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Foundations of Celestial Motion
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1
Question
What did Galileo recognize about falling bodies?
Page 1
Answer
All bodies, irrespective of their masses, accelerate towards the earth with constant acceleration.
2
Question
How did Galileo arrive at the value of acceleration due to gravity?
Page 1
Answer
Through experiments with bodies rolling down inclined planes.
3
Question
What is the geocentric model of planetary motion?
Page 1
Answer
A model where all celestial objects revolve around the earth.
4
Question
What motion was thought possible for celestial objects in the geocentric model?
Page 1
Answer
Motion in a circle.
5
Question
How did Ptolemy describe planetary motions?
Page 1
Answer
Planets move in circles with centers moving in larger circles.
6
Question
Who proposed the heliocentric model earlier than Copernicus?
Page 1
Answer
Aryabhatta in the 5th century A.D.
7
Question
What is the heliocentric model?
Page 1
Answer
A model where the sun is the center around which planets revolve.
8
Question
Why was Copernicus's theory discredited?
Page 1
Answer
By the church, leading to prosecution of supporters like Galileo.
9
Question
Who recorded planetary observations with the naked eye?
Page 1
Answer
Tycho Brahe (1546-1601).
10
Question
Who analyzed Tycho Brahe's data to formulate Kepler's laws?
Page 1
Answer
Johannes Kepler (1571-1640).
11
Question
How did Kepler's laws enable Newton's universal law of gravitation?
Page 1
Answer
They provided the basis for Newton to propose the law.
12
Question
What is Kepler's first law (law of orbits)?
Page 2
Answer
All planets move in elliptical orbits with the sun at one focus.
13
Question
What is the perihelion in a planetary orbit?
Page 2
Answer
The closest point to the sun.
14
Question
What is the aphelion in a planetary orbit?
Page 2
Answer
The farthest point from the sun.
15
Question
What is the semi-major axis of an ellipse?
Page 2
Answer
Half the distance between perihelion and aphelion.
16
Question
How is an ellipse drawn using a string?
Page 2
Answer
Fix string ends at two foci, stretch taut with a pencil.
17
Question
What are the foci of an ellipse?
Page 2
Answer
Two points F1 and F2 such that sum of distances from any point on ellipse is constant.
18
Question
What is Kepler's second law (law of areas)?
Page 2
Answer
The line joining planet to sun sweeps equal areas in equal time intervals.
19
Question
Why do planets appear to move slower when farther from the sun?
Page 2
Answer
According to the law of areas, to sweep equal areas in equal time.
20
Question
What is Kepler's third law (law of periods)?
Page 3
Answer
Square of time period is proportional to cube of semi-major axis.
21
Question
What does the table of planetary data confirm?
Page 3
Answer
Kepler's law of periods, as Q = T²/a³ is nearly constant.
22
Question
Why is the law of areas a consequence of conservation of angular momentum?
Page 3
Answer
For central forces, angular momentum L is constant, so dA/dt is constant.
23
Question
What is a central force?
Page 3
Answer
A force directed along the vector joining the two bodies.
24
Question
How is the area swept derived from angular momentum?
Page 3
Answer
ΔA = (1/2) (r × v) Δt = L / (2m) Δt, so dA/dt = L/(2m) constant.
25
Question
In Example 7.1, how does speed at perihelion relate to aphelion?
Page 3
Answer
v_p / v_A = r_A / r_p, so v_p > v_A since r_A > r_p.
26
Question
Why does the planet take longer to traverse BAC than CPB?
Page 3
Answer
Area SBAC > area SBPC, and equal areas swept in equal times.
27
Question
What inspired Newton's universal law of gravitation?
Page 3
Answer
Observation of an apple falling and the moon's orbit.
28
Question
What is the centripetal acceleration of the moon?
Page 3
Answer
a_m = V_m² / R_m = 4π² R_m / T².
29
Question
Why is a_m much smaller than g on earth's surface?
Page 3
Answer
Gravitational force decreases with distance from earth.
30
Question
How does assuming inverse square law relate a_m and g?
Page 3
Answer
g / a_m = (R_m / R_E)^2 ≈ 3600.