
  5. A roller coaster car of mass 800 kg goes through a circular loop of radius 10 m at the top of the loop with speed 15 m/s. Compute the centripetal force.
  6. A 1 kg object moves in a circle of radius 0.25 m at 5 m/s. Find the angular speed.
  7. A car of mass 1200 kg travels at 25 m/s on a circular track of radius 80 m. Find the required centripetal force.
  8. A bucket of water of mass 2 kg is swung in a vertical circle of radius 1 m at 4 m/s. Find the centripetal force at the bottom.
  9. A motorcyclist of mass 70 kg moves in a horizontal circular path of radius 30 m at 12 m/s. Find centripetal acceleration and force.
  10. A 3 kg ball rotates in a horizontal circle of radius 0.6 m with speed 5 m/s. Find tension in the string.
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