physics : class11 Equal Areas Law : Second Law of Kepler : Gravitation MCQ Questions and Answers
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Equal Areas Law
MCQ questions and answers.
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MCQ Questions
1. A planet sweeps out an area of 3.0 × 10^10 m² with respect to the Sun in 2.0 days. Assuming Kepler's equal areas law, what area will it sweep in 5.0 days?
A.
6.0 × 10^10 m²
B.
7.5 × 10^10 m²
C.
9.0 × 10^10 m²
D.
1.5 × 10^11 m²
Correct Answer:
7.5 × 10^10 m²
Explanation:
Kepler's second law states that the areal velocity is constant. Therefore, swept area is proportional to time: A2 = A1(t2/t1) = 3.0 × 10^10 × (5/2) = 7.5 × 10^10 m².
2. A planet is at distances r1 and r2 from the Sun at two points in its orbit. If its velocity is perpendicular to the radius vector at both points and its speeds are v1 and v2 respectively, which relation follows from the equal areas law?
A.
r1v1 = r2v2
B.
r1²v1 = r2²v2
C.
r1v1² = r2v2²
D.
v1/r1 = v2/r2
Correct Answer:
r1v1 = r2v2
Explanation:
For perpendicular velocity, the areal velocity is (1/2)rv. Since the areal velocity remains constant, (1/2)r1v1 = (1/2)r2v2, giving r1v1 = r2v2.
3. At a certain point in its orbit, a planet has position vector magnitude 4.0 × 10^10 m, speed 3.0 × 10^4 m/s, and the angle between its position and velocity vectors is 60°. At another point, its position vector magnitude is 6.0 × 10^10 m and the angle between the vectors is 30°. What is the speed at the second point?
A.
1.73 × 10^4 m/s
B.
2.00 × 10^4 m/s
C.
3.00 × 10^4 m/s
D.
3.46 × 10^4 m/s
Correct Answer:
1.73 × 10^4 m/s
Explanation:
The areal velocity is (1/2)rv sin θ. Hence r1v1 sin 60° = r2v2 sin 30°. Substituting values gives v2 = (4 × 3.0 × 10^4 × sin 60°)/(6 × sin 30°) = 1.73 × 10^4 m/s.
4. A planet sweeps out 1/8 of the total area enclosed by its orbit in 15 days. According to Kepler's equal areas law, what is its orbital period?
A.
60 days
B.
90 days
C.
120 days
D.
240 days
Correct Answer:
120 days
Explanation:
Because the areal velocity is constant, the fraction of total area swept equals the fraction of the orbital period elapsed. Thus 15/T = 1/8, so T = 120 days.
5. During two equal time intervals, a planet sweeps triangular areas A1 and A2 with respect to the Sun. If the distances from the Sun to the planet during the corresponding intervals are approximately in the ratio 2:3, what is the ratio of the average transverse speeds in those intervals?
A.
2:3
B.
3:2
C.
4:9
D.
9:4
Correct Answer:
3:2
Explanation:
Equal areas in equal times imply equal areal velocities. Since areal velocity is approximately (1/2)rv_transverse, r1v1 = r2v2. Therefore v1:v2 = r2:r1 = 3:2.
6. The areal velocity of a planet with respect to the Sun is 2.5 × 10^15 m²/s. How much area does it sweep in 8.0 hours?
A.
1.8 × 10^19 m²
B.
3.6 × 10^19 m²
C.
7.2 × 10^19 m²
D.
1.44 × 10^20 m²
Correct Answer:
7.2 × 10^19 m²
Explanation:
Areal velocity is swept area per unit time. In 8.0 hours, the time is 28,800 s, so A = (2.5 × 10^15)(28,800) = 7.2 × 10^19 m².
7. At two points in an orbit, the distances of a planet from the Sun are 5R and 2R. The angles between the radius vectors and the corresponding velocity vectors are 30° and 90°, respectively. If the speed at the first point is v, what is the speed at the second point?
A.
v/2
B.
5v/4
C.
5v/2
D.
5v
Correct Answer:
5v/4
Explanation:
Equal areas law gives r1v1 sin θ1 = r2v2 sin θ2. Thus 5R·v·sin 30° = 2R·v2·sin 90°, so (5R v)/2 = 2R v2 and v2 = 5v/4. Wait—using the stated values, the correct result is 5v/4, not 5v/2.
8. A planet's angular speed about the Sun at one location is four times its angular speed at another location. What is the ratio of its distances from the Sun at the first and second locations?
A.
1:2
B.
1:4
C.
2:1
D.
4:1
Correct Answer:
1:2
Explanation:
The areal velocity is (1/2)r²ω and is constant. Therefore r1²ω1 = r2²ω2. With ω1 = 4ω2, r1²/r2² = 1/4, giving r1:r2 = 1:2.
9. A planet sweeps areas A, 2A, and 3A during three successive time intervals. If the first interval lasts 4 days, what are the durations of the second and third intervals?
A.
2 days and 3 days
B.
4 days and 4 days
C.
8 days and 12 days
D.
6 days and 8 days
Correct Answer:
8 days and 12 days
Explanation:
Equal areas law gives a constant ratio between swept area and elapsed time. Since area A is swept in 4 days, area 2A requires 8 days and area 3A requires 12 days.
10. A planet moves from point P to point Q in the same time that it moves from point R to point S. The sectors swept by the Sun-planet line in the two intervals have different shapes. What must be true according to Kepler's equal areas law?
A.
The distances PQ and RS must be equal
B.
The angular displacements in the two intervals must be equal
C.
The areas of the two swept sectors must be equal
D.
The speeds at P and R must be equal
Correct Answer:
The areas of the two swept sectors must be equal
Explanation:
Kepler's second law states that the line joining the Sun and the planet sweeps equal areas in equal intervals of time, regardless of the shapes of the sectors or the distances traveled.
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