Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Chapter 5: Circular Motion
•Uniform Circular Motion
•Radial Acceleration
•Banked and Unbanked Curves
•Circular Orbits
•Nonuniform Circular Motion
•Tangential and Angular Acceleration
•Artificial Gravity
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§5.1 Uniform Circular Motion
y
Consider an object moving
in a circular path of radius r
at constant speed.
v
v
x
Here, v0. The direction
of v is changing.
v
If v0, then a0. The
net force cannot be zero.
v
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Conclusion: to move in a circular path, an object must have
a nonzero net force acting on it.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
y
 is the angular position.
f

Angular displacement:
i
x
   f   i
Note: angles measured CW are negative and angles
measured CCW are positive.  is measured in radians.
2 radians = 360 =1 revolution
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The average and instantaneous angular velocities are:


av 
and   lim
t 0 t
t
 is measured in rads/sec.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
y
arclength = s = r
f
r

i
x
s
 
r
 is a ratio of two lengths; it is
a dimensionless ratio!
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
y
An object moves along a
circular path of radius r; what
is its average speed?
f
r

i
x
total distance r
  
vav 

 r
  r av
total time
t
 t 
Also,
v  r
(instantaneous values).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The time it takes to go one time around a closed path is
called the period (T).
total distance 2r
vav 

total time
T
2
Comparing to v=r:  
 2f
T
f is called the frequency, the number of revolutions (or
cycles) per second.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§5.2 Centripetal Acceleration
The velocity of a particle is tangent to its path.
For an object moving in uniform circular motion, the
acceleration is radially inward.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The magnitude of the radial acceleration is:
v2
ar 
 r 2  v
r
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 5.14): The rotor is an amusement
park ride where people stand against the inside of a cylinder.
Once the cylinder is spinning fast enough, the floor drops out.
(a) What force keeps the people from falling out the bottom of
the cylinder?
y
fs
Draw an FBD for a person
with their back to the wall:
N
x
w
It is the force of static friction.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) If s = 0.40 and the cylinder has r = 2.5 m, what is the
minimum angular speed of the cylinder so that the people
don’t fall out?
1  Fx  N  mar  m 2 r
2  Fy  f s  w  0
Apply Newton’s 2nd Law:
From (2):
From (1)
fs  w
 s N   s m 2 r   mg
9.8 m/s 2
 

 3.13 rad/s
0.402.5 m 
s r
g
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 5.79): A coin is placed on a record
that is rotating at 33.3 rpm. If s = 0.1, how far from the
center of the record can the coin be placed without having it
slip off?
y
Draw an FBD for the coin:
N
fs
x
Apply Newton’s 2nd Law:
1  Fx  f s  mar  m
2  Fy  N  w  0
2
w
r
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
From 1 : f s  m 2 r
From (2)
f s   s N   s mg   m 2 r
s g
Solving for r: r  2

What is ?
rev  2 rad  1 min 
  33.3


  3.5 rad/s
min  1 rev  60 sec 
 s g 0.19.8 m/s 2 
r  2 
 0.08 m
2

3.50 rad/s 
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§5.3 Unbanked and Banked Curves
Example (text problem 5.20): A highway curve has a radius
of 122 m. At what angle should the road be banked so that
a car traveling at 26.8 m/s has no tendency to skid sideways
on the road? (Hint: No tendency to skid means the frictional
force is zero.)
Take the car’s motion
to be into the page.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
y
FBD for the car:

N
x
w
Apply Newton’s Second Law:
v2
1  Fx  N sin   mar  m
r
2  Fy  N cos   w  0
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
Rewrite (1) and (2):
v2
1 N sin   m
r
2 N cos   mg
Divide (1) by (2):
2


v
26.8 m/s
tan  

 0.6007
2
gr 9.8 m/s 122 m 
2
  31.0
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§5.4 Circular Orbits
r
Earth
Consider an object of mass m in a
circular orbit about the Earth.
The only force on the satellite is the force
of gravity:
Gms M e
v2
 F  Fg  r 2  ms ar  ms r
Solve for the speed of the satellite:
Gms M e
v2
 ms
2
r
r
GM e
vCopyright
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r
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: How high above the surface of the Earth does a
satellite need to be so that it has an orbit period of 24 hours?
GM e
v

