Planet X of mass m1 orbits a Sun in uniform circular motion at a distance r1 and speed v1. The mass of the Sun is MS1 and its radius is RS1 as shown in the figure below.

https://online-s.physics.uiuc.edu/cgi/courses/shell/common/showme.pl?courses/phys100/fall10/hwb/10/02/orbit.gif

1. If the planet's mass were doubled, how would a, the magnitude of its acceleration, change?

a would decrease.
a would remain the same.
a would increase.

Consider another solar system in which Planet Y of mass m2 = 2 m1 orbits a Sun of mass MS2 = 2 MS1 and radius RS2 = 2RS1 at a distance r2 = 2r1. Further, suppose R2, the radius of Planet Y is twice R1, the radius of Planet X.

2. How would w2, your weight on Planet Y, compare to w1, your weight on Planet X?

w2 < w1
w2 = w1
w2 > w1

3. How does P2, the period of the orbit of the Planet Y, compare to P1, the period of the orbit of Planet X

P2 = (1/2) P1
P2 = P1
P2 = 2 P1
P2 = 4 P1
We do not have enough information to answer this question. We need to be given v2, the speed of the second planet in its orbit.



A block of mass M rests on a frictionless inclined plane of angle θ as shown in the diagram below. Two springs of equal length are connected to the block and to two posts as shown. The separation between the posts is equal to the sum of relaxed lengths of the springs and the length of the block.

https://online-s.physics.uiuc.edu/cgi/courses/shell/common/showme.pl?courses/phys100/fall10/hwb/10/02/springs.gif

4. The force the bottom spring exerts on the block is in the opposite direction of the force that the top spring exerts on the block.

TRUE
FALSE

5. Suppose the top (bottom) spring is stretched (compressed) from its relaxed length by an amount δx = 0.082 m. If the spring constant of the top spring is twice that of the bottom spring and M = 0.25 kg and θ = 30o, what is k, the spring constant of the bottom spring?

k = 1.66 N/m
k = 2.22 N/m
k = 3.33 N/m
k = 4.98 N/m
k = 6.44 N/m

here is what I thought, but I am not sure

1. a would be the same because the mass of the planet would not be matter, it is only the mass of the center being orbited matter.
2. weight 2 is smaller than weight 1 because gravity on planet X to the gravity of planet Y is 2, and mass remains constant, so weight 2 is also smaller than weight 1
3. my concern is "will velocity of a planet be different if the orbital radius is different"?
4. I think both forces are at the same direction by looking at the restoring force
5. I drew a free body diagram for the block, and it is F(spring bottom)+F(spring up)-Mgsin(angle)=0, is this the right equation? So the answer would be 4.98?

1. Right.

But it's actually because:

F_g = F_c

\frac{GM_{s1}m_1}{r_1\ ^2} = \frac{m_1v_1\ ^2}{r_1}

m cancels, so, the acceleration depends on M, r and v. (M

2. Right.

3. I'll use another equation for this part just so not make use of everything we have.

\frac{GMm}{r^2} = mr\omega_1\ ^2

\omega = \sqrt{\frac{GM}{r^3}}

For planet 2:

\frac{G(2M)(2m)}{(2r)^2} = (2m)(2r)\omega_2\ ^2

\omega_2 = \sqrt{\frac{GM}{4r^3}} = \frac12 \sqrt{\frac{GM}{r^3}}

So,

\omega_2 = \frac12 \omega_1

\frac{1}{T_2} = \frac{1}{2T_1}

2T_1 = T_2

4. Right.

5. Right!

Well done :)

What is T1 and T2, and how does that related to weight 1 and weight 2?

T1 and T2 are the periodic Times of planet 1 and planet 2. I just like to use the same notation where I use them. In the context of the question, T1 is the same as P1 and T2 that same as P2. :)

I still don't get question three because isn't it the speed of planet Y is not given, then how can you find out the relationship?

I just showed you that you don't need it's speed.

What is important, is that you have the force which provides the centripetal force, which is gravity.

Once you have that force, there is only one speed that the planet can have to remain in circular motion.

For this question, I didn't make use of the linear speed of the planet, but it's angular speed. (denoted by omega) which is directly related to the periodic time.

Look back at how you solved Q1 and remember that for circular motion, if you know acceleration and radius, then you know the speed.

For question 4, is it true or false?
I think it's true because the restoring force of the bottom spring pulls box down
While the restoring force of the top spring pulls box up...

Am I correct?

That's what western50 said and I agreed :)