Transcript HP UNIT 4 - student handout

CIRCULAR MOTION
& GRAVITATION
Circular Motion - velocity
Direction of
velocity
Velocity is not
constant. Why?
T=
Circular Motion –
Force & Acceleration
A ball attached to string is whirled in horizontal circle by hand.
What force is responsible for ball changing direction?
What is direction of force on ball?
What would occur to ball if force vanished?
Circular Motion –
Force & Acceleration
If force on ball is directed INWARD to prevent ball
from flying outward (inertia), then net force
This force
A net force produces an acceleration.
Centripetal Force causes objects to navigate a
circle. Objects do not move towards center
because why?
Objects in circular motion at constant speed
are not balanced. Why?
Centrifugal Force
• Centrifugal = center-fleeing
Centripetal Force eqns
2
v
ac 
r
1
T
f
F  ma
2
v
Fc  mac  m
r
Example1
A washing machine drum makes 5
rotations per second during the spin cycle.
The inside drum has a radius of 25cm.
a) Determine speed of the drum.
b) What acts as centripetal force on clothes?
c) Determine the normal force on a 1.0kg pair of
jeans in the washing machine during this rotation.
Example 2:
A 1,200 kg car rounds a corner of radius
r = 45.0 m. If the coefficient of friction
between the tires and the road is ms = 0.82,
what is the maximum speed the car can have
on the curve without skidding?
Example3
An amusement park ride consists of a rotating circular
platform 8.00m in diameter from which 10.0kg seats are
suspended at the end of 2.50m light chains.
a) When the system rotates, the
chains make an angle of 28.0o
with the vertical. What is the
speed of each seat?
b) Find the tension in the chain
when a 40.0 kg child is riding
in a seat.
Example4
An early major objection to the idea that the
Earth is spinning on its axis was that Earth
would turn so fast at the equator that people
would be thrown off into space.
a) Show the error in this logic by solving for the NET inward
force necessary to keep a 97.0 kg person standing on the
equator with speed 444m/s and Rearth = 6400km.
b) What force(s) are responsible for composing the
centripetal force on person?
c) Determine the length of day in hours so that a person
would just be ‘weightless’ (scale can’t push on you).
Vertical Circles
A rollercoaster executes a loop
moving at 30m/s at the bottom and
20m/s at the top. The radius of
loop is 10m. Negligible friction.
a) What does a scale read on 50kg
passenger at bottom of loop?
b) What is the slowest speed coaster can
go at top of loop so as not to fall?
Example 2:
Tarzan, whose mass is 85 kg, tries to
cross a river by swinging from a vine.
The vine is 10.0 m long, and his speed
at the bottom of the swing (as he just
clears the water) is 8.00m/s. Tarzan
doesn't know that the vine has a
breaking strength of 1000 N. Does he
make it safely across the water?
Water & board
demo
Example 3
A fighter jet is in a vertical
dive and pulls up into a
vertical loop. The speed of
the plane is 230m/s.
a) What provides inward force on pilot? On plane?
b) What is the minimum radius of the loop so that the
pilot never feels more than 3x his weight?
A car rounds a curve at
angle θ. The radius of the
curve is R. Assuming
negligible friction,
determine the expression
for the speed it could
negotiate the curve
without sliding.
Newton’s Law of Universal Gravitation
Newton realized that ALL objects
Example1
Two bowling balls each have a mass of 6.8 kg. They
are located next to one another with their centers 21.8
cm apart. What amount of gravitational force does
one exert on the other?
Example2
The Earth exerts a pull of gravity on the moon as
does the less massive moon on the Earth. Which
planetary body pulls harder?
The acceleration due to gravity doesn’t change
that much for altitudes that are much less than
radius of Earth.
Once the altitude becomes comparable to the radius
of the Earth (RE) , the decrease in the acceleration of
gravity is much larger:
Acceleration due
to gravity (g)
drops off as 1/r2
Example3: A 50kg astronaut climbs a
ladder that is 6400km high. He stands on a
scale on the top step.
a) Determine scale reading on her
at that point if the mass of earth is
6.0x1024kg and RE = 6.4x106m.
