Rotational 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
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.
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.
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
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
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.
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.
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
