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Lecture 11
General Physics (PHY 2130)
• Rotational kinematics and uniform circular motion Angular displacement Angular speed and acceleration
http://www.physics.wayne.edu/~apetrov/PHY2130/
Lightning Review
Last lecture: 1. Forces; laws of motion
field forces and contact forces force of friction and gravitational force tension
Review Problem: Calculate gravitational attraction between two students (say, 70 kg and 90 kg) that are 1 meter apart.
Example:
Question: Calculate gravitational attraction between two students (say, 70 kg and 90 kg) that are 1 meter apart
( )N
mkgkg
kgmN
rmmGF 7
22
211
221 102.4
190701067.6 −− ×≈×==
Extremely small
Compare: weight of the 70 kg (154 lb) person
NmgF 686==
Angular Displacement
► Recall for linear motion: displacement, velocity, acceleration
► Need similar concepts for objects moving in circle (CD, merry-go-round, etc.)
► As before: need a fixed reference system (line) use polar coordinate system
tva
trvrrr if Δ
Δ=
Δ
Δ=−=Δ ,,
Angles measured CW are negative and angles measured CCW are positive.
Angular Displacement
► Every point on the object undergoes circular motion about the point O
► Angles generally need to be measured in radians
► Note:
rs
=θ
°=°
= 3.5723601π
rad
]degrees[180
]rad[ θπ
θ°
=
length of arc
radius
Example
A wheel has a radius of 4.1 m. How far (path length) does a point on athe circumference travel if the wheel is rotated through angles 30°, 30 rad and 30 rev respectively?
Angular Displacement
► The angular displacement is defined as the angle the object rotates through during some time interval
► Every point on the disc undergoes the same angular displacement in any given time interval
if θθθ −=Δ
Angular Velocity
► The average angular velocity (speed), ω, of a rotating rigid object is the ratio of the angular displacement to the time interval
ttt if
if
Δ
Δ=
−
−=
θθθω
Angular Speed
► The instantaneous angular velocity (speed) is defined as the limit of the average speed as the time interval approaches zero
► Units of angular speed are radians/sec (rad/s)
► Angular speed will be positive if θ is increasing
(counterclockwise) negative if θ is decreasing
(clockwise)
tt Δ
Δ=
→Δ
θω
0lim
Angular Acceleration
► What if object is initially at rest and then begins to rotate?
► The average angular acceleration, α, of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:
► Units are rad/s² ► Similarly, instant. angular accel.:
ttt if
if
Δ
Δ=
−
−=
ωωωα
tt Δ
Δ=
→Δ
ωα
0lim
Notes about angular kinematics: When a rigid object rotates about a fixed axis,
every portion of the object has the same angular speed and the same angular acceleration
► i.e. θ, ω, and α are not dependent upon r, distance form hub or axis of rotation
Examples:
1. Bicycle wheel turns 240 revolutions/min. What is its angular velocity in radians/second?
secradians1.25secradians8rev1rads2
sec60min1
minrev240 ≈=××= π
πω
2. If wheel slows down uniformly to rest in 5 seconds, what is the angular acceleration?
2secrad5sec5
secrad250−=
−=
Δ
−=
tif ωω
α
Examples:
Given: 1. Angular velocity: 240 rev/min 2. Time t = 5 s Find: 1. θ = ?
3. How many revolution does it turn in those 5 sec?
( ) ( )( )
srevolution102rev1rad5.62)(
rad5.62sec5secrad521sec5secrad25
21
2
20
=×=
=−+=
+=
πθ
αωθ
rev
tt
Recall that for linear motion we had: Perhaps something similar for angular quantities?
20 2
1 attvx +=
Analogies Between Linear and Rotational Motion
Rotational Motion About a Fixed Axis with Constant Acceleration
Linear Motion with Constant Acceleration
ti αωω +=
2
21 tti αωθ +=Δ
θαωω Δ+= 222i xavv i Δ+= 222
2
21 attvx i +=Δ
atvv i +=
Relationship Between Angular and Linear Quantities
► Displacements
► Speeds
► Accelerations
rsΔ
=Δθ
vr
ts
rt
1or
1
=
Δ
Δ=
Δ
Δ
ω
θ
ra α=
Relationship Between Angular and Linear Quantities
► Displacements
► Speeds
► Accelerations
► Every point on the rotating object has the same angular motion
► Every point on the rotating object does not have the same linear motion
rs θ=
rv ω=
ra α=
ConcepTest A ladybug sits at the outer edge of a merry-go-round, and
a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second.The gentleman bug’s angular speed is
1. half the ladybug’s. 2. the same as the ladybug’s. 3. twice the ladybug’s. 4. impossible to determine
ConcepTest A ladybug sits at the outer edge of a merry-go-round, and
a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second.The gentleman bug’s angular speed is
1. half the ladybug’s. 2. the same as the ladybug’s. 3. twice the ladybug’s. 4. impossible to determine
Note: both insects have an angular speed of 1 rev/s
19
The time it takes to go one time around a closed path is called the period (T).
