Rotational Motion

8.1

Circular Motion:

• Linear speed → distance travelled per unit time
• Tangential speed → linear speed of an object moving along a circular path
  • Called so because the direction of motion is tangential to the circumference of the circle
• Consider a rotating object:
  • Points further away from the center travel a longer distance than points closer to the center during one full rotation.
• Units of linear/tangential speed → m/s or km/h

Linear speed and tangential speed are interchangeable terms in circular motion.

• Rotational speed → number of rotations or revolutions per time.

  • In a rotating object, all points on the object turn about the axis of rotation in the same amount of time.
  • All parts have the same rate of rotation.
  • Meaning → same number of rotations or revolutions per unit of time.

• Rotational rates are expressed in RPM.

  • RPM → Revolutions Per Minute

• Eg: Phonograph turntables:

  • They rotate at 33.33 RPM.
  • A bug sitting anywhere on the surface of the table rotates at 33.33 RPM.

• Tangential and rotational speed are related.

• Tangential speed is directly proportional to rotational speed at any fixed distance from the axis of rotation.

  • more RPMs → more speed in meters per second

• Tangential speed depends on radial distance.

  • Radial distance → distance from the axis.

• Rotational speed does not.

• Consider a rotating platform:

  • At the center, tangential speed = 0 m/s
  • Moving towards the edge, tangential speed increases

• Tangential speed is directly proportional to distance from the axis for any rotational speed.

• Tangential speed ~ radial distance x rotational speed

~ → approximately equal to

• Symbolic form: v ~ rω

ω → greek letter omega, symbolizes rotational speed

• Tangential speed increases with an increase in radial distance, rotational speed, or both.

• Tangential acceleration → when tangential speed undergoes change

• Change in tangential speed means there is an acceleration parallel to tangential motion.

• Eg: person on a rotating platform

  • When it speeds up or slows down, the person undergoes tangential acceleration

• Centripetal acceleration → any object moving in a curved path undergoes this type of acceleration.

  • This is directed towards the center of curvature

Wheels on Railroad Trains:

• Think of rolling a tapered cup across a surface.
  • It makes a curved path.
  • The wider part of the cup has a larger radius and rolls a greater distance.
  • It thus has a greater tangential speed than the narrower part.
• If you consider a pair of cups fastened together at their wider ends and made to roll on a pair of parallel tracks, the cups stay on the track.
  • When they stray off center, the re-center themselves.
• Reason → if the pair shifts towards the left, the wider part of the left cup rides on the left, and the narrower part of the right cup rides on the right.
  • Vice versa for the cup shifting towards the right.
• This is how the pair of cups stays towards the center.
• This is how a moving railroad train stays on the tracks.
• A tapered shape like this is needed on the curves of railroad tracks.
  • On any curve, distance along the other part > distance along the inner part.
• Opposite wheels have the same RPM, but the tangential speed differs slightly based on whether the part of the wheel is wider or narrower.
  • Wider part → higher tangential speed
• When the train rounds a curve, the wheels on the outer track ride on the wider part.
• If the wheels aren’t tapered, scrapings happens and the wheels screech when the train rounds a curve.

8.2

Rotational Inertia:

• An object at rest stays at rest.
• An object moving in a straight line remains in that state of motion.
• An object rotating about an axis remains rotating about the same axis unless affected by some sort of external influence.
  • This is rotational inertia.

Rotating bodies stay rotating. Non-rotating bodies stay non-rotating. (in the absence of any external influence)

• Rotational inertia depends on mass.

• More specifically, it depends on the distribution of mass about the axis of rotation.

  • Greater the distance between the concentration of mass of the object and the axis of rotation, greater the rotational inertia.

• Consider industrial flywheels.

  • Most of their mass is concentrated at the rim.
  • It’s difficult to make them start rotating.
  • Once they do, they have a higher chance of staying that way.

• They are used to store energy in electric power plants.

  • The unwanted energy that is generated is diverted to the flywheels.
  • The wheels are connected to generators that release the power when needed.

• The flywheels are combined in banks of ten or more and connected to power grids.

  • This offsets fluctuations between supply and demand and keeps things running smoothly.

Flywheels are the counterpart of electric batteries, but without the toxic metals and hazardous waste.

• Greater the rotational inertia of an object, the more difficult it is to change its rotational state.

• Consider a tightrope walker:

  • They carry long poles to aid balance.
  • Most of the mass of the pole is away from the axis of rotation.
  • This increases the rotational inertia of the pole.
  • The tightrope walker’s grip on the pole helps them maintain balance. In case they topple, the rotational inertia resists.
  • Longer the pole, better the balance.
  • Stretching arms out along the sides of the body also increases rotational inertia.

• Rotational inertia depends on axis of rotation.

• Consider a rotating pencil:

  • Axis 1 → about central core, parallel to length of pencil.
  • It’s easy to rotate the pencil between your fingertips between the mass is very close to the axis.
  • Axis 2 → about the perpendicular midpoint axis
  • Rotational inertia is greater.
  • Axis 3 → perpendicular to one end
  • Rotational inertia is even greater, the pencil swings like a pendulum when rotated from here.

• A long baseball bat held near the narrow end has more rotational inertia than a short bat.

  • When the long bat is swinging, it is more likely to keep swinging.

• When you run with your legs bent, rotational inertia reduces, and it is easier to rotate them back and forth faster.

  • Longer legs → slower movement

• A solid cylinder will roll down an incline faster than a hoop.

  • Reason → rotational inertia
  • Mass of hoop is further away from the axis of rotation, so the rotational inertia is greater.
  • Meaning → It is harder for the hoop to start moving. It is harder for the hoop to stop moving.
  • Any cylinder would out-roll any hoop on the same incline.

• A rotating object with the greater rotational inertia relative to its own mass has the greater resistance to motion.

8.3

Torque:

• If you held a meterstick horizontally with your hand and dangling a weight from it, the stick would twist.

  • If you slid the weight further away, the stick would twist more.
  • The weight and the force acting on your hand would never change.
  • The torque is what changes.

• Torque → rotational counterpart of force

  • Forces change motion
  • Torque changes rotational motion

• Torque = lever arm x force

  • lever arm → shortest distance between applied force and axis of rotation
  • force → the magnitude of the force that is producing the rotation

• Consider a girl and a boy sitting on a seesaw.

  • The girl is 3m away from the fulcrum. The boy is 1.5m away from the fulcrum.
  • If the torque procuded