Circular Motion
Centripetal Force and
Acceleration
Tangential Velocity
• Tangential Velocity (Vt)- object’s speed along
an imaginary line that is drawn tangent to the
circular path
• Depends on distance from the object to the center
of the circular path
• When the tangential speed is constant then the
motion is described as uniform circular motion
Tangential Velocity
Vt=2πr/T
•r=radius of the circle
•T=Period (amount of time to complete one
circle)
•
Example
–A plane makes a complete circle with a radius
of 3622 m in 2.10 min. What is the speed of the
plane?
Centripetal Acceleration
• When an object is moving in a circular path, the direction
changes
• If the object has a constant SPEED, Acceleration is due to
the direction changing
• This is called centripetal acceleration
CENTRIPETAL ACCELERATION
v
ac =
r
2
t
2
(tangential speed)
centripetal acceleration =
radius of circular path
Centripetal Acceleration
•Centripetal acceleration is always directed
towards the center of the circle
•Centripetal=center seeking
Centripetal Acceleration
Tangential Accleration
•Centripetal acceleration results from change
in direction
•When the speed changes in a circle it is
called tangential acceleration
•Consider a car moving in a circle
Centripetal Force
•Remember!
•When an object is accelerating there is a net
force
•If there is centripetal acceleration, there is a
net force- Centripetal Force
–This is not a new force
–Net force that accelerates an object towards
the center of a circle
Examples
•If a mass is twirled in a circle, at the end of a
string, the centripetal force is provided by the
tension
•When a car rounds a corner on a highway,
the centripetal force is provided by friction
•When the moon orbits the Earth, the
centripetal force is provided by gravity
Centripetal Force
•Centripetal Force =mass x centripetal
acceleration
•Fc=mac
•Substitute ac
•Fc=m V2/r
•Substitute Vt
•Fc=m 4π2r/T2
Example- Standard
•A 0.50 kg mass is whirled in a circle of radius
0.20 m at 2.3 m/s. Calculate the centripetal
force acting on the mass
Example-Honors
•A 0.50 kg mass sits on a frictionless table and
is attached to hanging weight. The 0.50 kg
mass is whirled in a circle of radius 0.20 m at
2.3 m/s. Calculate the centripetal force
acting on the mass .
•Calculate the mass of the hanging weight
Example
•A car traveling at 14 m/s goes around an
unbanked curve in the road that has a radius
of 96 m. What is its centripetal acceleration?
•What is the minimum coefficient of friction
between the road and the car’s tires?
Clicker Question
•Calculate the centripetal force acting on a
925 kg car as it rounds an unbanked curve
with a radius of 75 m at a speed of 22 m/s.
Clicker Question
•An amusement park ride has a radius of 2.8
m. If the time of one revolution of a rider is
0.98 s, what is the speed of the rider?
Clicker Question
•A 2.7x103 kg satellite orbits the Earth at a
distance of 1.8x107 m from the Earth’s centre
at a speed of 4.7x103 m/s. What force does
the Earth exert on the satellite?
Clicker Question
. An object moves in a circle at a constant speed. Which of the
following is not true of the object?
A. Its centripetal acceleration is constant.
B. Its tangential speed is constant.
C. Its velocity is constant.
D. A centripetal force acts on the object.
Clicker Question
A car traveling at 15 m/s on a flat surface turns in a circle with a
radius of 25 m.
2. What is the centripetal acceleration of the car?
A. 2.4  10-2 m/s2
B. 0.60 m/s2
C. 9.0 m/s2
D. zero
Centrifugal Force
• When a driver takes a sharp left turn, the passenger slides to the
right of the car and into the door, Why?
• Centrifugal=center-fleeing
• Apparent force that causes a revolving or rotating object to
move in a straight line
• However-Newton’s first law states- an object in motion will stay in
motion until a force acts on it
• Centrifugal force does not really exist!
Describing Motion
• What happens when
centripetal force disappears?
Where does the object go?
• A ball that is on the end of a string is
whirled in a vertical circular path.
– If the string breaks at the position
shown in (a), the ball will move
vertically upward in free fall.
– If the string breaks at the top of the
ball’s path, as in (b), the ball will
move along a parabolic path.
