PHYSICS 231
Lecture 15: Rotations
Remco Zegers Walk-in hour Tue 4-5 pm helproom
PHY 231
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Chapter 6 in one slide
Momentum p=mv
F=p/t
Impulse (the change in momentum) p= Ft
Types of collisions
Inelastic collisions
Elastic collisions
•Momentum is conserved
•Some energy is lost in the
collision: KE not conserved
•Perfectly inelastic: the
objects stick
together.
•Momentum is conserved
•No energy is lost in the
collision: KE conserved
Conservation of momentum:
m1v1i+m2v2i=m1v1f+m2v2f
Conservation of KE:
Conservation of momentum: ½m v 2+½m v 2=½m v 2+½m v 2
1 1i
2 2i
1 1f
2 2f
m1v1i+m2v2i=(m1+m2)vf
(v1i-v2i)=(v2f-v1f)
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PHY 231
Radians & Radius
Circumference=2r
Part s=r
r
s

=s/r
 in radians!
360o=2 rad = 6.28… rad
 (rad) = /180o  (deg)
PHY 231
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Angular speed and acceleration
 f i



t f  ti
t

  lim
t 0 t
Average angular
velocity
Instantaneous
Angular velocity
Angular velocity : rad/s
rev/s
rpm (revolutions per
minute)
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example
What is the angular velocity of earth around the sun?
Give in rad/s, rev/s and rpm
Answer: in 1 year, the earth makes one full orbit around the
sun.
 f   i 

rad/s
t f  ti
rev/s

t
rpm
= 2 rad/1 year
1 rev/1 year
1 rev/1 year
= 2 rad/(3.2E+7 s) 1 rev/(3.2E+7 s) 1 rev/(5.3E+5 min)
= 2.0E-07 rad/s
3.1E-8 rev/s
1.9E-6 rpm
PHY 231
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Angular acceleration
Definition: The change in angular velocity per time unit

 f  i
t f  ti

  lim
t  0  t
 Average angular

t acceleration
Instantaneous angular
acceleration
Unit: rad/s2
PHY 231
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Equations of motion
Linear motion
Angular motion
X(t)=x(0)+v(0)t+½at2
(t)= (0)+(0)t+½t2
V(t)=V(0)+at
(t)= (0)+t
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example
=0
A person is rotating a wheel. The
handle is initially at =90o. For 5s
the wheel gets an constant angular
acceleration of 1 rad/s2. After that
The angular velocity is constant.
Through what angle will the wheel
have rotated after 10s.

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question
An object is rotating over a circle with radius of 2 m.
Its speed is 1.5 rad/s. After 4 seconds, how many
rotations did the object make?
a) 2
b) 3
c) 4
d) 8
e) 12
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Angular
s
 
r
 1  s

t r t
v

r
v  r
Linear velocities
V
V is called the tangential velocity
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Angular
v  r
v    r
v 

r
t
t
a  r
linear acceleration
velocity
Change in velocity
Change in velocity per time unit
acceleration
The linear acceleration equals the angular acceleration
times the radius of the orbit
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a
example
b
5m
The angular velocity of a is 2 rad/s.
a) What is its tangential velocity?
If b is keeping pace with a,
b) What is its angular velocity?
c) What is its linear velocity?
6m
PHY 231
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A rolling coin
l
d
0=18 rad/s
=-1.80 rad/s
d=0.02 m
a) For how long does the coin roll?
b) What is the average angular velocity?
c) How far (l)does the coin roll before coming to rest?
PHY 231
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question
What is the angular speed about the rotational axis of the
earth for a person standing on the surface?
a)7.3x10-5 rad/s
b)3.6x10-5 rad/s
c)6.28x10-5 rad/s
d)3.14x10-5 rad/s
e) ????
PHY 231
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question
peddles
rear
gear
front
gear
a)
b)
c)
d)
e)
rear wheel
equal to, equal to
equal to, larger than
larger than, larger than
smaller than, larger than
smaller than, equal to PHY 231
A boy is riding his bike.
If the front gear is rotating
with linear velocity v1 and
angular velocity 1 then
the rear gear has a linear
velocity ??????? v1 and
angular velocity ???????
1. The front gear is larger
in radius than the rear gear.
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Next quiz will be like this: bring calculator
gears
If 1=3 rad/s, how fast
Is the bike going?
b) What if r1=0.1 m ?
r3=0.7 m
r1=0.3
r2=0.15
b)
PHY 231
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