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ROTATIONAL MOTION
Center of Mass
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Newton's Laws and Conservation Laws
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But systems can be made up of many parts
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Describe and predict the motion of physical systems
In these laws, a system has only one acceleration
Each moving and accelerating in different directions
How to decide on one position, velocity, acceleration?
Center of Mass (“CM”)
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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
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Box of air molecules
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CM
Terminology
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Many different positions and velocities
CM is at center of box
CM of air molecules does not move
Each molecule is said to moving “relative to the CM”
Momentum and Energy
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This system has a total momentum of zero
This system does have Kinetic Energy → Temperature
“Rigid” Objects
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The atoms in a rigid object are fixed in position relative
to each other
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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”
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We pretend the “atomic springs” are infinitely stiff
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A pretty good approximation
Rigid objects and the CM
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Atoms can move relative to the CM only if the object rotates
Circular Motion
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Rotation always occurs around an axis
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Axis: a line around which the rotation occurs
When a rigid object rotates around an axis:
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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
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When a rigid object rotates, every atom takes the
same amount of time to complete one circle
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This time is called the period of the circular motion
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Earth's rotational period is 1 day
Different atoms move around circles of different radius
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So the distance traveled by each atom is different
For any given atom:
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Speed =
Circumference of circle
Period
Atoms which are far from the axis move faster!
Angular Speed
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Every circle covers 360° of angle
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Angular Speed ( symbol: ω )
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Every atom in a rotating object covers 360° in one period
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:
degrees revolutions radians
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,
sec
sec
sec
Tangential Speed
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Tangential speed is the actual speed of an atom in a
rotating object
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It is called “tangential speed” so it doesn't get confused with
“angular speed”
Angular speed is the same for every atom
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Tangential speed varies between atoms; it depends on the
distance an atom is from the axis of rotation
v = tangential speed
v = r
r = distance from axis
 = angular speed
Rotational Inertia
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Concept of “inertia” applies to rotational motion
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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
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Distance from mass to rotation axis is also a factor
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Rotational inertia depends on shape
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It also depends on the chosen axis
Rotational Inertia: An Example
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A spinning figure skater
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Begins the spin with arms extended away from the body
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Mass of arms is far from axis of rotation
Has a large rotational inertia
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To spin faster, bring arms inward
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Decreases rotational inertia
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Angular speed increases
Torque
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Newton's Laws: Forces can cause changes in speed
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To change angular speed:
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Force must be applied (no surprise)
Location and direction of force are important
The combination of force, location, and direction is
called Torque
Small
Torque
mg
mg
Large
Torque
Calculating Torque – “Lever Arm”
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Torque has three ingredients:
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Strength of the applied force
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Direction of the applied force
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Location of the applied force
Direction and location are combined to
form a “lever arm”
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Distance from the axis of rotation to the
location of the force
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Must be measured perpendicular to the
direction of the force
Torque = F d
F
d
Torque Example: See-Saw
d1
CCW
CW
d2
m2g
m1g
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The torque produced by a person's weight depends on
the lever arm
Torques come in two directions: Clockwise (CW) and
Counter-clockwise (CCW)
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To balance see-saw, CCW and CW torques must balance
perfectly ( zero net torque! )
Torque Example: Accelerating Car
CM
CM
F
F
Accelerating Car
Decelerating Car
Forward force makes
CCW torque
Backward force makes
CW torque
Front of car tips upward
Front of car tips downward
Equilibrium: Revised
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A better definition of equilibrium:
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Fnet = 0 ( zero net force → constant velocity )
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Tnet = 0 ( zero net torque → constant angular velocity )
It is possible to have a net force without a net torque!
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And vice-versa!
Equilibrium and Balance
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A tall object needs a base in order to stand
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Balance of tall objects requires equilibrium
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Edges of the base are made up of the parts
of the object in contact with the ground
The CCW torque must cancel the CW torque
Balance: The CM must be above the base
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If the CM moves outside the base:
The object begins to rotate...
And fall down!
CM
Using Force to Change Direction
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An object's natural state is a constant velocity
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Velocity includes both speed and direction
A net force changes the velocity of an object
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The effect depends on the direction of the net force and
the direction of the velocity
v
v
F
v
F
Speeding up
F
Slowing down
Changing direction
Centripetal Force
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Circular motion requires changing direction at all times
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Therefore it requires a force at all times!
Centripetal means “toward the center”
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The required direction of force for circular motion
v
F
F
v
v
F (road on tires)
CCW – constant left turn
CW – constant right turn
Calculating Centripetal Force
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Centripetal force has three ingredients:
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The mass of the object moving along a curve
The tangential speed of the object
The radius of the curve ( tight curves → small radius )
m v2
F cent =
r
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Note: Centripetal force doesn't just “happen”
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Must be provided by some other force
Tension, friction, gravity, etc.
Centrifugal Force (Fictitious)
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Newton's Laws must be used carefully when the
observer is accelerating!
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Example: The driver of an accelerating car
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If the observer accelerates, “fictitious forces” appear
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 )
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Feel an unexplained push toward the outside of the circle
Centrifugal Force (Example)
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Soda can on dashboard of car
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When car turns left, it is accelerating
Observers in car measure a fictitious force pushing
the can to the right
Car
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A more “Newtonian” view
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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
Can
Newton's Laws – Rotational Motion
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Analogy: Linear Motion → Rotational Motion
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x → θ;
v → ω;
a → α;
m → I;
F→T
Newton's Laws can be applied to rotational
motion by replacing the appropriate quantities
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1st Law: Zero net torque → rotation at constant ω
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2nd Law: T = I α
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3rd Law: Action torque → opposite reaction torque
Angular Momentum
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The concept of momentum can also be
translated into rotational motion:
Momentum = m v
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Angular Momentum = I 
Angular momentum ( like ω ) can point in the
CW or CCW direction
Total angular momentum is conserved
Angular Momentum Conservation Example
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As the Moon orbits the Earth:
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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
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So the Moon's angular momentum must increase!
Every year, the Moon gets further away by about an inch
Summary
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Rigid objects can rotate around their CM
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ω is the same for all atoms, v is different
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Rotational Inertia depends on distance from the
axis of rotation
Rotational motion laws can be made from linear
motion laws by simple replacement