Physics 121,Lecture 11,Pg 1
Physics 121,Sections 9,10,11,and 12
Lecture 11
Today’s Topics:
? Homework 5,Due Friday Oct,7 @ 6:00PM
? Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Chapter 5,Rotation
? Angular variables
? Centripetal acceleration
?Kepler’s laws
? Artificial gravity
Physics 121,Lecture 11,Pg 2
Chapter 5
Uniform Circular Motion
? What does it mean?
? How do we describe it?
? What can we learn about it?
Physics 121,Lecture 11,Pg 3
Recap:
R
v
s
???t
(x,y)
x = R cos(?)?= R cos(?t)?
y = R sin(?)?= R sin(?t)
? = tan-1 (y/x)
? = ?t
s = v t
s = R? = R?t
v = ?R
Physics 121,Lecture 11,Pg 4
Centripetal Acceleration
? UCM results in acceleration:
?Magnitude,a = v2 / R = ?? R since v = ?R
?Direction,- r (toward center of circle)
R
a
?
^
Physics 121,Lecture 11,Pg 5
Problem,Rotating puck & weight.
? A mass m1 slides in a circular path with speed v
on a horizontal frictionless table,It is held at a
radius R by a string threaded through a
frictionless hole at the center of the table,At the
other end of the string hangs a second mass m2.
?What is the tension in the string T?
?What is the speed of the sliding mass v?
m1
m2
v
R
Physics 121,Lecture 11,Pg 6
Problem,Rotating puck & weight...
? Draw FBD of hanging mass:
?Since R is constant,a = 0.
so T = m2g
m2
m2g
T
m1
m2
v
R
T
Physics 121,Lecture 11,Pg 7
Problem,Rotating puck & weight...
? Draw FBD of sliding mass:
m1
T = m2g
v gR mm? 2
1
m1g
N
m1
m2
v
R
T
Use F = T = m1a
where a = v2 / R
m2g = m1v2 / R
Physics 121,Lecture 11,Pg 8
Forces Causing Centripetal Acceleration
r
va t
c
2
?
r
vmaF t
c
2
??
v
F
r
mg
m
f
n
Physics 121,Lecture 11,Pg 9
Forces Causing Centripetal Acceleration
? Turn on a sloped road …
?
mg
n
?s in
2
nrvmaF tc ???
mg
n sin ?
n cos ?
Physics 121,Lecture 11,Pg 10
Accelerated Reference Frames:
The Accelerometer
? Your first job is with General Motors,You are
working on a project to design an accelerometer,
The inner workings of this gadget consist of a
weight of mass m that is hung inside a box that is
attached to the ceiling of a car,You design the
device with a very light string so that you can
mathematically ignore it,The idea is that the
angle the string makes with the vertical is
determined by the car’s acceleration,Your
preliminary task is to think about calibration of the
accelerometer,First you calibrate the
measurement for when the car travels on a flat
road,
Physics 121,Lecture 11,Pg 11
Accelerated Reference Frames:
The Accelerometer
a
?
i
1
We need to solve for the angle the plum bob makes with
respect to vertical,
First we will solve by using Newton’s Second Law and checking
x and y components,Then we will consider other possible
solution methods,
Physics 121,Lecture 11,Pg 12
Accelerometer...
2,Draw a free-body diagram for the mass:
We wish to solve for ? in terms of the acceleration a.
We will use SF = ma in two cartesian dimensions.
m
T
mg (gravitational force)
?
a
x
y
?
Physics 121,Lecture 11,Pg 13
Accelerometer...
3,Solving,
i,FX = TX = T sin ? = ma
j,FY = TY - mg
= T cos ??- mg = 0
? Eliminate T
T
mg
m
ma
j
i
?
?
T sin ???= ma
T cos ???= mg
t a n ? ? ag
Physics 121,Lecture 11,Pg 14
Accelerometer...
