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Practice Exam 5 - practicework 05 BAUTISTA ALDO Due May 8...

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practicework 05 – BAUTISTA, ALDO – Due: May 8 2006, 4:00 am 1 Version number encoded for clicker entry: V1:1, V2:4, V3:2, V4:4, V5:3. Question 1 Part 1 of 6. 10 points. Asteroids X , Y , and Z have equal mass. They orbit in the same angular direction around a planet with much greater mass (lo- cated at the black dot, at one of the foci of each of the two ellipsical orbits and at the center of the circular orbit). 3 2 1 X Y Z Figure 1: The orbits of the aster- oids are in the plane of the figure and are drawn to scale. Let the subscripts denote the asteroid’s po- sition. The magnitude of the angular momen- tum of the asteroid X at positions 2 and 1 are related by 1. 2 X < ‘ 1 X 2. Not enough information is available. 3. 2 X > ‘ 1 X 4. 2 X = 1 X correct Explanation: The net force on asteroid X is central; e.g. , ~ F = G M m r 2 ˆ r , therefore angular momentum is conserved; i.e., = constant . Angular momentum ~ = m~v × ~r = m r 2 is conserved; thus 2 X = 1 X . Question 2 Part 2 of 6. 10 points. The magnitude of the angular velocity ω of the asteroid X at positions 2 and 1 are related by 1. ω 2 X = ω 1 X 2. Not enough information is available. 3. ω 2 X < ω 1 X 4. ω 2 X > ω 1 X correct Explanation: Angular momentum ~ = m r 2 is con- served. Since r 2 X < r 1 X , then ω 2 X > ω 1 X . Question 3 Part 3 of 6. 10 points. Hint: 1) The semi-major axis a X of the orbit of asteroid X is equal to the radius r Y of the orbit of asteroid Y . 2) The semi-minor axis b X of asteroid X is equal to the semi-minor axis b Z of asteroid Z . 3) And the area of the orbit of asteroid Y ( A = π r 2 Y ) is equal to the area of the orbit of asteroid Z ( A = π a Z b Z ) . X Y Z Figure 2: The orbits of the aster- oids are shown for comparison pur- poses. The elliptical orbits of the as-
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practicework 05 – BAUTISTA, ALDO – Due: May 8 2006, 4:00 am 2 teroids are relocated and rotated in order that they have a common cen- ter and their major axes are aligned. The period T of asteroids X and Y are related by 1. T X < T Y 2. T X > T Y 3. T X = T Y correct 4. Not enough information is available. Explanation: Geometry of Ellipses: r 1 r 2 a b a a F 1 F 2 C Figure 3: The geometrical ele- ments of an ellipse An ellipse is defined as the curve traced by a particle moving so that the sum of its distances from two fixed points F 1 and F 2 is constant. The points F 1 and F 2 are called the foci of the ellipse and C is the center of the ellipse. The planet is located at one of these foci and the orbit of the asteroid is designated by the ellipse. Using the notation indicated, we have r 1 + r 2 = 2 a , where a is the semi- major axis, b is the semi-minor axis, ε is the eccentricity, ε a is the distance from the center of the ellipse to a focus of the ellipse, the ratio b a = p 1 - ε 2 , and the area A = π a b .
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