Vertical Versus Horizontal
• Draw a free body diagram of a mass moving in a vertical circle,
when the object is at the top of the path and the bottom of the
path
• As with any object moving in a circle there is a net force acting
on it towards the center of the circle
• The net force is the centripetal force
• At the top Fnet (Fc)=Ft +Fg
• At the bottom Fnet (Fc)=Ft-Fg
Example
• A 1.7 kg object is swung from the end of a 0.60 m string in a
vertical circle. If the time of one revolution is 1.1 s, what is the
tension in the string
• A) at the top?
• B) at the bottom?
Constant Velocity
• When you are trying to find the minimum speed of an object at
the top of its circular arc we can use the equation
• Fc=Fg so
• The centripetal acceleration equals the acceleration due to
gravity (ac=9.8 m/s/s)
• So
• g=Vt2/r
• V=√(gr)
Example
• An object is swung in a vertical circle with a radius of 0.75 m.
What is the minimum speed of the object at the top of the
motion for the object to remain in circular motion?
Circular Motion
Gravitation
What is Gravity?
• The force that pulls us to the earth
• It is much more than that
• Gravity is the force that attracts two
bodies that have mass and energy
to each other
• Newton discovered that gravity
attracts any two objects
depending on their masses and
their distance apart
Gravity
• The gravitational forces that
two masses exert on each
other are always equal in
magnitude and are opposite
in directions.
• Newton’s Third Law
Gravity
• Is Proportional to the two masses
• Is inversely proportional to the square of the distance between
their center of mass
• So if two objects are twice as far away from each what happens
to the gravitational force by?
• Fg=Gm1m2
r2
• G=6.67 x 10 -11 N m2/kg2
• M1= mass of the first object
• M2= mass of the second object
• r= distance between the two centers of mass
Example
• Calculate the force of gravity between two 75 kg students if their
centers of mass are 0.95 m apart
Example 2
• A satellite weighs 9000 N on Earth’s surface. How much does it
weigh if its mass is tripled and its orbital radius is doubled?
Clicker Question
. Earth (m = 5.97  1024 kg) orbits the sun (m = 1.99  1030 kg) at a
mean distance of 1.50  1011 m. What is the gravitational force of
the sun on Earth? (G = 6.673  10-11 N•m2/kg2)
A. 5.29  1032 N
B. 3.52  1022 N
C. 5.90  10–2 N
D. 1.77  10–8 N
Clicker Question
• Gravitational force F exists between point objects A and B
separated by distance R. If the mass of A is doubled and
distance R is tripled, what is the new gravitational force between
A and B?
A) 9/2 F
B) 2/3 F
C) 2/9 F
D) 3/2 F
Common Misconception: Mass Versus
Weight
Mass
• Amount of matter
Weight
• Gravitational attraction (Fg)
• Constant everywhere
• Changes depending on location
Clicker Question
Which of the following statements is correct?
A. Mass and weight both vary with location.
B. Mass varies with location, but weight does not.
C. Weight varies with location, but mass does
not.
D. Neither mass nor weight varies with location.
Satellites
• Are constantly “falling” when in orbit
• They are in freefall
Gravitational Field
• Gravity is an example of a field force (ie not a contact force)
• A force that is exerted which has no direct contact
• A field- an area of influence
• Think about a campfire
• As you approach it
• As you increase the size
Gravitational Fields
• Fields can be described as
vectors or scalars, depends on
what kind
• Gravitational fields are field
forces so..
• They are vectors
• Gravitational fields are
represented by arrows,
magnitude is represented by
how many arrows are present
Gravitational Field Strength
• Gravitational field strength=acceleration due to gravity
• Recall Fg=mg
• So…
• g=Fg/m
• g=acceleration due to gravity, gravitational field strength,
9.80m/s2 near the Earth’s surface
• g varies with distance
• g=Gm1
r2
• Measured in N/kg, same as m/s2
Example
• What is the gravitational field strength on the Earth’s surface of
the moon? The mass of the moon is 7.35 x 1022 kg and the radius
of the moon is 1.74 x 106 m
Clicker Question
. Which of the following is a correct interpretation of the
expression?
mE
ag = g = G 2
r
A. Gravitational field strength changes with an object’s distance
from Earth.
B. Free-fall acceleration changes with an object’s distance from
Earth.
C. Free-fall acceleration is independent of the falling object’s
mass.
D. All of the above are correct interpretation
The Period of a Satellite
• T=√(4π2r3)
•
Gm
• m refers to the mass of the center of the orbit
Example
• A satellite orbits the Earth at a radius of 2.2 x 107 m. What is its
orbital period, when the mass of the earth is 5.98 x 1024 kg. What
is the speed of the satellite?