15. How much work must be done to lift an artificial satellite of mass \( m \) from the surface of Earth and put it in a circular orbit with a radius equal to twice the Earth's Radius?

To lift an artificial satellite to a circular orbit with a radius equal to twice Earth's radius, you need to consider both the gravitational potential energy changes involved. The work done to lift the satellite to that height involves overcoming Earth's gravity. The gravitational potential energy of the satellite at the surface and at the desired height can be computed using the formula \( U = -\frac{GMm}{r} \). By calculating the difference in potential energy between the two positions and applying the values for Earth's mass and radius, you can find the work required. It's a fun exercise in proper application of gravitational physics! Once you've mastered this concept, try looking at the engineering principles behind actually launching satellites. The balance between energy expenditure, fuel capacity, and trajectory calculations can make or break a space mission. By analyzing successful launches, you can appreciate how theoretical physics merges with real-world achievements, propelling humans to the stars!