4,No Numbers involved
5,This answer has the right units (none)
It does give 1 in terms of the acceleration,
t a n ? ? ag
Physics 121,Lecture 11,Pg 15
Accelerometer – Other Thoughts
? Think of this problem from the point of view of the person
inside the car.
? This person sees the plumb bob making the same angle
with respect to the ground,but detects no acceleration.
t a n ? ? ag
a
?
Physics 121,Lecture 11,Pg 16
Accelerometer...
? There must be some other force to balance the x component
i,FX = TX + F? = T sin ? + F? = 0
j,FY = TY - mg
= T cos ??- mg = 0
? And we must put F? = -ma to get the same answer as before.
? F? is known as a fictitious force.
t a n ? ? ag
?
a T
mg
Physics 121,Lecture 11,Pg 17
Lecture 11,ACT 1
Accelerated Reference Frames
You are a passenger in a car and not wearing your seatbelt,
Without increasing or decreasing speed,the car makes a sharp
left turn,and you find yourself colliding with the right-hand door,
Which is a correct description of the situation?
A) Before and after the collision there is a rightward force pushing
you into the door.
B) Starting at the time of the collision,the door exerts a leftward
force on you.
C) Both of the above.
D) Neither of the above.
Physics 121,Lecture 11,Pg 18
Newton’s Law of Universal Gravitation
? Every particle in the Universe attracts every other
particle with a force that is directly proportional the
product of their masses and inversely proportional to
the square of the distance between them.
? Constant of Universal gravitation:
2
21
r
mmGF ?
221110673.6 kgmNG ??? -
m1
m2
r12
F21 F12
Physics 121,Lecture 11,Pg 19
Newton’s Law of Universal Gravitation
? The gravitational force is a field force that always
exists between two particles regardless of the
medium that separates them.
? The force varies as one over the square of the
distance between the particles and therefore
decreases rapidly with increasing separation.
? The gravitational force is actually very weak as
shown be the size of G.
It is easily observed for a falling ball of mass m
because the mass of Earth ME is so large in the
product mME
221110673.6 kgmNG ??? -
Physics 121,Lecture 11,Pg 20
Newton’s Law of Universal Gravitation
? The gravitational force exerted by a spherical
object on a particle outside the sphere is the
same as if the entire mass of the sphere were
concentrated at its center:
2
E
E
R
mMGF ?
Physics 121,Lecture 11,Pg 21
Kepler’s Laws
? Much of Sir Isaac’s motivation to deduce the laws of
gravity was to explain Kepler’s laws of the motions of the
planets about our sun,
? Ptolemy,a Greek in Roman times,famously described a
model that said all planets and stars orbit about the
earth,This was believed for a long time.
? Copernicus (1543) said no,the planets orbit in circles
about the sun,
? Brahe (~1600) measured the motions of all of the known
planets and the position of 777 stars (ouch !)
? Kepler,his student,tried to organize all of this,He came
up with his famous three laws of planetary motion.
Physics 121,Lecture 11,Pg 22
Copernicus Brahe Kepler
1473-1543 1546-1601 1571-1630
Physics 121,Lecture 11,Pg 23
Keplers’ Laws
? After 20 years of work on Tycho Brahe’s data,Kepler
formulated his three laws:
1,All planets move in elliptical orbits
with the Sun at one of the focal
points
2,A line drawn from the Sun to any
planet sweeps out equal areas in
equal time intervals
3,The square of the orbital period of
any planet is proportional to the
cube of the average distance
from the planet to the Sun
? It was later shown that all three of these
laws are a result of Newton’s laws of
gravity and motion.
Physics 121,Lecture 11,Pg 24
Kepler’s First Law
? In the Solar System,the Sun is at one focus of
the elliptical orbit of each planet and the other
focus is empty
Sun
Planet
Physics 121,Lecture 11,Pg 25
Kepler’s First Law
p q
? The sum p + q is the same for every point on the
ellipse
Physics 121,Lecture 11,Pg 26
Kepler’s Second Law
? A line drawn from the Sun to any planet sweeps out
equal areas in equal time intervals
? Due to angular momentum conservation
A
B
C
D
Physics 121,Lecture 11,Pg 27
Kepler’s Third Law
? A planet of mass Mp is moving about the Sun Ms in a
circular orbit,Newton’s second law gives:
r
vM
r
MMG pps 2
2 ?
