An object can have external forces acting on it that are balanced. For instance, the frictionless cart in the class demonstration had the force of gravity acting down and the normal force of the table acting up

HW #2 (Due 9/9)

1. 
   Newton’s first law states: An object at rest (or in motion) will stay at rest (or in motion) unless acted on by a net external force. Explain what “net external force means”. Why not just an external force?
   

An object can have external forces acting on it that are balanced. For instance, the frictionless cart in the class demonstration had the force of gravity acting down and the normal force of the table acting up. Both are external forces. But the cart did not move. This is because these two forces balance each other and therefore cancel, giving no NET external force.

2. 
   Use Newton’s second and third laws to explain why the Earth orbits the Sun instead of the Sun orbiting the Earth.
   

Newton’s 3^rd law tells us that the force the Sun exerts on the Earth has to be equal and opposite of the force that the Earth exerts on the Sun. In other words, the Sun can only exert as much force on the Earth as the Earth exerts back. Newton’s 2^nd Law states that the force exerted on an object is equal to the mass of the object times the acceleration. (F = ma) So we can write the following:

F_SE = -F_ES

Where F_SE is the force exerted by the Sun on the Earth and F_ES is the force exerted by the Earth on the Sun. Using F=ma we can write

m_Ea_E = m_Sa_S

Since the mass of the Sun is much greater than the mass of the Earth, the acceleration that the Earth feels is much greater than what the Sun feels. As a result the Earth goes around the Sun.

3. 
   Kepler found that the farther a planet was from the Sun, the slower it moved in its orbit. Use what you have learned about an orbit and the Newton’s law of Gravity to explain why Jupiter cannot orbit the Sun as fast as the Earth.
   

Jupiter is farther from the Sun than the Earth. Because of this, the Sun’s gravitational force is stronger on the Earth than it is on Jupiter. Both Jupiter and the Earth are in nearly circular orbits, which means they are both falling around the Sun. They do not fall into the Sun because their orbital velocity is such that the Sun curves away under them at the same rate that they are falling. (This is what we discussed in class, as to why the astronauts in the ISS are weightless. They are falling.) Earth is falling faster than Jupiter, because it is closer to the Sun. Therefore, in order to maintain its circular orbit, the Earth has to move faster in its orbit than Jupiter does.

4. 
   Mass is a very important stellar parameter for understanding stars. Explain why astronomers use binary stars in order to determine the mass of different types of stars. (Hint: Look at Kepler’s third law which is in the stellar toolbox to help you decide how stellar mass could possibly be determined.)
   

Kepler’s 3^rd law can be written as:



Where M is the mass of one object and m is the mass of the other. In order to probe the mass of an star we need to observe two objects interacting gravitationally. This is why we need to observe binary stars and see how fast they orbit in order to determine their mass. In order to determine the quantity all we need to know is the orbital period of the stars and the distance between them. (In this cases it is actually the distance between their center of mass). We will need to develop a little more physics to actually find the individual masses, M and m.