From previous slide:
r
Also need,
2r
v
T
 GM e 2 
T 
Combine these expressions and solve for r: r  
2
 4




 6.67 10 Nm /kg 5.98 10 kg
2
86400 s  
r  
2
4


 4.225 107 m
11
2
2
24
1
1
3
3
r  Re  h  h  r  Re  35,000 km
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
 GM e 2 
r 
T 
2
 4

1
3
is Kepler’s Third Law.
It can be generalized to:
 GM 2 
r  2 T 
 4

1
3
Where M is the mass of the central body. For example, it
would be Msun if speaking of the planets in the solar system.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§5.5 Nonuniform Circular Motion
Here, the speed is not constant.
a
at
There is now an acceleration
tangent to the path of the particle.
ar
v
The net acceleration of the body is a 
ar  at
2
2
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
a
ar
at
at changes the magnitude of v.
ar changes the direction of v.
Can write:
 F  ma
 F  ma
r
r
t
t
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: What is the minimum speed for the car so that it
maintains contact with the loop when it is in the pictured
position?
FBD for the car at
the top of the loop:
r
y
Apply Newton’s 2nd Law:
x
N
w
v2
 Fy   N  w  mar  m r
v2
N wm
r
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
The apparent weight at the top of loop is:
N = 0 when
v2
N wm
r
 v2


N  m  g 
 r

 v2

N  m  g   0
 r

v  gr
This is the minimum speed needed to make it around the
loop.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
Consider the car at the bottom of the loop; how does the
apparent weight compare to the true weight?
FBD for the car at the
bottom of the loop:
y
N
x
w
Apply Newton’s 2nd Law:
v2
 Fy  N  w  mac  m r
v2
N wm
r
 v2

N  m  g 
 r

Here, N  mg
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§5.6 Angular Acceleration
The average and instantaneous angular acceleration
are:


 av 
and   lim
t 0 t
t
 is measured in rads/sec2.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The kinematic equations:
Angular
Linear
v  v0  at
   0  t
1 2
x  x0  v0t  at
2
2
2
v  v0  2ax
1 2
   0   0 t  t
2
2
2
  0  2
With
vt  r and at  r
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 5.66): A high speed dental drill is
rotating at 3.14104 rads/sec. Through how many degrees
does the drill rotate in 1.00 sec?
Given:  = 3.14104 rads/sec;  = 0
Want .
1 2
   0  0 t  t
2
   0  0t


   0t  3.14  10 4 rads/sec 1.0 sec 
 3.14  10 rads  1.80  10 degrees
4
6
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 5.81): Your car’s wheels are 65 cm in
diameter and are spinning at =101 rads/sec. How fast in
km/hour is the car traveling, assuming no slipping?
v
X
total distance 2r N 2r
v


 r
T N T
total time
 101 rads/sec 32.5 cm 
 3.28 103 cm/sec  118 km/hr
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§5.7 Artificial Gravity
A large rotating cylinder in
deep space (g0).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
FBD for person at
the bottom position
FBD for person at
the top position
y
y
N
x
x
N
Apply Newton’s 2nd Law to each:
2
F

N

ma

m

r
 y
r
2
F


N


ma


m

r
 y
r
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 5.56): A space station is shaped like
a ring and rotates to simulate gravity. If the radius of the
space station is 120m, at what frequency must it rotate so
that it simulates Earth’s gravity?
Using the result from
the previous slide:
2
F

N

ma

m

r
 y
r
N
mg



mr
mr
g
 0.28 rad/sec
r
The frequency is f =(/2) = 0.045 Hz (or 2.7 rpm).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Summary
•A net force MUST act on an object that has circular motion.
•Radial Acceleration ar=v2/r
•Definition of Angular Quantities (, , and )
•The Angular Kinematic Equations
•The Relationships Between Linear and Angular Quantities
vt  r and at  r
•Uniform and Nonuniform Circular Motion
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