What is the circular velocity of an Earth satellite 1000. km above Earth's surface? (Note: Earth's average radius is 6371 km.)
To find the circular velocity of an Earth satellite 1000 km above Earth's surface, we can use the concept of centripetal force.
Centripetal force is the force that keeps an object moving in a circular path. In the case of a satellite orbiting the Earth, the centripetal force is provided by the gravitational force between the Earth and the satellite.
The formula for centripetal force is:
F = m*v^2 / r
Where F is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the satellite's circular orbit.
In this case, since the satellite's orbit is circular and we want to find the circular velocity, we can assume that the satellite is in uniform circular motion. This means that the gravitational force acting on the satellite provides the necessary centripetal force.
So we can set the gravitational force equal to the centripetal force:
F_grav = F_cen
G * (m_earth * m_sat) / r^2 = m_sat * v^2 / r
Where G is the gravitational constant, m_earth is the mass of the Earth, m_sat is the mass of the satellite, and r is the radius of the satellite's orbit.
Since the mass of the satellite cancels out on both sides of the equation, we can solve for v:
G * m_earth / r^2 = v^2
Taking the square root of both sides gives:
v = sqrt(G * m_earth / r)
Now we can plug in the values:
G = 6.67 * 10^-11 N(m/kg)^2 (the gravitational constant)
m_earth = 5.97 * 10^24 kg (the mass of the Earth)
r = 1000 km + 6371 km = 7371 km (the radius of the satellite's orbit)
Converting the values to SI units:
G = 6.67 * 10^-11 N(m/kg)^2
m_earth = 5.97 * 10^24 kg
r = 7371000 m
Plugging these values into the formula:
v = sqrt((6.67 * 10^-11 N(m/kg)^2 * 5.97 * 10^24 kg) / 7371000 m)
Calculating this expression gives the circular velocity of the satellite 1000 km above Earth's surface.
To calculate the circular velocity of an Earth satellite 1000 km above Earth's surface, we can use the formula for orbital velocity.
The formula for orbital velocity is:
v = √(GM/r)
Where:
v = orbital velocity (circular velocity)
G = gravitational constant (6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2)
M = mass of the Earth (5.972 × 10^24 kg)
r = distance from the center of the Earth to the satellite (6371 km + 1000 km = 7371 km)
Let's plug in the values and calculate:
v = √((6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2) * (5.972 × 10^24 kg) / (7371 km))
Convert the distance "7371 km" to meters by multiplying by 1000: r = 7371 km * 1000 = 7,371,000 meters
v = √((6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2) * (5.972 × 10^24 kg) / (7,371,000 m))
Calculating the equation, we get:
v ≈ 7,660 m/s
Therefore, the circular velocity of an Earth satellite 1000 km above Earth's surface is approximately 7,660 m/s.