Newton's Universal Law of Gravitation

Newton's Universal Law of Gravitation

The gravitational force that exists between two masses is given by

where - is the distance of separation between their centers.

- universal gravitation constant

𝑟

𝑚1 𝑚2𝐹 𝑔 𝐹 𝑔

Relationship between Force, mass and distance.

Relationship between Force, mass and distance.

𝑚1

𝑚2

𝑚3

𝑟1

𝑟2

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NET GRAVITATIONAL FORCE

For three or more objects, finding the net gravitational force is by component!

REVIEW YOUR COMPONENT METHOD!!!

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Cartesian Plane:

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.e 𝑥𝑎𝑚𝑝𝑙𝑒 :𝑔𝐸=9.8 m /𝑠2

Gravity Near The Earth’s SurfaceAt the Earth’s surface:

𝑔𝐸=𝐺𝑚𝐸

𝑟 𝐸2 =9.8m /s2

In general: 𝑔𝑝=𝐺𝑚𝑝

𝑟𝑝2

Gravitational field at the object’s surface

where

𝒓

The effective g, g’

As you go far from the Earth’s surface, the gravitational field decreases.

So, the effective g (g’):

where

𝑟=𝑟𝑝+𝑦

(𝑟>𝑟𝑝)𝑔 ′=𝐺

𝑚𝑝

𝑟 2

Since

𝑔 ′=𝑔𝑝( 𝑟 𝑝

𝑟 )2

Sample Problems:

1. Two objects attract each other with a gravitational force of magnitude when separated by 20.0 cm. If the total mass of the objects is 5.00 kg, what is the mass of each?

2. Calculate the effective value of g, at 3200 m and 3200 km above the earth’s surface.

3. 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

With 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

𝐹 𝑔=𝑚𝑣2

𝑟 𝐺 𝑚𝑀𝑟2 =𝑚𝑣2

𝑟

𝑣=√𝐺𝑀𝑟

Speed of satellite at orbit radius r

where

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, F1 and F2) remains constant.

Kepler’s Laws of Planetary Motion

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.

Sun

4

3

Kepler’s Laws of Planetary Motion

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 Problems1. 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.

2. Calculate the effective value of g, at 3200 m and 3200 km above the earth’s surface.

3. Calculate the velocity of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km.

4. 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

Fg =mg

1. Object about to start falling. V=0

W=mg

2. Object is falling. V>0

Friction

a 9.8 m/s2

Friction=Fg

a= m/s2

TERMINAL VELOCITY

3. The object now moves with TERMINAL VELOCITY.

An object is dropped from REST.

V = max

Fg =mg

1. Object about to start falling. V=0

=mg

2. Object is falling. V>0

Friction

a=10m/s2 The object accelerates towards

the earth. a<10 m/s2

Acceleration decreased!

Friction=

a= 0 m/s2

TERMINAL VELOCITY

3. The object now moves with TERMINAL VELOCITY.

An object is dropped from REST.

V = maxFres < Fg