A synchronous satellite is placed in a circular orbit above Earth, at an altitude of 36 400 km. Find the centripetal acceleration of the satellite.

Use significant digits and units

To find the centripetal acceleration of the satellite, we need to know the radius of the circular orbit. The altitude of the satellite is given as 36,400 km. However, this is the distance from the Earth's surface to the satellite. In order to find the radius of the orbit, we need to add the radius of the Earth to the given altitude.

The radius of the Earth is approximately 6,371 km. So the radius of the orbit can be calculated as:

Radius of orbit = radius of Earth + altitude
Radius of orbit = 6,371 km + 36,400 km
Radius of orbit = 42,771 km

Now that we have the radius of the orbit, we can calculate the centripetal acceleration of the satellite. The centripetal acceleration is given by the formula:

Centripetal acceleration = (velocity^2) / radius

In this case, the satellite is in a synchronous orbit, which means its orbital period is equal to the Earth's rotation period of 24 hours. The velocity of the satellite can be calculated using the relation:

Velocity = (2 * π * radius) / period

Using a period of 24 hours, the velocity can be calculated as:

Velocity = (2 * π * 42,771 km) / 24 hours

To convert the velocity to m/s, we need to convert km to m and hours to seconds using conversion factors:

1 km = 1000 m
1 hour = 3600 seconds

Velocity = (2 * π * 42,771,000 m) / (24 * 3600 seconds)

Now that we have the velocity, we can substitute the values into the centripetal acceleration formula:

Centripetal acceleration = (velocity^2) / radius
Centripetal acceleration = ((2 * π * 42,771,000 m) / (24 * 3600 seconds))^2 / 42,771,000 m

Calculating this equation gives the answer for the centripetal acceleration of the satellite in units of m/s^2 and with the appropriate significant digits.