Circular Motion and
Gravitation
Review
If an object travels at a constant velocity:
𝑎 = 0
𝐹𝑛𝑒𝑡 = 0
If an object travels with changing velocity:
𝑎 =
𝑣𝑓 −𝑣𝑜
𝑡
𝐹𝑛𝑒𝑡 = 𝑚𝑎
What if?
What if you do not speed up or slow down
BUT change direction?
Change in velocity occurs.
Acceleration occurs.
𝐹𝑛𝑒𝑡 = 𝑚𝑎
Centripetal Acceleration
‘Centripetal’ means ‘center seeking’
Points towards the center
Sometimes called radial acceleration
Dependent on v and r
2
𝑣
𝑎𝑐 =
𝑟
Example
A jet plane traveling at 525 m/sec pulls out of a
dive by moving in an arc of radius 6 km. What is
the plane’s centripetal acceleration?
Express the acceleration
in m/s2 and g’s.
Example
A ball at the end of string is revolving in a
horizontal circle of radius 0.6 m. The ball makes 2
revolutions in a second. What is the centripetal
acceleration?
Whoa! We need to figure
out the velocity first!
Time per revolution =
period (T)… measured in
seconds (s)
Example – You Try
The Moon’s nearly circular orbit about the Earth
has a radius of 384,000 km and a period of 27.3
days. Determine the acceleration of the Moon
toward the Earth.
Example – You Try
Taking the age of Earth to be about 4 ✕ 109 years and
assuming its orbital radius of 1.5 ✕ 1011 m has not
changed and is circular, calculate the approximate
total distance Earth has traveled since its birth (in a
frame of reference stationary with respect to the Sun).
Period and Revolutions
We will often see the terms ‘period’,
‘revolutions’, and ‘revolutions per unit time’
Period – the amount of time to make one
rotation (T)
Revolution – one rotation
Revolution per unit time – the inverse of the
period (1/T)
Dynamics of Circular Motion
If there is an acceleration, Newton’s 2nd Law says
there is a net external force.
Since the acceleration in centripetal, the net force
must also be centripetal.
𝐹𝑛𝑒𝑡 = 𝑚𝑎
𝐹𝑐 = 𝑚𝑎𝑐
𝑚𝑣
𝐹𝑐 =
𝑟
2
Centripetal Force
But that would mean centripetal force is not
a real force, but is a net force and caused
by something else.
Fn, Fg, Ff, Fp…
It is the sum of the forces acting on the radial
axis.
That “other” Force
Centrifugal - an outward
force acting on an object
moving in a circular path.
Everyone has “felt” it.
So, what do you feel?
Depends on who “you”
are.
“Centrifugal Force”
If the centrifugal force really existed what would
happen to the ball if you let go?
Example
A 10 kg mass is attached to a string that has a
breaking strength of 200 N. If the mass is whirled in
a horizontal circle of radius 0.8 m, what maximum
speed can it have?
Example
A flat puck of mass M is rotated in a circle on a
frictionless air-hockey tabletop, and is held in this orbit
by a light cord connected to a dangling block of
mass m. What is the speed of the flat puck if the
radius of the circle is R?
Example – You Try
A typical red blood cell has a mass of 3.0 ✕10-16
kg. A blood sample placed in a centrifuge is
subjected to a force of 5.0✕10-11 N when the
centrifuge is operated at 140 rev/s. What is the
diameter of the centrifuge?
Circular Motion and Friction
The centripetal force is
provided for by friction.
BUT… friction (static… no
sliding) has a maximum value
There is a maximum speed
you can take a turn.
Example
A 1000 kg car rounds a curve on a flat road of
radius 50 m at a speed of 15 m/sec. What
minimum coefficient of friction is needed for the
car to safely navigate the corner?
Example – You Try
An unbanked (flat) curve of radius 160 m is rated
for a maximum speed of 45.0 m/s. At what
maximum speed should a flat curve with radius of
75.0 m be rated?
Warm-Up
A child has a toy tied to the end of a string and
whirls the toy at constant speed in a horizontal
circular path of radius R. The toy completes each
revolution of its motion in a time period T. What is
the magnitude of the acceleration of the toy?
Agenda
 Circular Motion w/ Angles
 Tomorrow: Circular Motion Lab
 So far…
 Identify the direction of the velocity, acceleration, and force vectors of an
object in circular motion.
 Determine the velocity, acceleration, and centripetal force acting on an
object moving in a horizontal circle.
 Today…
 Identify the forces (components of forces) acting along the radial axis.
 Determine centripetal force and other unknowns of objects moving in circular
orbits with forces at angles.
Circular Motion and Angles!
Often times the force(s) providing
the centripetal force must also
balance other forces (mostly
gravity)
Since there is a net force acting,
we have to figure out which
forces are on the radial axis and
solve.
Example
A 300 g tetherball is attached to a massless rope 2
m long. If the rope makes an angle of 30° to the
vertical, what is the speed of the ball?
Example
A 4 kg object is attached to a vertical rod by two
strings. The object rotates in a horizontal circle at
constant speed 6 m/sec. Find the tension in each
of the strings.
Example – You Try
In a version of a “Giant Swing”, the seat is connected
to two cables as shown. The seat swings in a horizontal
circle at a rate of 15 rpm. If the seat weighs 255 N and
a 825 N person is sitting in it, find the tension in each
cable.