1. PHYS 2010 Nathalie Hoffmann University of Utah
Thursday 6/11
PHYS 2010
Nathalie Hoffmann
University of Utah

2. Newton’s Law of Universal Gravitation
The force of gravitation between two objects is given by
𝐹 𝑔𝑟𝑎𝑣 = 𝐺 𝑚 1 𝑚 2 𝑟 2
G is the gravitational constant (𝐺=6.67∙ 10 −11 𝑁∙ 𝑚 2 /𝑘 𝑔 2 )
𝑚 1 , 𝑚 2 are the masses of the two object
r is the center-to-center distance between the two objects

3. Acceleration due to gravity
The acceleration due to gravity on the surface of an object is
𝑔= 𝐺 𝑀 𝑜𝑏𝑗 𝑅 2
A particle’s weight on the surface of this object is
𝐹= 𝑚 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑔= 𝐺 𝑀 𝑜𝑏𝑗 𝑚 𝑝 𝑅 2 (Newton’s law of gravitation!)

4. Gravitational Potential Energy
This is the general/universal form of gravitational potential energy
𝑈 𝑔𝑟𝑎𝑣 =− 𝐺 𝑚 1 𝑚 2 𝑟
G is the gravitational constant
𝑚 1 , 𝑚 2 are the masses of the two objects
R is the distance between the centers of the two objects
N.B.: Gravitational potential energy is zero if the two objects are infinitely far apart
N.B.: 𝑈 𝑔𝑟𝑎𝑣 is always negative (or equal zero), never positive

5. Escape Speed
The minimum speed required at the surface of the planet in order to escape that planet.
1 2 𝑚 𝑣 2 + − 𝐺𝑀𝑚 𝑅 =0
𝑣 𝑒𝑠𝑐𝑎𝑝𝑒 = 2𝐺𝑀 𝑅
M is the mass of the planet
m is the mass of escaping object
R is the radius of the planet

6. Kepler’s Laws of Planetary Motion
1. The orbit of each planet is an ellipse with the Sun located at one focus of the ellipse.
2. A line joining the Sun and a planet sweeps out equal areas in equal intervals of time, regardless of the position of the planet in the orbit.
3. The square of the period of a planet’s orbit is proportional to the cube of the semimajor axis of the orbit.

7. 2nd Law: Law of Areas
A line joining the Sun and a planet sweeps out equal areas in equal intervals of time, regardless of the position of the planet in the orbit.
Conservation of angular momentum! Planet moves faster when closer to Sun/star, moves slower at farther distances.

8. 3rd Law: Law of Periods
The square of the period of a planet’s orbit is proportional to the cube of the semimajor axis of the orbit.
𝑇 2 ∝ 𝑎 3
T is the period
a is the semimajor axis
𝑇 2 = 4 𝜋 2 𝐺𝑀 𝑎 3
M is mass of the star

9. Circular Orbits Kepler’s 3rd law: Satellite Orbits 𝑇 2 ∝ 𝑟 3
r is the radius of the circular orbit
Satellite Orbits
𝐹 𝑔𝑟𝑎𝑣 =𝑚 𝑎 𝑐𝑒𝑛𝑡
𝐺𝑀𝑚 𝑟 2 =𝑚 𝑣 2 𝑟 (can solve for required satellite speed)

10. Problem 10
Eight stars, all with the same mass as our Sun, are located on the vertices of an astronomically sized cube with sides of 110 AU. Determine the net gravitational force on one of the stars due to the other seven. Give the value of each component of the gravitational force. (Give the magnitude only.)

11. Problem 10
Eight stars, all with the same mass as our Sun, are located on the vertices of an astronomically sized cube with sides of 110 AU. Determine the net gravitational force on one of the stars due to the other seven. Give the value of each component of the gravitational force. (Give the magnitude only.)

12. Problem 14
Find the net gravitational force exerted on a small, 4-kg object located at the point P by the three massive objects (see figure below). Each of the massive objects lies on a coordinate axis as shown.