Transcript N07 Rotational Motion and Law of Gravity (Notes)

Rotational Motion and
the Law of Gravity
Lecture Notes
Physics 2053
Rotational Motion and the Law of Gravity
Rotational Motion and the Law of Gravity
Topics
7-04 Centripetal Acceleration
7-05 Newtonian Gravitation
7-06 Kepler’s Laws
Rotational Motion and the Law of Gravity
Centripetal Acceleration
Uniform circular motion: motion in a circle of constant
radius at constant speed
Instantaneous velocity is always tangent to circle.
v2
v1
Rotational Motion and the Law of Gravity
Centripetal Acceleration
Radial Acceleration:
Similar Triangles
Δv v
vΔr

 Δv 
Δr r
r
v2
Dt
r2
Dq
-v1
Dr
r1
v2
Dv
v1
Divide by time
Δv

Δt
Centripetal
Acceleration
Dq
v  Δr 
 
r  Δt 
v2
ar 
r
Rotational Motion and the Law of Gravity
Centripetal Acceleration
In uniform circular motion the acceleration is called the
centripetal, or radial, acceleration. It is perpendicular to the
velocity and points towards the center of the circle.
v
a
v2
ar 
r
r
r
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
Is it possible for an object moving with a constant speed
to accelerate? Explain.
A) Yes, although the speed is constant, the direction
of the velocity can be changing.
B) No, if the speed is constant then the acceleration
is equal to zero.
C) No, an object can accelerate only if there is a
net force acting on it.
D) Yes, if an object is moving it can experience
acceleration
Rotational Motion and the Law of Gravity
Centripetal Acceleration Problem
A jet plane travelling 525 m/s pulls out of a
dive by moving in an arc of radius 6.00 km.
What is the plane’s acceleration?
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
An object moves in a circular path at a constant speed.
Compare the direction of the object's velocity and
acceleration vectors.
A) The vectors are perpendicular.
B) Both vectors point in the same direction.
C) The vectors point in opposite directions.
D) The question is meaningless, since the acceleration is zero.
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
What type of acceleration does an object moving with
constant speed in a circular path experience?
A) free fall
B) constant acceleration
C) linear acceleration
D) centripetal acceleration
Rotational Motion and the Law of Gravity
Centripetal Acceleration
For an object to be in uniform circular motion, there
must be a net force acting on it.
The radial force
on the ball is
provided by the string
v
a
 v2 
Fr  mar  m 
 r 
 v2 
Fr  m 
 r 
r
Fr
This radial force
is called a
centripetal force
There is no
centrifugal force
acting on the ball
Rotational Motion and the Law of Gravity
Centripetal Acceleration
The speed of an object in
Uniform Circular Motion
 F  ma
T  mar
T  Mg
mv 2
Mg 
r
v
Mgr
m
m
T
r
M
Mg
v2
ar 
r
Rotational Motion and the Law of Gravity
Centripetal Acceleration Problem
A 0.45 kg ball, attached to the end of a horizontal cord, is rotated
in a circle of radius 1.3 m on a frictionless horizontal surface. If
the cord will break when the tension in it exceeds 75 N, what is
the maximum speed the ball can have?
Rotational Motion and the Law of Gravity
Centripetal Acceleration
Motion in a vertical circle
T
The tension in the string
when the ball is at the top.
 F  ma
mg
r
v2
ar 
r
v2
mg  Ttop  m
r
mv 2
Ttop 
- mg
r
Rotational Motion and the Law of Gravity
Centripetal Acceleration
Motion in a vertical circle
The tension in the string when
the ball is at the bottom.
 F  ma
v2
ar 
r
r
T
v2
Tbottom - mg  m
r
mv 2
Tbottom 
 mg
r
mg
Rotational Motion and the Law of Gravity
Centripetal Acceleration Problem
A bucket of mass 2.00 kg is whirled in a vertical circle of
radius 1.10 m. At the lowest point of its motion the tension
in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket.
Rotational Motion and the Law of Gravity
Centripetal Acceleration Problem (con’t)
A bucket of mass 2.00 kg is whirled in a vertical circle of
radius 1.10 m.
(b) How fast must the bucket move at the top of the
circle so that the rope does not go slack?
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
A pilot executes a vertical dive then follows a semi-circular
arc until it is going straight up. Just as the plane is at its
lowest point, the force on him is
A) less than mg, and pointing up.
B) less than mg, and pointing down.
C) more than mg, and pointing up.
D) more than mg, and pointing down.
Rotational Motion and the Law of Gravity
Centripetal Acceleration
Fy  0
N - mg  0
N  mg
Maximum Speed in horizontal turn
Fx  ma x
2
v max
ax 
r
N
2
mv max
fmax 
r
2
mv max
mg 
r
vmax  gr
r
ax