The speed v = 2?r/T,with T is the period of the planet.
r
Tr
r
MG s 2
2
)/2( ??
3
21923322 1097.244
m
s
GMKrKrGMT ssss
-??????
?
?
???
?? ??
Physics 121,Lecture 11,Pg 28
Kepler’s Third Law
Ms
Mp
2
322 44
GT
r
GKM ss
?? ??
Kepler’s third law gives us a method for measuring
the mass of the Sun.
33
2
2 4 rKr
GMT ss ????
?
???
?? ?
Physics 121,Lecture 11,Pg 29
Geosynchronous Orbit
h
r
R
v
F
The gravitational force produces
the centripetal acceleration:
r
mv
r
mMG E 2
2 ?
T
r
T
dv ?2??
3/1
2
2
4 ?
?
?
?
???
??
EGM
Tr
?
T = 24h and ME = 5.98 ? 1024 kg
The height above Earth is app,22 000 miles
Physics 121,Lecture 11,Pg 30
Summary
(with comparison to 1-D kinematics)
Angular Linear
c o n s ta n t??
??
? ? ? 0 ??t
? ? ? ?? ? ?0 0 212t t
co n st a n ta ?
v v at? ?0
x x v t at? ? ?0 0 212
And for a point at a distance R from the rotation axis:
x = R????????????v = ?R ??????????a = ?R
Physics 121,Lecture 11,Pg 31
Recap of today’s lecture
? Homework 5,Due Friday Oct,7 @ 6:00PM
?Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Chapter 5,Rotation
?Angular variables
?Centripetal acceleration
?Kepler’s laws
?Artificial gravity
Physics 121,Sections 9,10,11,and 12
Lecture 11
Today’s Topics:
? Homework 5,Due Friday Oct,7 @ 6:00PM
? Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Chapter 5,Rotation
? Angular variables
? Centripetal acceleration
?Kepler’s laws
? Artificial gravity
Physics 121,Lecture 11,Pg 2
Chapter 5
Uniform Circular Motion
? What does it mean?
? How do we describe it?
? What can we learn about it?
Physics 121,Lecture 11,Pg 3
Recap:
R
v
s
???t
(x,y)
x = R cos(?)?= R cos(?t)?
y = R sin(?)?= R sin(?t)
? = tan-1 (y/x)
? = ?t
s = v t
s = R? = R?t
v = ?R
Physics 121,Lecture 11,Pg 4
Centripetal Acceleration
? UCM results in acceleration:
?Magnitude,a = v2 / R = ?? R since v = ?R
?Direction,- r (toward center of circle)
R
a
?
^
Physics 121,Lecture 11,Pg 5
Problem,Rotating puck & weight.
? A mass m1 slides in a circular path with speed v
on a horizontal frictionless table,It is held at a
radius R by a string threaded through a
frictionless hole at the center of the table,At the
other end of the string hangs a second mass m2.
?What is the tension in the string T?
?What is the speed of the sliding mass v?
m1
m2
v
R
Physics 121,Lecture 11,Pg 6
Problem,Rotating puck & weight...
? Draw FBD of hanging mass:
?Since R is constant,a = 0.
so T = m2g
m2
m2g
T
m1
m2
v
R
T
Physics 121,Lecture 11,Pg 7
Problem,Rotating puck & weight...
