How are these problems done? I feel like they are way easier than I am making th

Question

How are these problems done? I feel like they are way easier than I am making them out to be in my head... Please show how we figure it out. Thanks.

A disk has 2 points on it. Point P is near the outside edge, and Point Q is about halfway between the center and the edge. When the object is rotating around its center O, which of the following statements is correct about the motion of P and Q?

a. P & Q have the same velocity & the same angular velocity.

b. P & Q have the same velocity but different angular velocities.

c. P & Q have different velocities but the same angular velocity.

d. P & Q have different velocities and different angular velocities.

An Astronaut is resting motionless in outer space between two large asteroids. He is 8 km from the larger asteroid and 2 km from the smaller asteroid. If the large asteroid has 13 times the mass of the smaller one, what will happen to the astronaut?

a. He will remain between the two asteroids.

b. He will move towards the smaller asteroid.

c. He will move toward the larger asteroid.

Two identical masses are attached to the ends of two identical horizontal springs. The red mass is stretched 10 cm from equilibrium and released. The blue mass is stretched 5 cm from equilibrium and released. How do the periods of their oscillatory motions compare?

a. The red mass has quadruple the period of the blue mass.

b. The red mass has double the period of the blue mass.

c. The red mass has the same period of the blue mass.

d. The red mass has half the period of the blue mass.

Explanation / Answer

a)

vel of a point at a distance r is given by rw

So

Pand Q have same angular vel but P has higher linear vel as r is greater for P

b)

F=GMm/r^2

So

F=13GMm/64-GMm/4=-3GMm/64

So

as F comes negative so smaller body applied more force. So

astronaut moves towards smaller body as F is attractive

c)

T=2pi*sqrt(m/k)

So

they have same time period as T is not dependent on A

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