GRAVITATIONApplications of the Law of Gravity

Gravitational Potential Energy

Kepler’s Laws

Newton’s Universal Law of Gravitation

• The gravitational Force between masses is proportional to both masses.

• The Force Decreases as the inverse of the square of the distance between masses.

𝐹 =𝐺𝑚1𝑚2

𝑟2𝐺= 6.67 ∗ 10 − 11 𝑁 ∗ 𝑚2

𝑘𝑔2

Newton’s Universal Law of Gravitation• Newton: Gravity is property of matter. All matter is attracted to other matter.

• Gravity will only effect things that have mass.

• Einstein: Gravity is property of space. Large objects change the shape of space.

• Gravity can effect things without mass. (light)

Applications of the Law of Gravity• Weight = Force of gravity

𝑚𝑔 =𝐺𝑚𝑚

𝑟2

• Acceleration of Gravity

𝑔 =𝐺𝑚

𝑟2

• The acceleration of gravity on a planet depends on the mass and radius of the planet. (Density)

𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =𝑚𝑎𝑠𝑠

𝑣𝑜𝑙𝑢𝑚𝑒𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒 =

4

3𝜋𝑅3

Applications of the Law of Gravity• Orbital Velocity

• Objects orbiting a planet travel in a near circular path

• Circular motion requires Centripetal Force.

• Gravity provides centripetal force

𝐹𝑐 =𝑚𝑣2

𝑟=𝐺𝑚𝑚

𝑟2

𝑣2 =𝐺𝑚

𝑟

Applications of the Law of Gravity

• Calculating period of orbit.

• 𝑣2 =𝐺𝑚

𝑟

Applications of the Law of Gravity

• 𝑊𝑜𝑟𝑘 = −∆𝑈 → ∆𝑈 = −𝑊

• 𝐹𝑟 = −𝐺𝑚𝑚

𝑟2= 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑛𝑔 𝑜𝑢𝑡 𝑤𝑎𝑟𝑑 𝑓𝑟𝑜𝑚 𝑐𝑒𝑛𝑡𝑒𝑟.

• 𝑊𝐹𝑟 = −𝐺𝑚𝑚

𝑟2=

𝐺𝑚𝑚

𝑟

• −𝑊 = 𝑈 = −𝐺𝑚𝑚

𝑟→ Potential Energy is Negative!!!

𝐹𝑟

𝐹𝑔

Gravitational Potential Energy• Escape Velocity = the speed required to break free from the gravitational pull of

a body

• Total Energy = K+U

• After leaving a planet object will lose speed and its potential Energy will decrease as the distance form the planet increases.

• 𝐾1+ 𝑈1 = 𝐾2+ 𝑈2

• Eventually Total Energy will equal zero → 𝐾2+ 𝑈2 = 0

• 𝐾1+ 𝑈1 = 0 = 1

2mv2 + (−

𝐺𝑚𝑚

𝑟2)

𝑣 =2𝐺𝑀

𝑅

𝐾1+ 𝑈1

𝐾2+ 𝑈2 = 0

Gravitational Potential Energy• Conservation of Energy of a Planet

• 𝐾1+ 𝑈1 = 𝐾2+ 𝑈2

Keplers Laws• 1st Law: Planets move in ellipses with the Sun at one focus

Kepler’s Laws

• 2nd Law: The radius vector describes equal areas in equal times

Kepler’s Laws

• 3rd Law: The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semi-major axes.

𝑇2

𝑇2=𝑟3

𝑟3r=length of semi major axis =length of major axis divided by 2

Kepler’s Laws

•𝑇2

𝑟3= constant for orbiting objects → 𝑣2 =

𝐺𝑚

𝑟

•𝟐𝝅𝒓 𝟐

𝑻𝟐=

𝑮𝒎

𝒓→

𝟒𝝅𝟐𝒓𝟐

𝑻𝟐=

𝑮𝒎

𝒓→

𝑻𝟐

𝒓𝟑=

𝟒𝝅𝟐

𝑮𝒎

𝑻𝟐

𝒓𝟑=𝟒𝝅𝟐

𝑮𝒎