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Unformatted text preview: homework 27 BAUTISTA, ALDO Due: Apr 5 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 3. 10 points. Consider an Earth-like planet hit by an asteroid. The planet has mass M p = 6 . 35 10 23 kg and radius R p = 7060000 m, and you may ap- proximate it as a solid ball of uniform density. It rotates on its axis once every T = 39 hr. The asteroid has mass M a = 3 . 09 10 17 kg and speed v a = 35700 m / s (relative to the planets center); its velocity vector points = 74 below the Eastward horizontal. The impact happens at an equatorial location. The picture below shows the view from above the planets North pole: v m R First, calculate the planets angular mo- mentum (relative to its spin axis) before the impact. Correct answer: 5 . 66573 10 32 kg m 2 / s (tol- erance 1 %). Explanation: Basic Concept: A rigid body rotating around a fixed axis has angular momentum L = I where I is the bodys moment of inertia about the axis of rotation. Approximating the planet as a solid ball of uniform density, its moment of inertia is I p = 2 5 M R 2 = 1 . 26603 10 37 kg m 2 . Its angular velocity before the impact is = 2 T = 4 . 4752 10- 5 rad / s , so L = I = I = (1 . 26603 10 37 kg m 2 ) (4 . 4752 10- 5 rad / s) = 5 . 66573 10 32 kg m 2 / s . Question 2 Part 2 of 3. 10 points. Calculate the asteroids angular momen- tum relative to the planetary axis. Correct answer: 2 . 14669 10 28 kg m 2 / s (tol- erance 1 %). Explanation: Approximating the asteroid as a pointlike particle, its angular momentum is ~ L a = ~ R M a ~v a where ~ R is the asteroids radius vector and M a ~v a is its linear momentum vector. At the moment of impact, both the radius vector ~ R and the momentum vector M a ~v a of the asteroid lie in the planets equatorial plane. Consequently, their vector product is perpendicular to the equatorial plane and parallel to the planet axis. Because the hori- zontal component of the asteroids velocity is directed Eastward the same as the planets rotation the asteroids angular momentum ~ L a has the same direction as the planets an- gular momentum ~ L p . In magnitude, L a = R p M a horizontal component of v a = R p M a v a cos = (7060000 m) (3 . 09 10 17 kg) (35700 m / s) cos(74 ) = 2 . 14669 10 28 kg m 2 / s ....
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