Post on 15-Nov-2019
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