GRAVITATION Applications of the Law of Gravity Gravitational Potential Energy Kepler’s Laws
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