fmax
fmax  N
fmax  mg 
mg
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
A car goes around a curve of radius r at a constant speed v.
What is the direction of the net force on the car?
A) toward the curve's center
B) away from the curve's center
C) toward the front of the car
D) toward the back of the car
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
A car goes around a curve of radius r at a constant speed v.
Then it goes around a curve of radius 2r at speed 2v. What
is the centripetal acceleration of the car as it goes around the
second curve, compared to the first?
A) four times as big
B) twice as big
C) one-half as big
D) one-fourth as big
Rotational Motion and the Law of Gravity
Centripetal Acceleration Problem
How large must the coefficient of static friction be between
the tires and the road if a car is to round a level curve of
radius 85 m at a speed of 95 km/h?
Rotational Motion and the Law of Gravity
Centripetal Acceleration
N
Friction on a banked road
q
 Fy  0  N cosq  - mg - f sinq 
a
mg  f sinq 
N
cosq 
 Fx  ma
N sinq   f cosq  
v
q
f
q
mv
r
r
2
mg
mv 2
 mg  f sin q  

 sin q   f cosq  
cosq  
r

 v2 
m  cosq   mg sin q   f 
 r 
 
mv 2
f
cosq  - mg sinq 
r
Rotational Motion and the Law of Gravity
Centripetal Acceleration
mv 2
f
cosq  - mg sinq 
r
Friction on a banked road
N
v
q
f
a
When
mv 2
cosq   mg sinq 
r
f
When
mv 2
cosq   mg sinq 
r
q
mg
When
mv 2
cosq   mg sinq 
r
No Friction
Rotational Motion and the Law of Gravity
Centripetal Acceleration
N
Turning a banked curve
with no friction
mv 2
f
cosq  - mg sinq 
r
q
a
q
mv 2
0
cosq  - mg sinq 
r
mv 2
cosq   mg sinq 
r
2
v
tanq  
rg
v
mg
2

v
q  tan -1  
 rg 
 
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
The banking angle in a turn on the Olympic bobsled track
is not constant, but increases upward from the horizontal.
Coming around a turn, the bobsled team will intentionally
"climb the wall," then go lower coming out of the turn.
Why do they do this?
A) to give the team better control, because they are able
to see ahead of the turn
B) to prevent the bobsled from turning over
C) to take the turn at a faster speed
D) to reduce the g-force on them
Rotational Motion and the Law of Gravity
Centripetal Acceleration
Weight
 Fy  0
Reading on scale
is the normal force
N - mg  0
N
N  mg
Scale
mg
Rotational Motion and the Law of Gravity
Centripetal Acceleration
Apparent Weight at the Earth’s Surface
At the North Pole:
F  0
NN
N N - mg  0
mg
N N  mg
v
At the Equator:
NE
mg
 F  mar
mv 2
mg - N E 
R
mv 2
N E  mg R
Rotational Motion and the Law of Gravity
Centripetal Acceleration
A space station is in the shape of a hollow ring 450 m
in diameter. Gravity is simulated by rotating the ring.
Find the speed in revolutions per minute needed in
order to simulate the Earth’s gravity.
R
v
N
Rotational Motion and the Law of Gravity
Centripetal Acceleration
The speed in revolutions per minute
v2
ar 
r
 F  mar
mv 2
N  mg 
R
v  gR
v
N
2R
v
 2Rf
T
2Rf  gR
f
g
2
4 R

