Honors Physics
Mr. Fedell
Chapter 5 Notes – Circular Motion; Gravitation
(Student edition)
Chapter 5 Questions:
4, 5, 7, 8, 13, 14, 20, 24 Problems: 1, 2, 4, 9, 11, 17, 22, 24,
25, 28, 29, 30, 43, 56, 57,
Useful figures:
Every single figure and example has something that you can learn from in
this chapter.
5.1/5.2 Kinematics/Dynamics of Uniform Circular Motion
– the motion of an object traveling at a
constant speed on a circular path; magnitude of the velocity is constant while the
direction is always changing
Tangential Velocity
The velocity vector is said to branch from the circular path following a ___________
Magnitude:
a) The speed of an object in linear motion uses the formula:
𝑣=
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚)
𝑇𝑖𝑚𝑒 (𝑠)
While…
b) Circular motion and speed of an object uses:
𝑣=
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚) 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑚) 2𝑟
=
=
𝑇𝑖𝑚𝑒 (𝑠)
𝑃𝑒𝑟𝑖𝑜𝑑 (𝑠)
𝑇
Period (T) – the time it takes an object to travel the length of the ____________________
Unit:
Example #1: A Tire-Balancing Machine
The wheel of a car has a radius of r = 0.30 m and is being rotated at 830 revolutions per
minute (rpm) on a tire-balancing machine. Determine the speed (in m/s) at which the outer
edge of the wheel is moving. Determine the speed halfway to the outer edge (r = 0.15 m)
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Direction
Newton’s First Law: If there is any change in the velocity vector (direction or magnitude) it
is due to a
; with circular motion we call this the _______________________
Newton’s Second Law: F = ma; with every force comes an acceleration. With circular
motion we use the term
acceleration
Centripetal (Radial) Acceleration
– comes from Latin meaning “center-seeking”
Magnitude: An object moving with a speed v on a circular path of radius r has a magnitude
ar given by
𝑎𝑟 =
𝑣2
𝑟
Direction: The vector always points toward the center of the circle and continually changes
_____________________as the object moves
The linear and centripetal acceleration of an object are related via:
Example #2: The Effect of Radius on Centripetal Acceleration
The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii
of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s, a
speed that was achieved in the event. Express the answer in multiples of g = 9.8 m/s 2.
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The centripetal acceleration vector is inversely proportional to the radius only when v is
constant, so…
_____ Radius
_____ Centripetal acceleration
Centripetal Force
Magnitude: The
force required to keep an object of mass m, moving at a speed v,
on a circular path of radius r, and it has a magnitude of
𝐹𝑟 = 𝑚
Direction: Always points towards the
𝑣2
𝑟
of the circle and continually
direction
*Important – centripetal force is not a “new” or separate force we must account for, it is the
sum of all forces acting on an object pointing towards the center of the circular path*
Example #3: The Effect of Speed on Centripetal Force
The model airplane from the beginning of our notes we find has a mass of 0.90 kg and
moves at a constant speed on a horizontal circular path. Attached to the airplane is a
guideline which is taut and allows the plane to follow a circular path. Find the tension T in
the guideline (length = 17 m) for speeds of 19 m/s and 38 m/s.
*Speed and centripetal force have a direct relationship which
indicates as speed ______ centripetal force ______*
5.3 Highway Curves, Banked and Unbanked
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Circular Motion in the Real-World
When you go around a bend in a car you feel as though you are “pushed” into the car door,
and this is sometimes referred to as centrifugal force
*Centrifugal force is not a real force; it means “_______________” and is incorrect since
there is no outward force*
Forces acting on the car passenger:
Unbanked Curves
The force acting on the passenger from the door comes from the car; on unbanked curves,
this comes solely from frictional forces made between the __________ and __________
Forces acting on the car:
When the wheel is rolling without slipping the road exerts ____________ friction force;
when skidding, ____________ friction forces are present.
Mr. Fedell’s… why you should buy a car with antilock brakes
Example #4: Centripetal Force and Safe Driving
What is the maximum speed a car can safely negotiate an unbanked turn with (r = 51m) in
dry weather (coefficient of static friction = 0.95) and icy weather (coefficient of static
friction = 0.10)?
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Banked Curves
Putting a bank on a curve can reduce tire slippage; the normal force of the vehicle acts to
supplement the static friction force exerted on the tires. The angle of a bank can be made
such that no friction is required for the car to stay on the road at a certain speed.
It is possible to find the angle of the roadway which requires no frictional force at a certain
speed. The angle can be found using:
𝑣2
𝑡𝑎𝑛 𝜃 =
𝑟𝑔
Example #5: At the Daytona 500 the track has a radius of r = 316 m and are banked steeply,
with θ = 31̊ . Suppose these turns were frictionless. At what speed would the cars have to
travel around them?
Newton’s Law of Universal Acceleration
Newton realized that gravity worked over a distance (the apple?), he
proposed that gravity worked over large distances, even from the
__________________________________. He proposed his famous law of
_________________________________
The direction is toward the two objects.
What does this mean?
1. The force never dies out. Every body in the universe feels gravity from every other body
in the universe.
2. The larger the ___________, the stronger the _________.
3. The force between two objects is the same (_____________________). The accelerations
are different due to their different masses.
4. If the mass of an object doubles, the ______________________.
5. If the distance between two objects is doubled, the force is cut to one quarter of its value.
The value of G is G = 6.67 10-11 N-m2/kg2
We can measure the distance from the center of an object when the object is spherical.
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Example #6: What is the gravitational force on the moon from the earth?
mearth = 6 x 1024 kg
mmoon = 7.3x 1022 kg,
r = 3.84 x 105 km = is 3.84 x 108 m
F = 1.98 x 1020 N
This force is a vector, as are all forces, and must be added vectorially, by components.
Example #7: What is the magnitude of gravitational force that acts on each particle placed
horizontally and at a distance of 1.2 m from each other. The mass of the particles are 12 kg
and 25 kg.
Gravity Near the Earth
What is force of gravity near the surface of the earth? If I am standing on the
earth, we have said that the force is mg. That must come from Newton’s law of
universal gravitation. Let’s see, how this works. I need to take the distance from
the center of the earth when I am standing on the surface of the earth.
𝐺𝑚𝑚𝐸
𝐹 = 𝑚𝑔 =
𝑟2
𝐺𝑚𝐸
𝑔=
𝑟2
= (6.67 10-11 N-m2/kg2)(5.97 1024kg)/(6.38 106 m)2 = 9.78 m/s2
Example #8: How far above the earth must I go for the force I feel to be 1/2 of
what is on the surface of the earth?
Large objects near us influence what we perceive as the acceleration of gravity.
Near a mountain, there is mass pulling us toward the mountain. Most of the mass is still
underneath us, so the difference is very small, but it is there. Also if you go to higher
elevations, you are farther from the center of the earth so g goes down very slightly, as well.
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