A planet with a mass of \(7.00 \cdot 10^{21} \mathrm{~kg}\) is in a circular orbit around a star with a mass of \(2.00 \cdot 10^{30} \mathrm{~kg} .\) The planet has an orbital radius of \(3.00 \cdot 10^{10} \mathrm{~m}\). a) What is the linear orbital velocity of the planet? b) What is the period of the planet's orbit? c) What is the total mechanical energy of the planet?

Question: A planet with mass m_p orbits around a star with mass m_s. The distance between the planet and the star is r. Find the linear orbital velocity (v), the period of the planet's orbit (T), and the total mechanical energy (E) of the planet. Answer: To find the linear orbital velocity (v), the period of the planet's orbit (T), and the total mechanical energy (E) of the planet, follow these steps: 1. Find the gravitational force between the planet and the star using Newton's law of universal gravitation: \(F = G\frac{m_pm_s}{r^2}\) 2. Determine the centripetal force keeping the planet in a circular orbit: \(F_c = m_pv^2/r\) 3. Set the equations from steps 1 and 2 equal to each other and solve for v: \(v=\sqrt{\frac{Gm_s}{r}}\) 4. Calculate the period of the planet's orbit (T): \(T=\frac{2\pi r}{v}\) 5. Calculate the total mechanical energy of the planet (E): \(E=KE + PE = \frac{1}{2}m_pv^2 - G\frac{m_pm_s}{r}\) Using the given values for m_p, m_s, and r, along with the gravitational constant G, calculate the linear orbital velocity, the period of the planet's orbit, and the total mechanical energy of the planet.