Download Circular Motion

Circular Motion
(ΣF = ma for circles)
• Circular motion involves Newton’s Laws
applied to objects that rotate or revolve
about a fixed radius.
• This motion can be horizontal circles
(washing machine), vertical circles (ferris
wheel), partial circles (speed bump),
angled circles (banked curve), or satellites
about a planetary body.
Circular Motion - velocity
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 this 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 is inward
or CENTER-SEEKING.
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
Sources of Centripetal Force
You must identify the centripetal force. It may
be provided by the tension in a string, the normal
force, friction, among other sources.
Circular eqns
2
v
ac 
r
1
T
f
Fnet  ma
2
v
Fc  mac  m
r
Example 1
It takes a 615 kg car 14.3 s to travel at a uniform
speed around a circular racetrack of 50.0 m radius.
a) What is the speed of the car?
b) What is the acceleration of the car?
c) What amount of inward force must the track
exert on the tires to keep the car moving in the
circle?
example2
A washing machine drum makes 5 rotations per
second during the spin cycle. The inside drum
has a radius of 25cm.
a) Determine the period of rotation
b) Determine speed of the drum.
c) Determine the centripetal acceleration of the clothes
inside
d) What force provides inward force to clothes?
Example3: Rounding a Corner
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?
Example4:
A toy airplane (0.040kg) is suspended by a string and
flies in a circle. The diameter of the circle is 1.5m.
The string makes a 20o angle with the vertical.
a)Find the tension in the string
b) Find the speed of the plane
Vertical Circle
A rollercoaster executes a
loop moving at 30m/s at the
bottom and 20m/s at the top.
The radius of loop is 10m.
A) What does a scale read on
50kg passenger at bottom of
track?
B) What is the slowest speed at
coaster can go at top of loop
so as not to fall from track?
Example 2
a) What is the normal force
exerted by seat on 80.0 kg
passenger at the bottom of the
dip if you are moving at
17m/s? Radius of dip is 65m.
b) How fast can car go over a
hill of same radius so that it
doesn’t lose contact with road?
(critical velocity)
example3
A stunt plane is in a vertical
dive and pulls up into a vertical
loop. The speed of the plane
is 230m/s.
What is the minimum radius of
the loop so that the pilot never
feels more than 3x his weight?
example4
A bucket is whirled in a vertical
circle at a speed of 5m/s.
Determine the tension in the rope
if the mass of the bucket is 2.3kg.
Banked Curves
Banked curves provide extra
support towards the center. It
allows moving objects to
navigate a turn at a greater
rate of speed. The support
comes from a component of
the normal force
UNIVERSAL
GRAVITATION
Newton’s Law of Universal Gravitation
The acceleration of gravity decreases with
altitude…why? In diagram below, it changes very
gradually since altitude is minor compared to RE
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 drops
off as 1/d2
Example1
Determine the force of gravity between 2 apples that
are 0.11kg each and separated by 0.50m.
Example2
a) Determine the force of gravity between the center
of Earth (6.0x1024kg) and a 100kg person on the
surface where the radius is 6400km.
b) Compare this to Fg = mg
Example3
An 50kg astronaut climbs a ladder that is 6400km
high. He stands on a scale on the top step.
a) Determine his weight at that point if the mass of earth
is 6.0x1024kg
b) What would be the force of gravity on him if he
stepped off the ladder?
c) Determine the acceleration due to gravity (‘g’) at this
point.
Example4
The planet Venus has a mass of 4.86x1024 kg and a
diameter of 12,102km.
a) Determine the acceleration due to gravity on Venus
b) Determine how long it would take a 5.0kg object to fall
3.0m when dropped from rest near the surface of the planet.
Force of gravity
inside the Earth
What would force of
gravity be like if you were
at exact center of planet?
Your acceleration?
What would force of
gravity be like halfway
between center and
surface?
Consider a person standing on
the surface of the planet
What happens to a person’s weight if the
planet were to shrink? Mass of planet is still
same
What happens to a person’s weight if the
planet were to increase in mass?
By what factor is a person’s weight
changed if the planet were to increase
its radius by a factor of 2?
By what factor is a person’s weight
changed if the planet were to increase its
mass by a factor of 3 and its radius reduces
by a factor of 2?
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?
What happens to force of gravity
from the sun on the Earth if the sun
were to shrink down to the size of a
pea while still keeping all of its
mass?
TIDES
Tides
Tides are a result of differential gravity forces
exerted on near side of planet vs far side of planet
Differential Forces due to the Moon
Near side of earth
is pulled harder
than center and
center is pulled
harder than far
side by the moon.
Therefore, water at
(A) is pulled away
from earth (B). Earth
is pulled away from
water at (C)
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 maximal.
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.
Circular Orbits
Satellites
Satellites are fast-moving projectiles that continually
fall, but never hit the ground
The centripetal force that keeps them moving in
orbit is GRAVITY.
Important for weather, telecommunications, military,
& GPS. Satellites don’t crash because they have
sufficient tangential speed + no friction.
Astronauts are not weightless!
EXAMPLE: The astronauts orbit
about 200 miles (320km) above
the surface of the Earth.
a) What is their acceleration due to
gravity?
b) What is the speed of the shuttle?
c) What is the period of their orbit?
What would have to be true to be weightless?
Example 2
The acceleration due to gravity is 1.8m/s2 for an
object in orbit about the Earth. It’s speed is 5180m/s.
a)Find the period of the satellite.
b) Find the weight of satellite if it has a mass of 500kg
Example 3
On July 19, 1969, Apollo 11’s orbit around the Moon was
adjusted to an average altitude of 111 km. The mass of the
moon is 7.3x1022kg and its radius is 1.7x106m
(a) At that altitude how many minutes did it take to orbit once?
b) At what speed did it orbit the Moon?
A satellite takes 6.25 days to complete an orbit
about Saturn.
a) What is the distance of the orbit of the satellite
from the center of Saturn (M= 5.69x1026 kg)?
b) If the radius of Saturn is 6.027x107 m, at what
altitude was the satellite orbiting?
Assuming you are 17 Earth years old, how old
would you be if you instead had lived on Pluto
your whole life? Meaning, how old are you in
Plutonian years? The distance from the Sun to
Pluto is 5.9x1012 m. Mass of Sun = 1.99x1030 kg