The moon Europa, of the planet Jupiter, has an orbital period of 3.55 days and

an average distance from the center of the planet equal to 671,000 km. If the
magnitude of the gravitational acceleration at the surface of Jupiter is 2.36 times greater than that on the surface of the Earth, what is the radius of Jupiter?
(Hint: begin by calculating the rotation speed.)

radius of orbit of Europa, rm = 671000km

rotational period of Europa, T = 3.55 days
Rotational velocity of Europa, ω
= 2π/(3.55*86400) radians/second
Centripetal acceleration, a
=rmω²

radius of Jupiter, r = to be determined
Acceleration due to gravity on Jupiter
= GM/r² = 2.36g
Gravitational acceleration on Europa
=GM/(rm)²
=(GM/r²)*r²/(rm)²
=(2.36g)(r/rm)²

Equating gravitational acceleration with centripetal acceleration,
(2.36g)(r/rm)² = rmω²
r=(rm)³ω²/(2.36g)
=74,038 km

According to Google, r(Jupiter) = 71,492 km

Check my numbers and my thinking please.

To solve this problem, we can use the formula for the gravitational acceleration at the surface of a planet:

g = G * (M/r^2)

Where:
g is the gravitational acceleration
G is the gravitational constant (6.6743 × 10^-11 N m^2/kg^2)
M is the mass of the planet
r is the radius of the planet

We are given that the magnitude of the gravitational acceleration on Jupiter is 2.36 times greater than that on Earth. So we can say:

g_jupiter = 2.36 * g_earth

Now, let's calculate the radius of Jupiter step-by-step:

Step 1: Calculate the radius of the Earth
Given:
g_earth = 9.8 m/s^2 (approximate value)

Using the formula for the gravitational acceleration, we can rearrange it to solve for r:

r_earth = sqrt(G * M_earth / g_earth)

Let's assume that M_earth is the mass of the Earth, which is approximately 5.972 × 10^24 kg.

Plugging in the values, we get:

r_earth = sqrt((6.6743 × 10^-11 N m^2/kg^2) * (5.972 × 10^24 kg) / (9.8 m/s^2))

r_earth ≈ 6.371 × 10^6 m

Step 2: Calculate the rotation speed of Europa
Given:
T_europa = 3.55 days

The rotational speed (v) of a satellite can be calculated using the formula:

v = (2πr)/T

Where:
v is the rotational speed
r is the average distance from the center of the planet (671,000 km)
T is the orbital period (converted to seconds)

Converting the orbital period to seconds:

T_europa = 3.55 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute

T_europa ≈ 306,720 seconds

Plugging in the values, we get:

v_europa = (2π * 671,000,000 m) / 306,720 s

v_europa ≈ 1,732.09 m/s

Step 3: Calculate the radius of Jupiter
Given:
g_jupiter = 2.36 * g_earth
v_europa = (2πr_jupiter) / T

Using the formula for rotational speed, we can solve for r_jupiter:

r_jupiter = (v_europa * T) / (2π)

Plugging in the values, we get:

r_jupiter = (1,732.09 m/s) * (306,720 s) / (2π)

r_jupiter ≈ 2,914,309 m

Therefore, the radius of Jupiter is approximately 2,914,309 meters.

To find the radius of Jupiter, we need to calculate the rotation speed first. The rotation speed can be calculated using the formula:

Speed = Distance / Time

Given that the orbital period of Europa is 3.55 days and the average distance from the center of Jupiter is 671,000 km, we can calculate the rotation speed as follows:

Rotation Speed = (2 * π * Distance) / Time

Plugging in the values:

Rotation Speed = (2 * 3.14 * 671,000 km) / 3.55 days

To simplify the calculation, we need to convert the units. Let's convert km to meters and days to seconds:

Rotation Speed = (2 * 3.14 * 671,000 * 1,000) / (3.55 * 24 * 60 * 60 seconds)

Simplifying further:

Rotation Speed = (2 * 3.14 * 671,000,000) / (3.55 * 86,400 seconds)

Now, we can calculate the rotation speed of Europa.

Rotation Speed ≈ 12,642 m/s

Next, let's calculate the gravitational acceleration at the surface of Jupiter. We are given that it is 2.36 times greater than that on the surface of the Earth. The gravitational acceleration on Earth is approximately 9.8 m/s^2, so the acceleration on Jupiter can be calculated as:

Gravitational Acceleration on Jupiter = 2.36 * 9.8 m/s^2

Gravitational Acceleration on Jupiter ≈ 23.128 m/s^2

Finally, we can use the equation for the centripetal force to find the radius of Jupiter:

Centripetal Force = (Rotation Speed)^2 / Radius

Rearranging the equation:

Radius = (Rotation Speed)^2 / Centripetal Force

Plugging in the values:

Radius = (12,642 m/s)^2 / 23.128 m/s^2

Simplifying the calculation:

Radius ≈ 6894 km

Therefore, the radius of Jupiter is approximately 6894 km.