Transcript Rotational Motion Honors 2016x

Chapter 7
Rotational Motion
Angles, Angular Velocity and
Angular Acceleration
Rotational Motion

Rotational motion: motion of a body that spins
about an AXIS

Three different measures of angles:
1.
2.
3.
Degrees
Revolutions (1 rev. = 360 deg.)
Radians (2p rad.s = 360 deg.)
Angular Displacement

Angular Displacement: change in distance
of a point

Arc Length: the distance of sweep for an angular
displacement
- Symbol: S
- Unit: meter
- Equation: S = rθ
r = radius/distance from axis of
rotation
Θ = angle of rotation
Angular Displacement
Finish
S or arc
length
θ
Start
Clockwise = negative direction
Counterclockwise = positive direction
Revolution vs Rotation

Revolution: movement of an object through
360º about an external axis


Ex: the Earth revolves around the sun once every
365.25 days
Rotation: movement of an object through
360º about an internal axis

Ex: the Earth rotates about itself approximately once
every 24 hours
Revolution in terms of Problem
Solving

Radians and Degrees

Example
A girl on a merry-go-round moves clockwise through an
arc length of 2.50 m. If the girl’s angular displacement is
1.67 radians, how far is she from the center of the merrygo-round?
Solution
θ
θ= 1.67
radians
S= 2.50 meters
Angular Speed

h
Example
A car tire rotates with an average angular speed of 29 rad/s. In
what time interval will the tire rotate 3.5 times?
Note: 1 rotation = 2π
Angular Acceleration
Rate of change of angular speed
 Denoted by a

w f  wi w
a

t
t
Units of angular acceleration are rad/s²
 w must be in radians per sec.


Every portion of the object has same
angular speed and same angular
acceleration
Example
A car tire rotates with an initial angular speed of 21.5 rad/s. The
driver then accelerates and after 3.5 s the tires angular speed
is 28.0 rad/s. What is the tires average angular acceleration?
Analogies Between Linear and
Rotational Motion
Rotational Motion
1 2
  w i t  a t
2
ωf  ωi  αt
Linear Motion
1 2
x  vi t  at
2
v f  vi  at
ω  ω  2a  v f  v  2 ax
2
f
2
i
2
2
i
Tangential Speed

Tangential Speed- the speed of something
linearly that is moving in rotational motion

WHAT?!




Think of tangential speed as this….you are swinging
a ball that is attached to a string around in a circle.
If you cut the line during mid-swing, which
direction and with what velocity will the ball fly?
Simulation
http://www.mrwaynesclass.com/teacher/circular/TargetPractice/home.html
https://www.youtube.com/watch?v=Lwo9AvTA5xY

Who Spins Faster?

Which child has greater rotational speed?
Which child has greater tangential speed?
Child
#2
Child
#1
http://www.mrwaynesclass.com/teacher/circular/disc/home.html
Track and Field…Throwing
Events
http://www.youtube.com/watch?v=pT1o4K7GdzU
 http://www.youtube.com/watch?v=zQoEPPPJGCw
 http://www.youtube.com/watch?v=SpMVwN-cXiI
 http://www.youtube.com/watch?v=Q-96j3T3Dh0

Example
A discus thrower spins in a circle before
releasing the disc with a tangential speed of 9.0
m/s. If the radius the disc from the center of
the throwers mass is 0.75m, what is the angular
speed of the thrower?
Example

Tangential Acceleration

Tangential acceleration is a measure of how the
tangential velocity of a point at a certain radius
changes with time.



Symbol at
Unit: m/s2
Equation:
a t =rα
For previous equations ensure that , w and
a are in radians!
Centripetal Acceleration
Moving in a circle at constant SPEED does
not mean constant VELOCITY
 Centripetal acceleration results from
CHANGING DIRECTION of the velocity

Centripetal Acceleration, cont.

Acceleration is directed toward the center of
the circle of motion
Symbol: ac
 Unit: m/s2

Equation
v t (vf -vi )
2
ac 

 rw
r
r
2
2
Example


A race car moves along a circular track
at an angular speed of 0.512 rad/s. If
the car’s centripetal acceleration in
15.4 m/s2 , what is the radius of the
track?
𝑎𝑐 = 𝑟𝜔2 , 𝑟 =
𝑎𝑐
𝜔2
=
15.4 𝑚/𝑠 2
(0.512 𝑟𝑎𝑑/𝑠)2
= 58.7 m
Centripetal Force

Centripetal Force: force directed toward the fixed
center that causes an object to follow a circular path


What provides the centripetal force for the whirling
ball example from the previous slides?
Motorcycles in a cage:

http://www.youtube.com/watch?v=gJPPftZYe_M
Centripetal Force
Symbol: Fc
 Unit: N
 Equation:

𝐹𝑐 =
𝑚𝑉𝑡 2
𝑟
Due to the centripetal acceleration
caused by the change in direction
during rotation
 Not included in a free body diagram as
it is the NET FORCE

Centrifugal Force

Centrifugal Force: fictitious force, apparent outward
force on a rotating or revolving body

Why does it feel like there is a force acting on you as
you spin in the gravitron (rotor)?

Centrifugal Force is really inertia. What?!
 When an object is spun in a circle, it resists a change
in its motion. This includes direction! If there is no
support or frictional force acting on an object as it
spin, the object will fly off in a straight line path.
Example

Forces Causing Centripetal
Acceleration

Newton’s Second Law
F  mac
Radial acceleration requires radial force
 Examples of forces

Spinning ball on a string
 Gravity
 Electric forces, e.g. atoms

Newton’s Law of Universal
Gravitation
Force is always attractive
 Force is proportional to both masses
 Force is inversely proportional to
separation squared

Gravitation Constant
Determined experimentally
 Henry Cavendish, 1798
 Light beam / mirror amplify motion

END