ASTR 1010 – Homework Assignment 4 – Spring 2009

Question 1 - #2. Acceleration is the change in velocity over a period of time,

Delta v/Delta t

The units of acceleration are m/s² The acceleration of gravity is the acceleration of a falling body. At the surface of the Earth, this is at 9.8 m/s² .

Question 2 - #3. Momentum is the mass x velocity. Applying a force to an object changes its momentum. However, the force has to be an external force. Internal forces like those arising from Newton’s Third Law do not change the momentum of a system.

Question 3. - #4. Free fall is a condition or state in which an object is falling without any resistance. Objects in free fall are weightless.

Question 4. - #5. Newton’s First Law: An object moves at constant velocity if there is no net force acting upon it. Newton’s Second Law: F = ma. Newton’s Third Law: For any force, there is always and equal and opposite force.

Question 5. - #6. Conservation of momentum states that the total momentum of interacting objects cannot change as long as no external force is acting on them. Rocketships are an example of conservation of momentum. Conservation of angular momentum states that, in the absence of external turning or twisting forces (torques), the total angular momentum of a set of interacting objects cannot change. Kepler’s Second Law is an example of conservation of angular momentum. Conservation of energy states that objects lose energy or gain energy only by exchanging energy with other objects. An example of conservation of energy in astronomy is looking at the interplay of kinetic and gravitational potential energy as a planet orbits the Sun.

Question 6. - #7. Kinetic energy is the energy that moving objects have ( 0.5mv²)

Radiative energy is the energy carried by light.

Potential energy is the energy stored in as system that can later be converted into kinetic and radiative energy.

Question 7. - #8. Temperature is a measure of the average Kinetic Energy of a system of particle. Thermal energy represents the collective Kinetic Energy of the many individual particles. Thermal energy measures the total kinetic energy of all randomly moving particles in a substance while the temperature measures the average Kinetic Energy of the particles.

Question 8. - #11. Newton’s Law of Universal Gravitation states that the force between two masses is proportional to the product of the two masses and inversely proportional to the square of the distance between them. If you choose to use meters, kilograms, and seconds to describe the quantities involved, then the calibration constant between Newtons (the unit of force) and the other quantities is called G and it has a value of 6.67 x 10^-11 Nt m²/kg² .

The Law of Universal Gravitation can be expressed as F = GM₁M₂/d²

Question 9. - #17. Tidal friction is the frictional force on an object that is caused by a tidal interaction between that object and another body. A tidal force is produced when gravity pulls the nearer side of an object more than the farther side causing the object to stretch. Tidal forces between the Earth and the Moon are slowly increasing the Earth’s rotational period (the length of the day). Synchronous rotation is when an orbiting object’s rotational period has been slowed down so much by tidal friction that this rotational period becomes equal to the period of the object’s revolution about the main body. Tidal friction slowed down the Moon’s rotational period till it was the same as its period of revolution.

Question 10. - #45. a) GM₁M₂/d²

If we make d 4 times larger, we decrease F by a factor of (4)² = 16

b) It would double

c) If we make d 3 times smaller, we increase F by a factor of (3)²

Question 11. - #55. The mass of the planet is negligible compared to the star so the situation is just like that of the Earth moving around the Sun. Thus, the orbital period is one year.

N.B. – in the problems below, I just cannot get the font on the html page to do greek letters, so I’m writing 3.1415927…as “pi”, do not confuse it with the period, p…sorry about that, and if anybody has a solution to this stupid issue, please let me know…

Question 12. - #56.

Let p_E be the Earth’s period of revolution about the Sun

a_E is the average distance of the Earth from the Sun, so:

p_E² = 4pi²a_E³ / (GM_sun)

describes Kepler’s Third Law (Newton’s version) for the Earth going around the Sun. Now, if we have a planet (use the subscript “p” to identify those quantities associated with the planet) going around a star with 4 times the mass of the Sun and at an average distance of 1 AU (like the Earth), we can write those two conditions as:

a_p = a_E

M_star = 4M_sun

So, Kepler’s Third Law in this case becomes:

p_p² = 4pi²a_p³ / (GM_star)

now, substitute for a_p and M_star:

p_p² = 4pi²a_E³ / (G4M_sun)

p_p² = 4pi²a_E³ / 4(GM_sun) = (1/4) 4pi²a_E³ / (GM_sun)

p_p² = (1/4) p_E²

p_p = (1/2) p_E

So the period of the planet around the star is ½ that of the Earth around the Sun, or, half a year.

Question 13. - #57. (a)

P_Moon² = 4pi²a_Moon³ / (GM_Earth)

P_Moon = 27.3 days = 2.36 x 10⁶ s

a_Moon = 3.84 x 10⁵ km = 3.84 x 10⁸ m

M_Earth = 6.02 x 10²⁴ kg

(b) 1.90 x 10²⁷ kg

(c) 4.6 x 10¹⁰ m

Question 14. - #59. (a)

Use v_esc = (2GM/r)^1/2

5.1 x 10³ m/s

(b) 11 m/s

(c) 5.96 x 10⁴ m/s

(d) 4.2 x 10⁴ m/s

(e) 1.4 x 10⁴ m/s

Question 15. - #60. (a)

Use a = GM/r² to get the acceleration on that world, and then multiply by your mass to get the weight in Newtons (Nt). I’m going to use my mass because it is a nice round number: 100 kg.

a_Mars = 3.9 m/s²

Weight = 390 Nt

(b) a_Venus = 8.9 m/s²

Weight = 890 Nt

(c) a_Jupiter = 25 m/s²

Weight = 2500 Nt

(d) a_Europa = 1.3 m/s²

Weight = 130 Nt

(e) a_Phobos = 0.0051 m/s²

Weight = 0.51 Nt