? Draw FBD of sliding mass:
m1
T = m2g
v gR mm? 2
1
m1g
N
m1
m2
v
R
T
Use F = T = m1a
where a = v2 / R
m2g = m1v2 / R
Physics 121,Lecture 11,Pg 8
Forces Causing Centripetal Acceleration
r
va t
c
2
?
r
vmaF t
c
2
??
v
F
r
mg
m
f
n
Physics 121,Lecture 11,Pg 9
Forces Causing Centripetal Acceleration
? Turn on a sloped road …
?
mg
n
?s in
2
nrvmaF tc ???
mg
n sin ?
n cos ?
Physics 121,Lecture 11,Pg 10
Accelerated Reference Frames:
The Accelerometer
? Your first job is with General Motors,You are
working on a project to design an accelerometer,
The inner workings of this gadget consist of a
weight of mass m that is hung inside a box that is
attached to the ceiling of a car,You design the
device with a very light string so that you can
mathematically ignore it,The idea is that the
angle the string makes with the vertical is
determined by the car’s acceleration,Your
preliminary task is to think about calibration of the
accelerometer,First you calibrate the
measurement for when the car travels on a flat
road,
Physics 121,Lecture 11,Pg 11
Accelerated Reference Frames:
The Accelerometer
a
?
i
1
We need to solve for the angle the plum bob makes with
respect to vertical,
First we will solve by using Newton’s Second Law and checking
x and y components,Then we will consider other possible
solution methods,
Physics 121,Lecture 11,Pg 12
Accelerometer...
2,Draw a free-body diagram for the mass:
We wish to solve for ? in terms of the acceleration a.
We will use SF = ma in two cartesian dimensions.
m
T
mg (gravitational force)
?
a
x
y
?
Physics 121,Lecture 11,Pg 13
Accelerometer...
3,Solving,
i,FX = TX = T sin ? = ma
j,FY = TY - mg
= T cos ??- mg = 0
? Eliminate T
T
mg
m
ma
j
i
?
?
T sin ???= ma
T cos ???= mg
t a n ? ? ag
Physics 121,Lecture 11,Pg 14
Accelerometer...
4,No Numbers involved
5,This answer has the right units (none)
It does give 1 in terms of the acceleration,
t a n ? ? ag
Physics 121,Lecture 11,Pg 15
Accelerometer – Other Thoughts
? Think of this problem from the point of view of the person
inside the car.
? This person sees the plumb bob making the same angle
with respect to the ground,but detects no acceleration.
t a n ? ? ag
a
?
Physics 121,Lecture 11,Pg 16
Accelerometer...
? There must be some other force to balance the x component
i,FX = TX + F? = T sin ? + F? = 0
j,FY = TY - mg
= T cos ??- mg = 0
? And we must put F? = -ma to get the same answer as before.
? F? is known as a fictitious force.
t a n ? ? ag
?
a T
mg
Physics 121,Lecture 11,Pg 17
Lecture 11,ACT 1
Accelerated Reference Frames
You are a passenger in a car and not wearing your seatbelt,
Without increasing or decreasing speed,the car makes a sharp
left turn,and you find yourself colliding with the right-hand door,
Which is a correct description of the situation?
A) Before and after the collision there is a rightward force pushing
you into the door.
B) Starting at the time of the collision,the door exerts a leftward
force on you.
C) Both of the above.
D) Neither of the above.
Physics 121,Lecture 11,Pg 18
Newton’s Law of Universal Gravitation
? Every particle in the Universe attracts every other
particle with a force that is directly proportional the
product of their masses and inversely proportional to
the square of the distance between them.
? Constant of Universal gravitation:
2
21
r
mmGF ?
221110673.6 kgmNG ??? -
m1
m2
r12
F21 F12
Physics 121,Lecture 11,Pg 19
Newton’s Law of Universal Gravitation
? The gravitational force is a field force that always
exists between two particles regardless of the
medium that separates them.
? The force varies as one over the square of the
distance between the particles and therefore
decreases rapidly with increasing separation.
? The gravitational force is actually very weak as
shown be the size of G.
