(a) Calculate the magnitude of the gravitational force exerted by Venus on a 75 kg human standing on the surface of Venus. (The mass of Venus is \( 4.9 \times 10^{24} \mathrm{~kg} \) and its radius is \( 6.1 \times 10^{6} \mathrm{~m} \).) 66.5 (b) Calculate the magnitude of the gravitational force exerted by the human on Venus. 66.5 (c) For comparison, calculate the approximate magnitude of the gravitational force of this human on a similar human who is standing 2.5 meters away. 5.996e-11 × N (d) What approximations or simplifying assumptions must you make in these calculations? (Note: Some of these choices are false because they are wrong physics!) Treat Venus as though it were spherically symmetric. \( \checkmark \) Ignore the effects of the Sun, which alters the gravitational force that one object exerts on another. \( \checkmark \) Use the same gravitational constant in (a) and (b) despite its dependence on the size of the masses. Treat the humans as though they were points or uniform-density spheres.

Let's go through each part of the problem step by step. ### (a) Calculate the gravitational force exerted by Venus on a 75 kg human. The gravitational force \( F \) can be calculated using Newton's law of universal gravitation: \[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \] Where: - \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \) - \( m_1 \) is the mass of Venus, \( 4.9 \times 10^{24} \, \text{kg} \) - \( m_2 \) is the mass of the human, \( 75 \, \text{kg} \) - \( r \) is the radius of Venus, \( 6.1 \times 10^{6} \, \text{m} \) Substituting the values into the formula: \[ F = \frac{(6.674 \times 10^{-11}) \cdot (4.9 \times 10^{24}) \cdot (75)}{(6.1 \times 10^{6})^2} \] Calculating the denominator: \[ (6.1 \times 10^{6})^2 = 3.721 \times 10^{13} \, \text{m}^2 \] Now substituting back into the equation: \[ F = \frac{(6.674 \times 10^{-11}) \cdot (4.9 \times 10^{24}) \cdot (75)}{3.721 \times 10^{13}} \] Calculating the numerator: \[ (6.674 \times 10^{-11}) \cdot (4.9 \times 10^{24}) \cdot (75) \approx 2.465 \times 10^{16} \] Now, calculating \( F \): \[ F \approx \frac{2.465 \times 10^{16}}{3.721 \times 10^{13}} \approx 66.5 \, \text{N} \] ### (b) Calculate the gravitational force exerted by the human on Venus. By Newton's third law, the gravitational force exerted by the human on Venus is equal in magnitude and opposite in direction to the force exerted by Venus on the human. Therefore, the gravitational force exerted by the human on Venus is also: \[ F \approx 66.5 \, \text{N} \] ### (c) Calculate the gravitational force of this human on a similar human standing 2.5 meters away. Using the same formula for gravitational force: \[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \] Where: - \( m_1 = 75 \, \text{kg} \) - \( m_2 = 75 \, \text{kg} \) - \( r = 2.5 \, \text{m} \) Substituting the values: \[ F = \frac{(6.674 \times 10^{-11}) \cdot (75) \cdot (75)}{(2.5)^2} \] Calculating the denominator: \[ (2.5)^2 = 6.25 \] Now substituting back into the equation: \[ F = \frac{(6.674 \times 10^{-11}) \cdot (75) \cdot (75)}{6.25} \] Calculating the numerator: \[ (6.674 \times 10^{-11}) \cdot (75) \cdot (75) = 3.75075 \times 10^{-8} \] Now calculating \( F \): \[ F = \frac{3.75075 \times 10^{-8}}{6.25} \approx 6.000 \times 10^{-9} \, \text{N} \] So, the approximate magnitude of the gravitational force of this human on another human standing 2.5 meters away is: \[ F \approx 5.996 \times 10^{-11} \, \text{N} \] ### (d) What approximations or simplifying assumptions must you make in these calculations? 1. **Treat Venus as though it were spherically symmetric.** \( \checkmark \) 2. **Ignore the effects of the Sun, which alters the gravitational force that one object exerts on another.** \( \