Chapter 11 GRAVITATION

Newton's Universal Law of Gravitation

Newton's Universal Law of Gravitation
𝑟

𝑚1

𝑚2

𝐹𝑔

𝐹𝑔

The gravitational force that exists between two masses 𝑚1 and 𝑚2 is given by
𝑭𝒈=𝑮𝒎𝟏𝒎𝟐𝒓𝟐
where 𝑟 - is the distance of separation
between their centers.
𝐺=6.67 𝑥 10−11 Nm2/kg2
- universal gravitation constant

Relationship between Force, mass and distance.

NET GRAVITATIONAL FORCE
For three or more objects, finding the net gravitational force is by component!
REVIEW YOUR COMPONENT METHOD!!!
Cartesian Plane:
𝑟1

𝑚1

𝑚3

𝐹31

𝐹31

𝐹21

𝑟2

𝐹21

𝐹1

𝑚2

Gravitational Field Strength & Gravitational Acceleration
A gravitational field (popularly known as acceleration due to gravity) is created by an object causing masses inside it to experience the gravitational force.
e𝑥𝑎𝑚𝑝𝑙𝑒: 𝑔𝐸=9.8 m/𝑠2

Gravity Near The Earth’s Surface
At the Earth’s surface:
𝐹𝑔=𝑤
𝐺𝑚𝑚𝐸𝑟𝐸2=𝑚𝑔𝐸

𝑔𝐸=𝐺𝑚𝐸𝑟𝐸2=9.8 m/s2

𝑚𝐸=mass of the earth
𝑟𝐸=radius of the earth

Gravitational field at the object’s surface
In general:
𝑔𝑝=𝐺𝑚𝑝𝑟𝑝2

where 𝑚𝑝=mass of the object having the field
𝑟𝑝=radius of that object

The effective g, g’
As you go far from the Earth’s surface, the gravitational field decreases.
𝑟=𝑟𝑝+𝑦

𝒓

where 𝑦=distance above object′s surface

(𝑟>𝑟𝑝)

So, the effective g (g’):
𝑔′=𝐺𝑚𝑝𝑟2

Since 𝑔𝑝=𝐺𝑚𝑝𝑟𝑝2

𝑔′=𝑔𝑝𝑟𝑝𝑟2

Sample Problems:
Two objects attract each other with a gravitational force of magnitude 1.00 𝑥 10−8 N when separated by 20.0 cm. If the total mass of the objects is 5.00 kg, what is the mass of each?
Calculate the effective value of g, at 3200 m and 3200 km above the earth’s surface.
Calculate the velocity of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km.

Satellite Motion and Weightlessness
without gravity
Artificial satellite is put into orbit by accelerating it to a sufficiently tangential speed with the use of the rocket.
If the speed is too high, the satellite will escape.
If the speed is too low, it will fall back to earth.
Fg
With gravity

𝐹𝑔=𝑚𝑣2𝑟

𝐺𝑚𝑀𝑟2=𝑚𝑣2𝑟

𝑣=𝐺𝑀𝑟

Speed of satellite at orbit radius r
where
𝑀=mass of the object/planet that the satellite 𝑚 is orbiting

Satellite Motion and Weightlessness
The “weightlessness” experienced by a person in a satellite orbit close to Earth is the same apparent weightlessness experienced in a freely falling elevator.

Kepler’s Laws and Newton’s Synthesis

Kepler’s Laws of Planetary Motion
Kepler’s First Law:
The path of each planet about the Sun is an ellipse with the Sun at one focus
An Ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, F1and F2) remains constant.

Kepler’s Second Law:
Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.

4
Sun
3

Kepler’s Third Law:
The ratio of the squares of the periods of any two planets revolving around the Sun is equal to the ratio of the cubes of their mean distances from the Sun.

Sample Problems
Four 7.5-kg spheres are located at the corners of a square of side 0.60 m. Calculate the net gravitational force on one sphere due to the other three.
Calculate the effective value of g, at 3200 m and 3200 km above the earth’s surface.
Calculate the velocity of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km.
Neptune is an average distance of 4.5 x 109 km from the Sun. Estimate the length of the Neptunian year given that the Earth is 1.50 x 108 km from the Sun on the average.

TERMINAL VELOCITY

TERMINAL VELOCITY
An object is dropped from REST.
Object about to start falling. V=0
2. Object is falling. V>0
3. The object now moves with TERMINAL VELOCITY.
Friction
Friction=Fg
V = max
Fg =mg
W=mg
a 9.8 m/s2
a= m/s2

TERMINAL VELOCITY
An object is dropped from REST.
Object about to start falling. V=0
2. Object is falling. V>0
3. The object now moves with TERMINAL VELOCITY.
Friction
Friction=
Fg =mg
Fres < Fg
V = max
=mg
a=10m/s2 The object accelerates towards the earth.
a<10 m/s2
Acceleration decreased!
a= 0 m/s2

Chapter 11 GRAVITATION

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