It is easily observed for a falling ball of mass m
because the mass of Earth ME is so large in the
product mME
221110673.6 kgmNG ??? -
Physics 121,Lecture 11,Pg 20
Newton’s Law of Universal Gravitation
? The gravitational force exerted by a spherical
object on a particle outside the sphere is the
same as if the entire mass of the sphere were
concentrated at its center:
2
E
E
R
mMGF ?
Physics 121,Lecture 11,Pg 21
Kepler’s Laws
? Much of Sir Isaac’s motivation to deduce the laws of
gravity was to explain Kepler’s laws of the motions of the
planets about our sun,
? Ptolemy,a Greek in Roman times,famously described a
model that said all planets and stars orbit about the
earth,This was believed for a long time.
? Copernicus (1543) said no,the planets orbit in circles
about the sun,
? Brahe (~1600) measured the motions of all of the known
planets and the position of 777 stars (ouch !)
? Kepler,his student,tried to organize all of this,He came
up with his famous three laws of planetary motion.
Physics 121,Lecture 11,Pg 22
Copernicus Brahe Kepler
1473-1543 1546-1601 1571-1630
Physics 121,Lecture 11,Pg 23
Keplers’ Laws
? After 20 years of work on Tycho Brahe’s data,Kepler
formulated his three laws:
1,All planets move in elliptical orbits
with the Sun at one of the focal
points
2,A line drawn from the Sun to any
planet sweeps out equal areas in
equal time intervals
3,The square of the orbital period of
any planet is proportional to the
cube of the average distance
from the planet to the Sun
? It was later shown that all three of these
laws are a result of Newton’s laws of
gravity and motion.
Physics 121,Lecture 11,Pg 24
Kepler’s First Law
? In the Solar System,the Sun is at one focus of
the elliptical orbit of each planet and the other
focus is empty
Sun
Planet
Physics 121,Lecture 11,Pg 25
Kepler’s First Law
p q
? The sum p + q is the same for every point on the
ellipse
Physics 121,Lecture 11,Pg 26
Kepler’s Second Law
? A line drawn from the Sun to any planet sweeps out
equal areas in equal time intervals
? Due to angular momentum conservation
A
B
C
D
Physics 121,Lecture 11,Pg 27
Kepler’s Third Law
? A planet of mass Mp is moving about the Sun Ms in a
circular orbit,Newton’s second law gives:
r
vM
r
MMG pps 2
2 ?
The speed v = 2?r/T,with T is the period of the planet.
r
Tr
r
MG s 2
2
)/2( ??
3
21923322 1097.244
m
s
GMKrKrGMT ssss
-??????
?
?
???
?? ??
Physics 121,Lecture 11,Pg 28
Kepler’s Third Law
Ms
Mp
2
322 44
GT
r
GKM ss
?? ??
Kepler’s third law gives us a method for measuring
the mass of the Sun.
33
2
2 4 rKr
GMT ss ????
?
???
?? ?
Physics 121,Lecture 11,Pg 29
Geosynchronous Orbit
h
r
R
v
F
The gravitational force produces
the centripetal acceleration:
r
mv
r
mMG E 2
2 ?
T
r
T
dv ?2??
3/1
2
2
4 ?
?
?
?
???
??
EGM
Tr
?
T = 24h and ME = 5.98 ? 1024 kg
The height above Earth is app,22 000 miles
Physics 121,Lecture 11,Pg 30
Summary
(with comparison to 1-D kinematics)
Angular Linear
c o n s ta n t??
??
? ? ? 0 ??t
? ? ? ?? ? ?0 0 212t t
co n st a n ta ?
v v at? ?0
x x v t at? ? ?0 0 212
And for a point at a distance R from the rotation axis:
x = R????????????v = ?R ??????????a = ?R
Physics 121,Lecture 11,Pg 31
Recap of today’s lecture
? Homework 5,Due Friday Oct,7 @ 6:00PM
?Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Chapter 5,Rotation
?Angular variables
?Centripetal acceleration
?Kepler’s laws
?Artificial gravity
