CIRCULAR MOTION
& GRAVITATION
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.
Uniform Circular Motion - velocity
Circular Motion:
Force & Acceleration
What force is responsible for car changing direction?
What is the direction of THIS force on the car?
What would car do if this force vanished?
Circular Motion –
Force & 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 =
• Name given to
• Net Force is always
Centripetal Force equations
Example 1: A washing machine drum makes 5
rotations per second during the spin cycle. The
inside drum has a radius of 25cm.
a) Determine the speed of the drum.
b) What acts as the centripetal force on the 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 45.0 m.
If the coefficient of friction between the tires and
the road is µs = 0.82, what is the maximum speed
the car can have on the curve without skidding?
Example 3: 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) What would happen if the
speed were to increase?
c) Would the chair eventually
become horizontal?
Example 4: 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 (Fc) 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
Example 1: A rollercoaster
executes a loop moving at 20m/s at
the bottom and 12m/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, assuming no underneath locking
wheels?
EXAMPLE 2: In an automatic
clothes drier, a hollow cylinder
moves the clothes on a vertical
circle (radius r = 0.39 m). It is
designed so that the clothes
tumble gently as they dry
where a piece of clothing
reaches an angle of θ=69o
above the horizontal, it loses
contact with the wall of the
cylinder and falls onto the
clothes below. How many
revolutions per second should
the cylinder make?
Example 3: A fighter jet is in a
vertical dive when it 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?
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
Example: 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.
UNIVERSAL GRAVITATION
Newton’s Law of Universal Gravitation
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:
Example3: A 50kg astronaut climbs a
ladder that is 6400km high. She 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) Determine force of gravity on
her if she steps off 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.
Force of gravity INSIDE
the earth (assume uniform density)
What happens to force of
gravity when you enter the
earth on way towards
center? Explain.
What would force of gravity
be like halfway between
center and surface?
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.
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?
Kepler's 1st Law:
The Law of Elliptical Orbits
Kepler’s 2nd Law:
The Law of Equal Areas
Kepler’s 3rd Law:
The Law of Periods
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
in terms of gravity!
Circular Orbits
In circular Earth orbit, gravity is perpendicular to the
velocity, responsible for changing the direction only.
Satellite Example1
The Magellan space probe was placed into circular 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?
Satellite Example 2
On July 19, 1969, Apollo 11’s circular orbit around the
Moon (7.36x1022kg) was at an altitude of 111km.
a) Determine the centripetal acceleration of the
spacecraft. (Rmoon=1.74x106m)
b) Determine the value of ‘g’ on the surface of the
moon.
Example 3: Determine the height IN MILES
above the earth of a geostationary (form of a
geosynchronous) satellite (stays over same location on
Earth).