Based on the following data about planet X (which orbits around the Sun):

Planet X's distance from Sun = 3.6*1012 m
Planet X's radius = 2*106 m
Planet X's mass = 8.2*1022 kg

a.) Find gx, the size of the acceleration due to gravity on the surface of Planet X. m/s2

b.) What is the weight of a 10 kg mass on the surface of Planet X? N
(How does this compare to its weight on Earth?)

c.) How long would it take for a ball dropped from a height of 8 m to hit the ground? s
(How does this compare to the time it would take on Earth?)

d.) At 3 of Planet X's radii above the planet's surface, what is gx? m/s2

e.) Find the orbital speed of Planet X around the Sun. m/s

f.) How long is a year on Planet X? Express your answers in both seconds and Earth years:
s
Earth years

I will be glad to critique your work. We are not a homework performing service. Here are some comments to get you started.

a) gx = GM/R^2, where R is the planet's radius. G was explained in one of my previous answers.

You won't need to know the distance from the sun until you get to e) and f)

a.) To find the size of the acceleration due to gravity on the surface of Planet X (gx), we can use the formula:

gx = (G * M) / r^2

where G is the gravitational constant (approximately 6.674 × 10^-11 N(m/kg)^2), M is the mass of Planet X, and r is the radius of Planet X.

Plugging in the values:
G = 6.674 × 10^-11 N(m/kg)^2
M = 8.2 * 10^22 kg
r = 2 * 10^6 m

gx = (6.674 × 10^-11 N(m/kg)^2 * 8.2 * 10^22 kg) / (2 * 10^6 m)^2

gx ≈ 10 m/s^2

b.) To find the weight of a 10 kg mass on the surface of Planet X, we can use the formula:

Weight = mass * gx

Plugging in the values:
mass = 10 kg
gx = 10 m/s^2 (from part a)

Weight = 10 kg * 10 m/s^2

Weight = 100 N

The weight of a 10 kg mass on the surface of Planet X is 100 N.
This is the same as its weight on Earth where the acceleration due to gravity is approximately 9.8 m/s^2.

c.) To find how long it would take for a ball dropped from a height of 8 m to hit the ground on Planet X, we can use the formula:

time = sqrt((2 * height) / gx)

Plugging in the values:
height = 8 m
gx = 10 m/s^2 (from part a)

time = sqrt((2 * 8 m) / 10 m/s^2)

time ≈ 1.264 s

On Earth, it takes approximately 1.264 s for a ball dropped from a height of 8 m to hit the ground.

d.) At 3 of Planet X's radii (3 * 2 * 10^6 m) above the planet's surface, the new distance (d) from the center of Planet X is:

d = r + 3 * r = 4 * r

Plugging in the values:
r = 2 * 10^6 m

d = 4 * 2 * 10^6 m

d = 8 * 10^6 m

To find gx at this new distance, we use the same formula:

gx = (G * M) / d^2

Plugging in the values:
G = 6.674 × 10^-11 N(m/kg)^2
M = 8.2 * 10^22 kg
d = 8 * 10^6 m

gx = (6.674 × 10^-11 N(m/kg)^2 * 8.2 * 10^22 kg) / (8 * 10^6 m)^2

gx ≈ 1.157 m/s^2

At 3 of Planet X's radii above the planet's surface, gx is approximately 1.157 m/s^2.

e.) To find the orbital speed of Planet X around the Sun, we can use the formula:

orbital speed = sqrt((G * M) / r)

Plugging in the values:
G = 6.674 × 10^-11 N(m/kg)^2
M = 8.2 * 10^22 kg
r = 3.6 * 10^12 m

orbital speed = sqrt((6.674 × 10^-11 N(m/kg)^2 * 8.2 * 10^22 kg) / (3.6 * 10^12 m))

orbital speed ≈ 3577 m/s

The orbital speed of Planet X around the Sun is approximately 3577 m/s.

f.) The length of a year on Planet X can be calculated using the formula:

year = (2 * pi * r) / orbital speed

Plugging in the values:
r = 3.6 * 10^12 m
orbital speed = 3577 m/s (from part e)

year = (2 * pi * 3.6 * 10^12 m) / 3577 m/s

year ≈ 6.3565 × 10^8 s ≈ 20.14 Earth years (1 Earth year ≈ 3.154 × 10^7 s)

Therefore, a year on Planet X is approximately 6.3565 × 10^8 seconds or 20.14 Earth years.