A planet of mass \( 8 \times 10^{24} \mathrm{~kg} \) is at location \( <-4 \times 10^{11}, 8 \times 10^{11}, 0>\mathrm{m} \). A star of mass \( 9 \times 10^{30} \mathrm{~kg} \) is at location \( <7 \times 10^{11} \), \( -5 \times 10^{11}, 0>\mathrm{m} \). What is the force exerted on the planet by the star? (It will probably be helpful to draw a diagram, including the relevant vectors.) \( \vec{F}_{\text {on planet }}=\left(-1.07 \cdot 10^{20}, 1.27 \cdot 10^{20}, 0\right) \quad \mathrm{N} \)

To calculate the gravitational force exerted on the planet by the star, we can use Newton's law of universal gravitation, which states: \[ F = G \frac{m_1 m_2}{r^2} \] where: - \( F \) is the gravitational force, - \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \mathrm{N \cdot m^2/kg^2} \), - \( m_1 \) and \( m_2 \) are the masses of the two objects, - \( r \) is the distance between the centers of the two objects. ### Step 1: Calculate the distance \( r \) First, we need to find the distance \( r \) between the planet and the star. The positions of the planet and star are given as: - Planet: \( \vec{r}_{\text{planet}} = \langle -4 \times 10^{11}, 8 \times 10^{11}, 0 \rangle \, \mathrm{m} \) - Star: \( \vec{r}_{\text{star}} = \langle 7 \times 10^{11}, -5 \times 10^{11}, 0 \rangle \, \mathrm{m} \) The distance vector \( \vec{r} \) from the planet to the star is: \[ \vec{r} = \vec{r}_{\text{star}} - \vec{r}_{\text{planet}} = \langle 7 \times 10^{11} - (-4 \times 10^{11}), -5 \times 10^{11} - 8 \times 10^{11}, 0 - 0 \rangle \] Calculating the components: \[ \vec{r} = \langle 7 \times 10^{11} + 4 \times 10^{11}, -5 \times 10^{11} - 8 \times 10^{11}, 0 \rangle = \langle 11 \times 10^{11}, -13 \times 10^{11}, 0 \rangle \] Now, we find the magnitude of \( \vec{r} \): \[ r = \sqrt{(11 \times 10^{11})^2 + (-13 \times 10^{11})^2} \] Calculating the squares: \[ r = \sqrt{(121 \times 10^{22}) + (169 \times 10^{22})} = \sqrt{290 \times 10^{22}} = \sqrt{2.9 \times 10^{24}} \approx 1.70 \times 10^{12} \, \mathrm{m} \] ### Step 2: Calculate the gravitational force \( F \) Now we can substitute the values into the gravitational force formula. The masses are: - \( m_1 = 8 \times 10^{24} \, \mathrm{kg} \) (mass of the planet) - \( m_2 = 9 \times 10^{30} \, \mathrm{kg} \) (mass of the star) Substituting into the formula: \[ F = G \frac{m_1 m_2}{r^2} = (6.674 \times 10^{-11}) \frac{(8 \times 10^{24})(9 \times 10^{30})}{(1.70 \times 10^{12})^2} \] Calculating \( r^2 \): \[ r^2 = (1.70 \times 10^{12})^2 = 2.89 \times 10^{24} \, \mathrm{m^2} \] Now substituting back into the force equation: \[ F = (6.674 \times 10^{-11}) \frac{(8 \times 10^{24})(9 \times 10^{30})}{2.89 \times 10^{24}} \] Calculating the numerator: \[ (8 \times 10^{24})(9 \times 10^{30}) = 72 \times 10^{54} \] Now substituting: \[ F = (6.674 \times 10^{-11}) \frac{72 \times 10^{54}}{2.89 \times 10^{24}} \approx (6.674 \times 10^{-11}) \times (24.9 \times 10^{30}) \approx 1.66 \times 10^{20} \, \mathrm{N} \] ### Step 3