Recap: Lecture 8
Lecture 9:
Uniform Circular
Motion, Radial
Acceleration
Concept: Rigid Body
• theoretical construct:
•
•
•
•
•
a body in which any two constituent parts are
always at a fixed distance from each other
by construction, an incompressible solid
useful for studies of rotational motion
fix points along a line in a rigid body - define an axis
rotate body around the axis
by construction all pieces will stay in the same
relative position with respect to each other
(fixed distances)
3
Angular Displacement, Angular Velocity
• complete analogy with linear displacement, velocity
Linear
Angular
✓f
xf
xi
x̂
x
Displacement
Avg.Velocity
x̂
x = xf
vavg =
Inst.Velocity vinst = lim
✓i
✓
xi
✓ = ✓f
x
t
t!0
!avg =
x
t
!inst = lim
✓i
✓
t
t!0
✓
t
4
Unit of measure: Radians
• up until now you are accustomed to measuring
angles in degrees [full circle = 360 ]
• in physics calculations, a more elegant unit are
o
radians, which connect the angle to the arc it spans
at a given radius:
arc length
s
s
✓[in radians] =
r
r
r
radius
• quick conversion formula:
⇡
✓[in radians] =
[in degrees]
180
✓
5
H-ITT1: Radians vs
Degrees
H-ITT 1a (1 min)
• An angle of π/2 radians corresponds to
A) 30 degrees
B) 60 degrees
C) 90 degrees ✔
D) 120 degrees
E) 150 degrees
7
Period, Frequency
• when the speed of a point moving around a circle is
•
•
•
•
constant, the motion is called uniform circular
motion
a process which repeats itself in regular intervals is
called periodic
the shortest amount of time after which the process
repeats itself is called the period [usually denoted T]
the frequency is the number of repetitions of a
process per second [usually denoted f, unit: Hz]
relationship between
1
f=
period and frequency:
T
8
H-ITT 2: Angular
Velocity
H-ITT 2a (1 min)
• A wheel is spinning at 3000 rotations per minute. Its
angular velocity is:
A) 100 π radians/sec ✔
B) 150 π radians/sec
C) 6000 π radians/sec
D) 9000 π radians/sec
E) This is not the correct answer
10
Linear vs Angular Speed
• circular motion, during a time Δt object traveled the
distance corresponding to the arc length s:
distance traveled = s = r| !|
average speed =
s
r| ✓|
=
= r|!avg |
t
t
• for infinitesimally short time intervals, the average
speed becomes the abs. value of the inst. velocity
|vinst | = lim
t!0
s
t
= lim r|!avg | = r|!inst |; v = r|!|
t!0
11
Rolling Without Slipping
• key phrasing in physics problems
• implies a fixed relationship between the translational
velocity of an object and its rotational velocity
• usually applied to wheels / cylinders rolling
• implies perfect contact between object and surface
• distance that the axis travels is exactly the same as
the arc length that has been rotated through
s
r s
Implied relationship:
|v| = r|!|
12
Problem: Measuring Car Speeds
• A car is driving at 60 mph. The diameter of each car
wheel is 2 feet.
• What is the angular velocity of a wheel, if the wheel
is rolling on the highway without slipping?
• How many rotations per minute is it performing?
(continued on next page)
13
Car Speeds, continued
14
Radial Acceleration
• recall, the direction of the instantaneous velocity
vector is tangential to the trajectory
• circular motion:
~v(t=2T /3)
~a(t=T /2)
~v(t=5T /6)
~a(t=T /3)
~r
~v(t=0)
~a(t=2T /3)
~a(t=5T /6)
~v(t=T /2)
~v(t=T /6)
~a(t=T /6)
~a(t=0)
~v(t=T /3)
~a is in the direction opposite to ~r
15
Uniform Circular Motion
• recall the relationship between the:
• abs. value of velocity |v|
(rate of change of the position vector)
• radius (r)
(length of the position vector)
~v(t=2T /3)
• angular velocity (ω)
|v| = r|!|
•
~v(t=T /2)
we can use this relationship
to calculate radial acceleration ~v(t=T /3)
~v(t=5T /6)
~r
~v(t=0)
~v(t=T /6)
16
Uniform Circular Motion
• we will now relate the
• abs. value of acceleration |a|
(rate of change of the velocity vector)
• abs. value of velocity (v)
~a(t=T /2)
(length of the velocity vector)
• angular velocity (ω) ~a(t=T /3)
|~a| = v|!|
|~v | = r|!|
|~a| = r|!|
2
|~v |
|~a| =
r
2
~a(t=T /6)
~a(t=2T /3)
~a(t=5T /6)
~a(t=0)
17
Banked vs Unbanked Curves
• banking a curve at an angle
•
produces force towards the
center of the curve
Example: a car moving at 10 m/s
is entering a curve of radius 20
m. There is no friction on the
road. At which angle does the
bank have to be for the car not
to slip in or out of the curve?
2
(use g = 10 m/s )
↵
N~x
~
N
N~y
↵
18