Transcript presentation source

Friday, October 16, 1998
Chapter 7: periods and frequencies
Centripetal acceleration & force
Newton’s Law of Gravity
Friday, October 23, 1998 in class
Chapters 5 - 7 inclusive
Hint: Be able to do the homework (both the
problems to turn in AND the recommended ones)
you’ll do fine on the exam!
You may bring one 3”X5” index card (hand-written
on both sides), a pencil or pen, and a scientific
calculator with you.
I will put any constants, math, and Ch. 1 - 4
formulas which you might need on a single
page attached to the back of the exam.
Angular Velocity


t
[  ]
[ ] 
[ t ]
rad
[ ] 
 s 1
s
Angular Acceleration
[  ]
[ ] 
[ t ]
rad / s 2
[ ] 
s
s


t
For an object moving in a circle with a constant
angular velocity (), we can define a frequency
and period associated with the motion.
The period of the
rotation is simply
the time required
for the object to go
around the circle
exactly one time.
T
2

The frequency is the
number of revolutions
completed each second.

f 
2
Notice that frequency
and period are inversely
related to one another…
That is...
1
f 
T
Period
[2 ]
[T ] 
T
[ ]

rad
[T ] 
s
rad / s
2
frequency

f 
2
[ ]
[f ]
[2 ]
rad / s 1
[f ]
s
rad
Equations for Systems Involving
Rotational Motion with Constant
Angular Acceleration
Again, these are completely analogous to what
we derived for the kinetic equations of a linear
system with constant linear acceleration!
   0   0t  t
1
2
   0  t
2
A wheel rotates with constant angular
acceleration of 0 = 2 rad/s2. If the
wheel starts from rest, how many
revolutions does it make in 10 s?
   0   0t  t
1
2
   0t  t  0  (2
1
2
rad
s2
)(10 s)
2
1 rev  2 rad
  100 rad
  (100 rad)
1
2
2
2
1 rev
2 rad

50

rev  159
. rev
We’ve seen how arc length
relates to an angle swept out:
s  r  r ( f   i )
Let’s look at how our angular velocity
and acceleration relate to linear quantities.
First, a question...
In which direction does the instantaneous
velocity of an object moving in a circle point?
For an object moving
in a circle with a constant
linear speed (a constant
angular velocity), the
instantaneous velocity
vectors are always tangent
to the circle of motion!
The magnitude of the tangential velocity can be
found from our relationship of arc length to angle...
s  r
s

r
t
t
vtan  r
For an object moving
in a circle with a constant
linear speed (a constant
angular velocity), the
instantaneous tangential
acceleration is ZERO!
The tangential acceleration tells us how the
tangential velocity changes. It will point
either parallel or anti-parallel to the tangential
velocity vector.
Under what circumstances will the tangential
acceleration be NON-ZERO?
For an object moving
around a circle with a
changing angular velocity,
and hence a changing
tangential velocity, the
instantaneous tangential
acceleration is NON-ZERO!
Let’s look at how the tangential velocity changes
with time in such a case:
vtan  r
vtan vtan f  vtan i r f  r i


t
t f  ti
t f  ti
vtan vtan f  vtan i r f  r i


t
t f  ti
t f  ti
a tan
r

 r
t
A wheel of radius 0.1 m rotates with
constant angular acceleration of
0 = 2 rad/s2. If the wheel starts from
rest, what is the tangential acceleration
at t = 10 s?
atan  r  (01
. m)(2 rad / s )  0.2 m / s
2
2
The tangential acceleration,
however, is not the only
acceleration we need to
consider in problems of
circular motion...
The blue arrows represent the direction of
the CENTRIPETAL ACCELERATION,
which always points towards the center
of the circle.
Although our objects moves
in a circle at constant speed,
it still accelerates.
WHY?
Recall that acceleration and velocity
are both vector quantities. Since
acceleration is a change in velocity
over some change in time, an object
whose velocity changes direction is
accelerating!
Recall our definition of
acceleration:
 
v f  vi

v

a

t f  ti
t
You can demonstrate
to yourself (and the
book provides a nice
proof on p. 186)
v
(r )
2
ac 

 r
r
r
2
t
2
So, tangential acceleration occurs when the
angular velocity of an object changes (i.e.,
the angular acceleration is not equal to 0).
Centripetal acceleration exists for any object
moving in a circle. The instantaneous
centripetal acceleration is related to the
instantaneous tangential velocity as
defined before. Centripetal acceleration
always points towards the center of the
circle.
Tangential
acceleration
The total acceleration will
be the vector sum of the
tangential acceleration and
the centripetal acceleration.
atot  a
2
tan
a
Centripetal
acceleration
vector
ac
2
c
atan
atot
When an object experiences a centripetal
acceleration, there must be a force acting
which results in such an acceleration.
Fc = m a c
For objects moving in a circle, Newton’s 2nd Law
takes the form
mv
Fnet  Fc  mac 
r
2
tan
What kind of forces might result in
a centripetal force?
Force of Tension
What makes this car
turn to the right?
Frictional Force
of Tires on Road
What if the car now goes around
an inclined bend?
Component of Normal Force
of Road on Car
Frictional Force
of Tires on Road
Speed of the car
Angle of the incline
Could point up or down
the road. On what will
the direction of the
frictional force depend?