9.8
4 225 
2
rev  60 s 
0.00332


s  min 
R = 225 m
 0.00332 rev/s
rev
 1.99
min
Rotational Motion and the Law of Gravity
Newtonian Gravitation
Gravitational Force:
Gravitational Force is the mutual force of attraction between
any two objects in the Universe.
m
F
F
R
FG
Mm
R2
M
Universal Gravitational Constant
2
Nm
G  6.67 x 10 -11
kg 2
Rotational Motion and the Law of Gravity
Newtonian Gravitation
Gravitational Potential Energy associated with an
object of mass m at a distance r from the center of
the Earth is
m
r
ME
m Em
PE  -G
r
[7.21]
Rotational Motion and the Law of Gravity
Newtonian Gravitation
Escape velocity
v esc
An object projected upward from the Earth’s
surface with a large enough speed will soar off
into space and never return.
This speed is called the Earth’s escape velocity.
KEi  PEi  0
R E ME
11.2 km/s
25,000 mi/h
2
 GM Em 
mv esc
0
2
RE 

v esc 
2GM E
RE
Rotational Motion and the Law of Gravity
Newtonian Gravitation Problem
Calculate the acceleration due to gravity on the Moon. The
Moon’s radius is 1.74 x 106 m and its mass is 7.35 x 1022 kg.
Rotational Motion and the Law of Gravity
Newtonian Gravitation Problem
A hypothetical planet has a mass 1.66 times that of Earth,
but the same radius. What is g near its surface?
Rotational Motion and the Law of Gravity
Newtonian Gravitation
Speed of a Satellite
v
FG
m
F
F
R
Mm
R2
v2
Fm
R
v2
G
m
2
R
R
Mm
M
GM
v
R
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
Two planets have the same surface gravity, but planet B
has twice the radius of planet A. If planet A has mass m,
what is the mass of planet B?
m
A)
2
B) m
C)
D) 4m
2m
Rotational Motion and the Law of Gravity
Newtonian Gravitation Problem
A certain neutron star has five times the mass of our Sun
packed into a sphere about 10 km in radius. Estimate the
surface gravity on this monster.
Rotational Motion and the Law of Gravity
Newtonian Gravitation
The satellite is kept in orbit by its speed – it is continually
falling, but the Earth curves from underneath it.
Without
gravity
Earth
With
gravity
Rotational Motion and the Law of Gravity
Rotational Motion and The Law of Gravity
Compared to its mass on the Earth, the mass of an object
on the Moon is
A) the same.
B) less.
C) more.
D) half as much.
Rotational Motion and the Law of Gravity
Newtonian Gravitation Problem
Calculate the force of Earth’s gravity on a spacecraft 12,800 km
(2 Earth radii) above the Earth’s surface if its mass is 1350 kg.
Rotational Motion and the Law of Gravity
Kepler’s Laws
Kepler’s Three Laws:
1. All planets move in elliptical orbits with the
Sun at one of the focal points.
Rotational Motion and the Law of Gravity
Kepler’s Laws
Kepler’s Three Laws:
2. A line drawn from the Sun to any planet sweeps
out equal areas in equal time intervals.
Area 1
Area 2
Area 1 = Area 2
Rotational Motion and the Law of Gravity
Kepler’s Laws
Kepler’s Three Laws:
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.
T
R
T2
R
3
 constant
Rotational Motion and the Law of Gravity
Kepler’s Laws
T
r
MS
F  ma
G
M Sm
r2
GM S 
v2
m
r
 2r 


M Sm
T 

G 2 m
r
r
F
2
r3
T
2

4  2r 3
T2
GM S
4 2
Rotational Motion and the Law of Gravity