Trv π2
timetotaldistance total
av ==
Comparing to v = rω: fT
ππ
ω 22==
f is called the frequency, the number of revolutions (or cycles) per second.
Period and frequency
Centripetal Acceleration
► An object traveling in a circle, even though it moves with a constant speed, will have an acceleration (since velocity changes direction)
► This acceleration is called centripetal (“center-seeking”).
► The acceleration is directed toward the center of the circle of motion
Centripetal Acceleration and Angular Velocity
► The angular velocity and the linear velocity are related (v = ωr)
► The centripetal acceleration can also be related to the angular velocity
ts
rva
tv
srvv
rs
vv
Δ
Δ=⇒
Δ
Δ=
Δ=Δ⇒Δ
=Δ
a
but,
rarva CC
22
or ω==Thus:
Similar triangles!
Total Acceleration
► What happens if linear velocity also changes?
► Two-component acceleration: the centripetal component of the
acceleration is due to changing direction
the tangential component of the acceleration is due to changing speed
► Total acceleration can be found from these components:
22Ct aaa +=
slowing-down car
Vector Nature of Angular Quantities
► As in the linear case, displacement, velocity and acceleration are vectors:
► Assign a positive or negative direction
► A more complete way is by using the right hand rule Grasp the axis of rotation
with your right hand Wrap your fingers in the
direction of rotation Your thumb points in the
direction of ω
Forces Causing Centripetal Acceleration
► Newton’s Second Law says that the centripetal acceleration is accompanied by a force
F stands for any force that keeps an object following a
circular path ► Force of friction (level and banked curves) ► Tension in a string ► Gravity
rvmmaF C
2
==∑
Example1: level curves
Consider a car driving at 20 m/s (~45 mph) on a level circular turn of radius 40.0 m. Assume the car’s mass is 1000 kg.
1. What is the magnitude of
frictional force experienced by car’s tires?
2. What is the minimum coefficient of friction in order for the car to safely negotiate the turn?
Example1:
Given: masses: m=1000 kg velocity: v=20 m/s radius: r = 40.0m Find: 1. f=? 2. µ=?
1. Draw a free body diagram, introduce coordinate frame and consider vertical and horizontal projections
mgNmgNFy
=
−==∑ 0
( ) Nmsmkg
rvmmaf
fmaFx
422
100.140201000 ×−=−=−=−=
−==∑
2. Use definition of friction force:
02.18.91000
101.0
thus,10
2
4
42
≈×
=
===
smkgN
Nrvmmgf
µ
µ
Lesson: µ for rubber on dry concrete is 1.00!
rubber on wet concrete is 0.2!
driving too fast…
ConcepQuestion
Is it static or kinetic friction that is responsible for the fact that the car does not slide or skid?
1. Static 2. Kinetic
Example2: banked curves
Consider a car driving at 20 m/s (~45 mph) on a 30° banked circular curve of radius 40.0 m. Assume the car’s mass is 1000 kg.
1. What is the magnitude of
frictional force experienced by car’s tires?
2. What is the minimum coefficient of friction in order for the car to safely negotiate the turn?
A component of the normal force adds to the frictional force to allow higher speeds rg
v2tan =θ
Example2:
Given: masses: m=1000 kg velocity: v=20 m/s radius: r = 40.0m angle: α = 30° Find: 1. f=? 2. µ=?
1. Draw a free body diagram, introduce coordinate frame and consider vertical and horizontal projections
Nmgrvmf
mgfrvmFx
376030sin30cos
30sin30cos
2
2
=−=
−−=−=∑
NmgrvmN
mgNrvmFy
42
2
103.130cos30sin
30cos30sin
×=+=
−==∑
2. Use definition of friction force:
28.0101.3
3760is minimalthus,
4
s
≈×
==
=
NN
NfNf
ss
s
µ
µµ
Lesson: by increasing angle of banking,
one decreases minimal µ or friction with which one can take curve!