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1.3 CIRCULAR MOTION AND GRAVITATION

Calculations –
A. Calculating the state of objects in circular motion

Centripetal Magnitude Direction

𝒗𝟐
Acceleration 𝒂𝒄 = Towards
𝒓
the center
𝐹𝑐 = 𝑚𝑎𝑐 of circular 𝑠 − 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
Force 𝒗𝟐 path 𝑣 − 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑭𝒄 = 𝒎( )
𝒓 𝑎𝑐 − 𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
Velocity 𝐹𝑐 − 𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
𝑟 − 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑝𝑎𝑡ℎ
𝒔
𝒗= 𝑇 − 𝑝𝑒𝑟𝑖𝑜𝑑
𝒕 𝟐𝝅𝒓 − 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑
𝟐𝝅𝒓 (𝑇𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑓𝑜𝑟 𝑜𝑛𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛)
𝑖𝑛 𝑜𝑛𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑐𝑖𝑟𝑐𝑙𝑒
𝒗=
𝑻
Banking Angle
𝑭𝑯 𝑣2
𝒕𝒂𝒏 𝜽 = tan 𝜃 =
𝑭𝑽 𝑟𝑔
𝟐
(𝒎𝒗 ⁄𝒓) 𝑣2
= 𝜽 = 𝐭𝐚𝐧−𝟏 ( )
𝒎𝒈 𝑟𝑔

B. Gravity equations

Force of
attraction 𝒎𝟏 𝒎𝟐
between 𝑭=𝑮
𝒓𝟐
two
𝐺 − 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑓 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛
masses
(𝟔. 𝟔𝟕 × 𝟏𝟎−𝟏𝟏 𝑵𝒎𝟐 𝒌𝒈−𝟐 )
Velocity
𝑚1 / 𝑚2 − 𝑚𝑎𝑠𝑠𝑒𝑠 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡 1 𝑜𝑟 2
of 𝑮𝑴 𝑀 − 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑜𝑏𝑗𝑒𝑐𝑡
orbiting 𝒗=√
𝒓 𝑏𝑒𝑖𝑛𝑔 𝑜𝑟𝑏𝑖𝑡𝑒𝑑
mass 𝑟 − 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡𝑤𝑜 𝑚𝑎𝑠𝑠𝑒𝑠
𝑓𝑟𝑜𝑚 𝑡ℎ𝑒𝑖𝑟 𝑐𝑒𝑛𝑡𝑒𝑟
2. 𝑅𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑔 − 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙
1. 𝐿𝑒𝑡 𝑣 = 𝑣,
4𝜋 2 𝑟 2 𝐺𝑀 𝑓𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
Period 2𝜋𝑟 𝐺𝑀 = 𝐹 − 𝑓𝑜𝑟𝑐𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒𝑑 𝑏𝑦 𝑜𝑏𝑗𝑒𝑐𝑡
𝑇2 𝑟
=√ 𝟒𝝅 𝟐 𝟑
𝒓 𝑚 − 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡
𝑇 𝑟
𝑻𝟐 =
𝑮𝑴

Field 𝑭
𝒈=
Strength 𝒎

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Theoretical understanding –
A. Uniform circular motion: When object moves in a circle with tangential (constant) velocity.

#1 The direction of velocity experienced at any point is at a tangent

to the circular path
#2 A centripetal acceleration is created by the changing velocity
that results with the constant change of direction, even though its
magnitude stays the same
∆𝑣
𝑎=
𝑡
#3 The centripetal acceleration is always pointed to center of
circular path/ at right angles to object’s velocity

B. Forces resulting in centripetal acceleration

Force Example Diagram

Swinging a mass on a
Tension
string

Roller-coaster with a
Normal
looped path

A car turning a corner

Frictional
on a flat road

The Moon orbiting the

Gravitational
Earth

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C. Banked curves

#1 Roads are banked at an angle so no frictional force is needed

in producing a centripetal acceleration
#2 Car experiences a centripetal acceleration via tension force
#3 This will allow the car to move around the corner at faster
speeds
𝑾 − 𝒘𝒆𝒊𝒈𝒉𝒕
𝑭 − 𝒏𝒐𝒓𝒎𝒂𝒍 𝒇𝒐𝒓𝒄𝒆
𝑭𝑽 − 𝒗𝒆𝒕𝒊𝒄𝒂𝒍 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕 𝒐𝒇 𝒏𝒐𝒓𝒎𝒂𝒍 𝒇𝒐𝒓𝒄𝒆
𝑭𝑯 − 𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕 𝒐𝒇 𝒏𝒐𝒓𝒎𝒂𝒍 𝒇𝒐𝒓𝒄𝒆

𝑾 = 𝒎𝒈 𝑭 = √𝑭𝑽 𝟐 + 𝑭𝑯 𝟐
𝑭𝑽 = 𝑭 𝐜𝐨𝐬 𝜽
𝒗𝟐
𝑭𝑯 = 𝑭 𝐬𝐢𝐧 𝜽 𝜽 = 𝐭𝐚𝐧−𝟏( )
𝒓𝒈

D. Gravitational field: a space experiencing forces acting radially towards the center of the
mass
𝒎𝟏 𝒎𝟐
Newton’s Law of Universal Gravitation: 𝑭 = 𝑮 𝒓𝟐

E. Kepler’s Laws of Planetary Motion: describes motion of planets and their satellites

Law of Ellipses

- There are two foci being orbited

All planets move in an by the planet.
elliptical orbit with the
Sun at one focus *The closer the foci to one another, the
closer the shape of ellipse is to a circle

Law of Equal Areas

A line connecting the Sun

- This is because the planet
to a planet sweeps equal
moves faster when it is closer to
areas in equal time
the Sun
intervals

𝑻𝟐 ∝ 𝒓𝟑
The period of satellites - The period of revolution
𝟒𝝅𝟐 𝒓𝟑
𝟐
𝑻 = depends on its radius of its squared is directly proportional
𝑮𝑴 orbit to the radius of orbit cubed

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F. Types of satellite orbits on Earth

Type of orbit Description Application

Satellite in such - Has a large coverage,
orbits remain fixed, but produces low
Geostationary
one point above the resolution images
Earth’s surface
- Used for continuous
Satellite will move
monitoring and
Equatorial directly above the
communication
equator of the Earth
satellites
- Low-altitude orbit
satellites are used to
Satellite will move
produce higher
over the North and
Polar resolution images
South pole as it
orbits the Earth
- Used for surveillance
and meteorology
*The center of these orbits must coincide with the center of the Earth because a centripetal
acceleration, which is given by the gravitational force, is needed for uniform circular motion to keep
the satellite in stable orbit.

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