Ignore rotation of Earth
b) What would be the force of
gravity on her if she stepped off
the ladder?
c) Determine the acceleration
due to gravity (‘g’) at this point.
Example 4: The planet Saturn has a mass of
5.68x1026 kg and a radius of 5.85x107m.
Determine the time it takes a 1.0kg object to fall
10.0m from rest near the surface of the planet.
EXAMPLE 5
Imagine an astronaut in
between the Earth and
moon. Would a 70kg
astronaut have to be
placed closer to the earth
or the moon (7.3x1022kg)
so that the net gravitational
force on them is zero?
Determine this distance if
the avg distance between
moon and earth is
384,403km (center to
center).
Force of gravity
INSIDE earth
What happens to force of
gravity when you enter the
earth on way towards
center?
What would force of gravity
be like halfway between
center and surface?
Tall buildings, but
not in a single bound
If you walked into the
lobby of a
skyscraper, what
would happen to
your weight or force
of gravity, technically
speaking?
Describe what would occur to orbit of Earth if
our Sun were to collapse into a very tiny
volume (while still maintaining same mass).
Tides
Tides are a result of
Differential Forces due to the Moon
Full cycle of 2 highs & 2 lows
is about 25hrs, not 24hrs
It takes more than 24hours for us to rotate through 2 tidal
bulges of 2 high and 2 lows. The reason is because by the
time we make one full rotation on Earth, the moon has
moved from its position it was at 24hrs prior. As a result, in
order to get back to the high tide bulge, we have to rotate a
little bit further than just one rotation. (see figure below)
We need about our hour's worth of extra rotation to
accomplish this. Therefore, there are typically two
high tides every 25 hours, rather than every 24
hours. This is why the times of high tides are about
an hour later each day.
Tides due to both Sun and Moon –
Spring Tides and Neap Tides
Spring tides are much
larger…this is when
both sun and moon are
lined up making
attractive pull greater
Neap tides are
smaller…this is when
sun and moon are
pulling earth at 90o with
respect to one another
The effect of the Sun is not as
great as the Moon’s effect
Sun’s pulls on earth
180x more than moon
does but only has ½
the effect. So, why
does moon have more
effect on tides?
No tides in a glass of
water…Why?
Kepler's 1st Law:
The Law of Elliptical Orbits
Each planet travels in an elliptical
orbit with the sun at one focus.
When the planet is located at
point A it is at the aphelion
position or apogee. (farthest)
When the planet is located at
point P it is at the perihelion
position or perigee. (closest)
Kepler’s 2nd Law:
The Law of Equal Areas
A line from the planet to
the sun sweeps out equal
areas of space in equal
intervals of time.
At the aphelion, the position farthest from the sun
along the planet’s orbital path, the planet’s speed is
minimal.
At the perihelion, the position closest to the sun
along the planet’s orbital path, the planet’s
speed is maximum.
Kepler’s 3rd Law:
The Law of Periods
The square of a planet’s orbital period is directly
proportional to the cube if its average distance from the sun.
We make an assumption that the orbits are circular since
they are only SLIGHTLY elliptical…
FC= Fg
Newton’s Mountain
Geometric
curvature
of Earth
Newton reasoned that if you fired a projectile fast
enough horizontally, it would continually fall from
its straight-line path but never hit the earth…”falling
around” or orbiting the earth.
Astronauts are not weightless!
Circular Orbits
Satellite Example1
The Magellan space probe was placed into orbit
around the planet Venus in 1992. The probe was to
orbit at an altitude of 4370km. The mass of Venus is
4.87x1024 kg and its radius is 6100 km.
a) What speed is required to maintain this orbit?
b) What was the orbital period in hours?
Example 2
On July 19, 1969, Apollo 11’s orbit around the Moon
(7.36x1022kg) was at an altitude of 111km.
a) How many minutes did it take to orbit once?
(Rmoon=1.74x106m)
b) At what speed did it orbit the Moon?
Example 3: Determine the height above the
earth of a geosynchronous satellite.