Section 2.7: Circular Motion and Tension

When an object moves in a circular path and is attached to a string or rope, the tension in the string provides the centripetal force necessary for circular motion. Analyzing tension is essential in vertical and horizontal circular motion problems.

Horizontal Circular Motion:
  • Tension provides the centripetal force: \( T = \frac{m v^2}{r} \)
Vertical Circular Motion:
  • At the top: \( T_{top} + mg = \frac{m v_{top}^2}{r} \)
  • At the bottom: \( T_{bottom} - mg = \frac{m v_{bottom}^2}{r} \)
  • Speed varies due to gravity along the vertical circle: \( v^2 = v_0^2 + 2 g h \)

Example 1

A 0.5 kg ball is tied to a string and swung in a horizontal circle of radius 1 m at 4 m/s. Find the tension in the string.

\( T = \frac{m v^2}{r} = \frac{0.5 * 4^2}{1} = 8 \text{ N} \)

Example 2

A 2 kg mass rotates in a vertical circle of radius 1.5 m. If its speed at the top is 3 m/s, find the tension in the string at the top and bottom.

Top: \( T_{top} + mg = m v^2 / r \Rightarrow T_{top} = (2*3^2/1.5) - 2*9.8 = 12 - 19.6 = -7.6 \text{ N} \) (string goes slack, so minimum speed needed!)

Bottom: \( T_{bottom} - mg = m v^2 / r \Rightarrow T_{bottom} = 12 + 19.6 = 31.6 \text{ N} \)

Practice Problems

  1. A 1 kg ball moves in a horizontal circle of radius 0.5 m at 3 m/s. Find the tension in the string.
  2. A mass of 0.75 kg rotates in a vertical circle of radius 1.2 m at a speed of 4 m/s at the bottom. Find the tension.
  3. A 2 kg mass rotates at the top of a vertical circle of radius 2 m with speed 5 m/s. Determine tension at the top and bottom positions.
  4. A stone is tied to a 1.5 m string and swung in a horizontal circle at 6 m/s. Find tension.
  5. A roller coaster car of mass 500 kg moves in a vertical loop of radius 10 m at the top with speed 12 m/s. Compute the tension in the track support.
  6. A 0.8 kg ball rotates in a horizontal circle of radius 0.4 m at 5 m/s. Find tension in the string.
  7. A mass of 1.5 kg rotates in a vertical circle of radius 1.5 m. If speed at bottom is 4 m/s, find tension at top and bottom.
  8. A bucket of mass 2 kg is swung in a vertical circle of radius 1 m at 4 m/s. Find tension at the top.
  9. A mass of 3 kg moves in a horizontal circle of radius 0.6 m at 5 m/s. Determine tension.
  10. A 1.2 kg object rotates in a vertical circle of radius 1 m. Speed at top is 3 m/s. Find tension at top and bottom.
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