Circular and rotational motion

NUST NET (Engineering) · Physics · Mechanics. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

A body moving in a circle is always accelerating towards the centre, so it needs a net inward force, the centripetal force. This is not a new force: it is supplied by tension, friction, gravity or a normal reaction.

Rotation mirrors straight-line motion. Moment of inertia I plays the part of mass, torque the part of force and angular momentum Iω the part of momentum. With no external torque, Iω stays constant.

For a satellite, gravity is the centripetal force. The satellite and everything in it are in free fall together, so objects inside press on nothing and appear weightless, although gravity still acts on them.

Rules to remember

Common mistake

Using the height above the ground as r in orbit formulas. r is measured from the centre of the Earth, so r = R + h.

Worked example

A car takes a level bend of radius 80 m. The coefficient of friction between tyres and road is 0.50. Taking g = 10 m s−2, what is the greatest speed at which it can take the bend without skidding?

  1. Friction supplies the centripetal force: at the limit μmg = mv2/r.
  2. The mass cancels: v2 = μgr = 0.50 × 10 × 80 = 400.
  3. v = √400 = 20.

Answer: 20 m s−1

Practice questions

Try each one, then open the answer.

1. What is the centripetal acceleration of a point on the rim of a wheel of radius 0.25 m that turns steadily at 6.0 rad s−1?

  1. A
    1.5 m s−2
  2. B
    36 m s−2
  3. C
    144 m s−2
  4. D
    9.0 m s−2
Show answer

Answer: D. ac = rω2 = 0.25 × 36 = 9.0 m s−2. 1.5 is rω, the rim speed, and 144 divides by r instead of multiplying.

2. A flywheel spinning at 10 rad s−1 has a rotational kinetic energy of 200 J. Its moment of inertia is

  1. A
    4.0 kg m2
  2. B
    2.0 kg m2
  3. C
    40 kg m2
  4. D
    20 kg m2
Show answer

Answer: A. K = ½Iω2, so I = 2K/ω2 = 400/100 = 4.0 kg m2. 2.0 kg m2 forgets the ½, and 40 kg m2 divides by ω instead of ω2.

3. The SI unit of angular momentum is

  1. A
    kg m2 s−1
  2. B
    kg m s−1
  3. C
    kg m2
  4. D
    N m
Show answer

Answer: A. L = Iω has the unit kg m2 × s−1 = kg m2 s−1 (the same as J s). kg m s−1 is linear momentum, kg m2 is moment of inertia and N m is torque.

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