Question 1 Asteroids X, Y, and Z have equal mass of 8.0 kg each. They orbit arou

Question

Question 1

Asteroids X, Y, and Z have equal mass of 8.0 kg each. They orbit around a planet with M=3.20E+24 kg. The orbits are in the plane of the paper and are drawn to scale. In the statements below, TE is the total mechanical energy, KE is the kinetic energy, and PE is the potential energy.

greater than less than equal to  The PE of Z at r is .... the PE of Y at i
greater than less than equal to  The TE of X is .... the TE of Z
greater than less than equal to  The PE of Z at c is .... the PE of X at i
greater than less than equal to  The PE of X at a is .... that at c
greater than less than equal to  The speed of Z at c is .... that at m
greater than less than equal to  The PE of Z at r is .... the PE of Y at r
greater than less than equal to  The TE of Y is .... the TE of X
greater than less than equal to  The KE of Y at i is .... that at r

Question 2

Asteroids X, Y, and Z have equal mass of 3.0 kg each. They orbit around a planet with M = 5.20×1024 kg. The orbits are in the plane of the paper and are drawn to scale. The three asteroids orbit in the same clockwise direction.

greater than less than equal to  The angular momentum of Z is .... that of X.
greater than less than equal to  The angular velocity of Z at n is .... that at e.
greater than less than equal to  The angular velocity of X at c is .... that at s.
greater than less than equal to  The angular momentum of X at s is .... that at e.
greater than less than equal to  The angular velocity of Z at e is .... that of X at e.
greater than less than equal to  The period of X is .... that of Y.
greater than less than equal to  The period of Z is .... that of X.

Explanation / Answer

Answer (1)

The total mechanical energy is the sum of kinetic enrgy and potential Energy. Potential energy increases as the distance from the attractive body (in this case the planet) increases and hence kinetic energy decreases with increase in distance.

The PE of Z at r is ..equal to .. the PE of Y at i (same mass, equidistant from the planet)

The TE of X is ..less than.. the TE of Z (Z goes more far away from the planet than X does)

The PE of Z at c is ..equal to.. the PE of X at i (same distance)

The PE of X at a is ..less than.. that at c (distance of a is lesser than c from the planet)

The speed of Z at c is ..greater than.. that at m (speed decreases as the distance from the planet increases)

The PE of Z at r is ..equal to.. the PE of Y at r (same distance)

The TE of Y is ..less than.. the TE of X (X reaches more distance from the planet than Y does)

The KE of Y at i is ..equal to.. that at r ( i and r at the same distance from the planet)

Answer (2)

The total angular momentum of a planetary object is given by I=mrv, r being the distance from the planet and v being the speed of the object. As r increases, v decreases or vice-versa. Hence angular momentum of a body orbiting an attractive body is always conserved. (This is from Keplar's second law of orbital motion).

The angular momentum of Z is ..equal to.. that of X (the product of vr remains same)

The angular velocity of Z at n is ..equal to.. that at e ( n and e are equidistant from the planet)

The angular velocity of X at c is ..greater than.. that at s (c is closer to planet than s and speed is greater at shorter distances)

The angular momentum of X at s is ..equal to.. that at e (momentum is conserved)

The angular velocity of Z at e is ..equal to.. that of X at e (same distance)

The period of X is ..less than.. that of Y (X has more angular velocity hence lesser time period)

The period of Z is ..less than.. that of X (X goes more distance and reducing speed)

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