Section 3.9: Advanced Problems
This section combines all circular motion and gravitation concepts into challenging multi-step problems, including satellites, planetary orbits, and tension forces in curved paths.
Tips for Solving Advanced Problems:
- Always identify forces acting along radial and tangential directions.
- Use centripetal acceleration formulas: \( a_c = \frac{v^2}{r} \).
- Apply Newton’s law of gravitation: \( F = G\frac{m_1 m_2}{r^2} \).
- Check units and convert distances to meters when necessary.
Example 1 — Satellite Speed and Tension
A satellite of mass 500 kg orbits 3 × 10^7 m from Earth's center. Calculate its orbital speed and the gravitational force acting on it.
Orbital speed: \( v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{6.674\times10^{-11}\cdot5.97\times10^{24}}{3\times10^7}} \approx 3.65 \times 10^3 \text{ m/s} \)
Gravitational force: \( F = \frac{GMm}{r^2} = \frac{6.674\times10^{-11}\cdot5.97\times10^{24}\cdot500}{(3\times10^7)^2} \approx 2.21 \times 10^3 \text{ N} \)
Practice Problems
- A satellite doubles its orbit radius. How does its speed change?
- Calculate the period of a satellite in a 10^7 m orbit.
- Determine tension in a string when a ball moves in vertical circle of radius 2 m at top with speed 4 m/s.
- A satellite in low Earth orbit has mass 1000 kg. Find centripetal acceleration if orbit radius is 6.7 × 10^6 m.
- Compute orbital speed for a geostationary satellite.