5) What is the gravitational potential energy (relative to infinite) of a 5.00times 10^3kg satellite that is in orbit with a radius of 9.90times 10^6 m around the Earth? (-2.0times 10^11J) 6) How much work is done against gravity in lifting the satellite in probler #5 to its orbital height? (1.11times 10^11J)

Here's how to solve these problems:<br /><br />**5) Gravitational Potential Energy**<br /><br />* **Formula:** The gravitational potential energy (U) of an object relative to infinity is given by: U = -GMm/r<br /><br /> Where:<br /> * G is the gravitational constant (6.674 x 10⁻¹¹ Nm²/kg²)<br /> * M is the mass of the Earth (5.972 x 10²⁴ kg)<br /> * m is the mass of the satellite (5.00 x 10³ kg)<br /> * r is the distance between the center of the Earth and the satellite (9.90 x 10⁶ m)<br /><br />* **Calculation:**<br /> U = -(6.674 x 10⁻¹¹ Nm²/kg²) * (5.972 x 10²⁴ kg) * (5.00 x 10³ kg) / (9.90 x 10⁶ m)<br /> U ≈ -2.02 x 10¹¹ J<br /><br />* **Answer:** The gravitational potential energy is approximately -2.0 x 10¹¹ J. The negative sign indicates that the satellite is bound to the Earth.<br /><br />**6) Work Done Against Gravity**<br /><br />* **Concept:** The work done against gravity to lift the satellite to its orbital height is equal to the *change* in gravitational potential energy. Since the initial potential energy (at the surface of the Earth) is more negative than the final potential energy (in orbit), the work done is positive.<br /><br />* **Formula:** Work (W) = ΔU = U_final - U_initial<br /><br />* **U_initial:** We need to calculate the gravitational potential energy at the Earth's surface. The radius of the Earth is approximately 6.371 x 10⁶ m.<br /><br /> U_initial = -(6.674 x 10⁻¹¹ Nm²/kg²) * (5.972 x 10²⁴ kg) * (5.00 x 10³ kg) / (6.371 x 10⁶ m)<br /> U_initial ≈ -3.13 x 10¹¹ J<br /><br />* **Calculation:**<br /> W = (-2.02 x 10¹¹ J) - (-3.13 x 10¹¹ J)<br /> W ≈ 1.11 x 10¹¹ J<br /><br />* **Answer:** The work done against gravity is approximately 1.11 x 10¹¹ J. This is the energy required to lift the satellite to that height.<br />