ROTATIONAL MOTION

Center of Mass● Newton's Laws and Conservation Laws

– Describe and predict the motion of physical systems– In these laws, a system has only one acceleration

● But systems can be made up of many parts– Each moving and accelerating in different directions– How to decide on one position, velocity, acceleration?

● Center of Mass (“CM”)– The average position of the mass in a system– Newton's Laws describe the motion of the CM– Sometimes, there is no mass at the CM(!)

Center of Mass: Example● Box of air molecules

– Many different positions and velocities– CM is at center of box– CM of air molecules does not move

● Terminology– Each molecule is said to moving “relative to the CM”

● Momentum and Energy– This system has a total momentum of zero – This system does have Kinetic Energy → Temperature

CM

“Rigid” Objects● The atoms in a rigid object are fixed in position relative

to each other– This is an idealization: Every real object will deform if a big

enough force is exerted on it

● In physics we often treat solid objects as being “rigid”– We pretend the “atomic springs” are infinitely stiff

– A pretty good approximation

● Rigid objects and the CM– Atoms can move relative to the CM only if the object rotates

Circular Motion● Rotation always occurs around an axis

– Axis: a line around which the rotation occurs

● When a rigid object rotates around an axis:– Every atom in the object moves in a circle– The radius of an atom's circle is the distance

between that atom and the axis

Circular Motion● When a rigid object rotates, every atom takes the

same amount of time to complete one circle– This time is called the period of the circular motion

– Earth's rotational period is 1 day

● Different atoms move around circles of different radius– So the distance traveled by each atom is different

● For any given atom:– Atoms which are far from the axis move faster!

Speed = Circumference of circlePeriod

Angular Speed● Every circle covers 360° of angle

– Every atom in a rotating object covers 360° in one period

● Angular Speed ( symbol: ω )– Measures how much angle is covered per second– Angular speed is the same for all atoms in a rigid object– ω is also called “rotational speed”

● Units: degreessec

, revolutionssec

, radianssec

Tangential Speed● Tangential speed is the actual speed of an atom in a

rotating object– It is called “tangential speed” so it doesn't get confused with

“angular speed”

● Angular speed is the same for every atom– Tangential speed varies between atoms; it depends on the

distance an atom is from the axis of rotation

v = rv = tangential speed

r = distance from axis

= angular speed

Rotational Inertia● Concept of “inertia” applies to rotational motion

– A rotating object's “natural state” is to continue rotating at a constant angular speed

– Applying a force can change rotational speed

● Rotational motion: inertia ≠ mass– Distance from mass to rotation axis is also a factor

– Rotational inertia depends on shape

– It also depends on the chosen axis

Rotational Inertia: An Example● A spinning figure skater

– Begins the spin with arms extended away from the body

– Mass of arms is far from axis of rotation– Has a large rotational inertia

● To spin faster, bring arms inward– Decreases rotational inertia

– Angular speed increases

Torque● Newton's Laws: Forces can cause changes in speed

● To change angular speed:– Force must be applied (no surprise)– Location and direction of force are important

● The combination of force, location, and direction is called Torque

SmallTorque mg

LargeTorquemg

Calculating Torque – “Lever Arm” ● Torque has three ingredients:

– Strength of the applied force– Direction of the applied force– Location of the applied force

● Direction and location are combined to form a “lever arm”

– Distance from the axis of rotation to the location of the force

– Must be measured perpendicular to the direction of the force

F

d

Torque = F d

Torque Example: See-Saw

● The torque produced by a person's weight depends on the lever arm

● Torques come in two directions: Clockwise (CW) and Counter-clockwise (CCW)

– To balance see-saw, CCW and CW torques must balance perfectly ( zero net torque! )

m1g

m2g

d1 d

2CCW

CW

Torque Example: Accelerating Car

F

CM

F

CM

Accelerating Car

Forward force makes CCW torque

Front of car tips upward

Decelerating Car

Backward force makes CW torque

Front of car tips downward

Equilibrium: Revised● A better definition of equilibrium:

– Fnet

= 0 ( zero net force → constant velocity )

– Tnet

= 0 ( zero net torque → constant angular velocity )

● It is possible to have a net force without a net torque!– And vice-versa!

Equilibrium and Balance● A tall object needs a base in order to stand

– Edges of the base are made up of the parts of the object in contact with the ground

● Balance of tall objects requires equilibrium– The CCW torque must cancel the CW torque

● Balance: The CM must be above the base– If the CM moves outside the base:– The object begins to rotate...– And fall down!

CMCM

Using Force to Change Direction● An object's natural state is a constant velocity

– Velocity includes both speed and direction

● A net force changes the velocity of an object– The effect depends on the direction of the net force and

the direction of the velocity

v

FSpeeding up

v

FFSlowing down

vF

FChanging direction

Centripetal Force● Circular motion requires changing direction at all times

– Therefore it requires a force at all times!

● Centripetal means “toward the center”– The required direction of force for circular motion

vF

CCW – constant left turn

Fv v

F (road on tires)

CW – constant right turn

Calculating Centripetal Force● Centripetal force has three ingredients:

– The mass of the object moving along a curve– The tangential speed of the object– The radius of the curve ( tight curves → small radius )

● Note: Centripetal force doesn't just “happen”– Must be provided by some other force– Tension, friction, gravity, etc.

F cent =mv2

r

Centrifugal Force (Fictitious)● Newton's Laws must be used carefully when the

observer is accelerating!– If the observer accelerates, “fictitious forces” appear

● Example: The driver of an accelerating car– The driver is pushed forward by the seat ( actual force )– Observers inside the car feel pushed back into their seat for no

reason ( fictitious force – the observer is accelerating )

● Centrifugal force is a fictitious force felt by observers who are in circular motion ( that is, accelerated )

– Feel an unexplained push toward the outside of the circle

Centrifugal Force (Example)

● Soda can on dashboard of car– When car turns left, it is accelerating– Observers in car measure a fictitious force pushing

the can to the right

● A more “Newtonian” view– Can's natural state: move forward in a straight line– When car turns, friction tries to keep can and

dashboard together – If friction is too weak to force the can to turn tightly:

can goes straight while car goes left

Car Can

Newton's Laws – Rotational Motion● Analogy: Linear Motion → Rotational Motion

– x → θ; v → ω; a → α; m → I; F → T

● Newton's Laws can be applied to rotational motion by replacing the appropriate quantities

– 1st Law: Zero net torque → rotation at constant ω– 2nd Law: T = I α– 3rd Law: Action torque → opposite reaction torque

Angular Momentum● The concept of momentum can also be

translated into rotational motion:

● Angular momentum ( like ω ) can point in the CW or CCW direction

● Total angular momentum is conserved

Momentum= mv Angular Momentum= I

Angular Momentum Conservation Example

● As the Moon orbits the Earth:– It pulls on the earth's oceans (causing the tides)– This pull creates friction between the oceans and rocks– This friction slows down the earth's spin

● Earth's angular momentum decreases over time– So the Moon's angular momentum must increase!– Every year, the Moon gets further away by about an inch

Summary● Rigid objects can rotate around their CM

● ω is the same for all atoms, v is different

● Rotational Inertia depends on distance from the axis of rotation

● Rotational motion laws can be made from linear motion laws by simple replacement