Question

A 58kg satellite orbits the earth at 27000 km/h in a circular orbit. If the force attracting it to the earth is found to be 493 N at what altitude is it orbiting? (Mean radius of earth is 6367 km).

Answer 1

The satellite is orbiting at an altitude of approximately 1,079 km.

Step-by-step explanation:

In order to determine the altitude at which the satellite is orbiting, we can utilize the gravitational force equation and centripetal force equation.

Firstly, the centripetal force acting on the satellite is given by
`\( F_c = \frac{{m \cdot v^2}}{{r}} \), where \( m \)` is the mass of the satellite,
`\( v \)` is its orbital velocity, and
`\( r \)` is the orbital radius. Rearranging this equation to solve for
`\( r \),` we get
`\( r = \frac{{m \cdot v^2}}{{F_c}} \).`

Given the force
`\( F_c = 493 \, N \)`, the mass
`\( m = 58 \, kg \)`, and the velocity
`\( v = 27000 \, km/h \),` we need to convert the velocity to m/s by multiplying by
`\( \frac{{1000}}{{3600}} \) to get \( v = 7500 \, m/s \).`

Substituting these values into the formula, we get
`\( r = \frac{{58 \, kg \cdot (7500 \, m/s)^2}}{{493 \, N}} \). Solving this gives \( r \approx 1,079,305 \, m \).`

However, this value represents the total distance from the center of the Earth to the satellite. To find the altitude above the Earth's surface, we subtract the Earth's mean radius
`(\( R = 6367 \, km \) or \( 6,367,000 \, m \)): \( \text{{Altitude}} = r - R \), yielding \( \text{{Altitude}} \approx 1,079,305 \, m - 6,367,000 \, m \),` resulting in an altitude of approximately 1,079 km.

Answer 2

The satellite is orbiting at an altitude of 449 kilometers above the Earth's surface.

Step-by-step explanation:

To determine the altitude of the satellite, we'll use the formula for gravitational force and centripetal force. The gravitational force acting on the satellite is given by Newton's law of universal gravitation:

`\[F_{\text{gravity}} = \frac{{G \cdot m_{\text{satellite}} \cdot m_{\text{Earth}}}}{{r^2}}\]`

Given
`\(F_{\text{gravity}} = 493 \, \text{N}\)` and the mass of the Earth, we can rearrange this equation to find the distance from the center of the Earth (r).

The centripetal force keeping the satellite in orbit is given by:

`\[F_{\text{centripetal}} = \frac{{m_{\text{satellite}} \cdot v^2}}{{r}}\]`

Here, v = 27000 ,km/h. We'll convert this to meters per second (m/s) for consistency in units.

Next, we equate the gravitational force to the centripetal force to find the altitude above the Earth's surface. Solving for r, we get:

`\[r = \sqrt[3]{\frac{{G \cdot m_{\text{Earth}}}}{{\left(\frac{{v^2}}{{r}}\right)^2}}}\]`

After substituting the known values and solving the equation, we find the altitude
`(\(r - \text{radius of Earth}\))` to be 449 kilometers.

Hence, the satellite orbits at an altitude of 449 kilometers above the Earth's surface. This altitude enables the gravitational force and centripetal force to balance, maintaining the circular orbit at the given speed.