Homework 27

# Homework 27 - homework 27 – BAUTISTA ALDO – Due Apr 5...

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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 planet’s center); its velocity vector points θ = 74 ◦ below the Eastward horizontal. The impact happens at an equatorial location. The picture below shows the view from above the planet’s North pole: v m θ ω R First, calculate the planet’s 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 × ω where I is the body’s 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 ω = 2 π T = 4 . 4752 × 10- 5 rad / s , so L = I × ω = I × ω = (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 asteroid’s 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 asteroid’s 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 planet’s 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 asteroid’s velocity is directed Eastward — the same as the planet’s rotation — the asteroid’s angular momentum ~ L a has the same direction as the planet’s 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 θ = (7060000 m) × (3 . 09 × 10 17 kg) (35700 m / s) cos(74 ◦ ) = 2 . 14669 × 10 28 kg m 2 / s ....
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Homework 27 - homework 27 – BAUTISTA ALDO – Due Apr 5...

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