Question

Asteroids X, Y, and Z have equal mass of 6.0 kg each. They orbit around a planet with M = 3.20×1024 kg. The orbits are in the plane of the paper and are drawn to scale. The three asteroids orbit in the same clockwise direction.

http://postimg.org/image/3yjgpptpp/

The angular momentum of Y is .... that of Z.

The angular momentum of Z at s is .... that at c.

The period of Y is .... that of Z.

The angular velocity of Z at c is .... that at i.

The angular velocity of Y at c is .... that at n.

The angular velocity of Z at c is .... that of Y at c.

The period of X is .... that of Y.

Options are Greater than, Less than, or Equal to

Answer 1

We know that the angular momentum L is given as

L=r xp or can be given as

L=r x mv ...(1)

Here, r is the radius vector, m is the mass and v is the velocity. It is given that all the asteriods have same mass and are moving in the same direction. We can think of X and Z as projectiles launched from the orbit of Y. The source gives another way of seeing the problem. So we can say that Z is orbiting around Y in an elliptical orbit so its angular mometum must be conserved.

Hence, the angular momentum of Y is equal to angular momentum of Z.

(ii) Eq. (1) states for a single orbit angular momentum L will dependent on the value of r. If r is more then L is more and vice -versa. Clearly, s is lying along the semi major axis while c is lying along the semi minor axis . Hence, the r is more for point s as compare to point c. So anular momentum of Z at s is greater than that at c.

(iii) According to Keppler's law the time period is dirrectly proportional to the cube of the semi major axis. Z reaches a greater height from the planet so the period of Y is less than the period of Z.

(iv) The angular speed omega is given as

omega=v/r ......(2)

The speed at point c is greater than at point i as it decreases with height. So angular velocity of Z at c is greater than at i.

(v) Asteriod Y is a circle . Hence, the angular velocity of Y at c is equal to that at n.

(vi) The angular velocity of Z at c is greater than that of Y at c because the value of v is more at pont c due to Z.

(vii) According to Keppler's law the time period is dirrectly proportional to the cube of the semi major axis. So the period of X is greater than that of Y.