physics : class11 Angular Momentum and Orbital Period : Second Law of Kepler : Gravitation MCQ Questions and Answers
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Angular Momentum and Orbital Period
MCQ questions and answers.
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MCQ Questions
1. A planet moves in an elliptical orbit around the Sun. If its distances from the Sun at perihelion and aphelion are in the ratio r_p:r_a = 1:4, what is the ratio of its angular speeds about the Sun, omega_p:omega_a?
A.
1:16
B.
4:1
C.
16:1
D.
1:4
Correct Answer:
16:1
Explanation:
By conservation of angular momentum, r^2 omega is constant because the areal velocity is constant. Hence omega_p/omega_a = (r_a/r_p)^2 = 4^2 = 16.
2. A planet takes time T to complete one elliptical revolution about the Sun. What fraction of the total area of its orbit is swept out by the radius vector during the first T/2 after passing through perihelion?
A.
One-fourth
B.
One-half
C.
Three-fourths
D.
It depends on the eccentricity
Correct Answer:
One-half
Explanation:
Kepler's second law states that equal areas are swept out in equal times. Since T/2 is half the orbital period, the swept area is half the total elliptical area, independent of eccentricity.
3. A planet of mass m has specific angular momentum h about the Sun. Its areal velocity, defined as the area swept per unit time by the Sun-planet line, is
A.
h/m
B.
h
C.
h/2
D.
2h
Correct Answer:
h/2
Explanation:
The areal velocity is dA/dt = (1/2)r^2 dtheta/dt. Since specific angular momentum h = r^2 dtheta/dt, the areal velocity equals h/2.
4. A planet has perihelion and aphelion distances r_p and r_a from the Sun. If its speeds at these points are v_p and v_a, respectively, which relation follows directly from conservation of angular momentum?
A.
v_p/v_a = r_p/r_a
B.
v_p/v_a = r_a/r_p
C.
v_p/v_a = (r_a/r_p)^2
D.
v_p/v_a = 1
Correct Answer:
v_p/v_a = r_a/r_p
Explanation:
At perihelion and aphelion the velocity is perpendicular to the radius vector, so angular momentum conservation gives m r_p v_p = m r_a v_a. Therefore v_p/v_a = r_a/r_p.
5. Two planets orbit the same star in elliptical orbits having the same semimajor axis but different eccentricities. Which statement about their orbital periods and specific angular momenta is correct?
A.
Both periods and specific angular momenta are equal
B.
Periods are equal, but specific angular momenta are different
C.
Periods are different, but specific angular momenta are equal
D.
Both periods and specific angular momenta are different in the same ratio
Correct Answer:
Periods are equal, but specific angular momenta are different
Explanation:
For a given central mass, the orbital period depends only on the semimajor axis: T^2 is proportional to a^3. However, h^2 = GM a(1 - e^2), so different eccentricities give different specific angular momenta.
6. For a planet in an elliptical orbit, the ratio of its angular speeds at perihelion and aphelion is 25:1. What is the ratio of its perihelion distance to its aphelion distance?
A.
1:25
B.
1:5
C.
5:1
D.
25:1
Correct Answer:
1:5
Explanation:
Since r^2 omega is constant, omega_p/omega_a = (r_a/r_p)^2. Thus 25 = (r_a/r_p)^2, giving r_a/r_p = 5 and therefore r_p:r_a = 1:5.
7. A planet sweeps out an area equal to one-third of the total area of its elliptical orbit. If it starts at perihelion, what fraction of its orbital period has elapsed?
A.
One-ninth
B.
One-third
C.
One-half
D.
It cannot be determined from Kepler's second law
Correct Answer:
One-third
Explanation:
Kepler's second law makes the swept area directly proportional to elapsed time. Therefore sweeping one-third of the total orbital area requires one-third of the orbital period.
8. For an elliptical orbit with semimajor axis a and semiminor axis b, a planet of mass m has angular momentum magnitude L about the central star. What is its orbital period T in terms of a, b, m, and L?
A.
T = L/(2pi m a b)
B.
T = 2pi m a b/L
C.
T = pi m a b/L
D.
T = 2pi L/(m a b)
Correct Answer:
T = 2pi m a b/L
Explanation:
The total orbital area is piab, while the areal velocity is L/(2m). Hence T = total area divided by areal velocity = piab divided by L/(2m) = 2pi m a b/L.
9. A planet moves from point P to point Q in an elliptical orbit in time Delta t. The area swept by the Sun-planet radius vector is A. If the planet's mass is doubled while its orbit and time interval remain unchanged, what happens to the angular momentum required for the same areal sweep rate?
A.
It becomes half as large
B.
It remains unchanged
C.
It becomes twice as large
D.
It becomes four times as large
Correct Answer:
It becomes twice as large
Explanation:
The areal velocity is A/Delta t = L/(2m). For the same orbit and time interval, the areal velocity is unchanged. Therefore L is proportional to m, so doubling the mass doubles the angular momentum.
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