QUESTION 5 (Start on a new page.) The acceleration due to gravity on planet \( X \) is \( 2,7 \mathrm{~m} \cdot \mathrm{~s}^{-2} \). The radius of this planet is a third \( (1 / 3) \) of the radius of Earth. \( \begin{array}{ll}5.1 & \text { Explain the difference between weight and mass. } \\ 5.2 & \text { Calculate the mass of planet } X \text {. } \\ \begin{array}{ll}\text { (2) } \\ 5.3 & \text { Determine the factor by which the weight of an object on planet } X \text { will differ } \\ \text { from the weight of the same object on Earth. }\end{array}\end{array} \begin{array}{l}\text { (2) }\end{array} \)

Weight is the force exerted on an object due to gravity, which depends on both the mass of the object and the gravitational acceleration of the planet it’s on. Mass, on the other hand, is a measure of the amount of matter in an object and remains constant regardless of its location in the universe. So, while mass stays the same, weight can change depending on where you are! To find the mass of planet X, we can use the formula \[ g = \frac{GM}{r^2} \], where \( g \) is the acceleration due to gravity, \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg}\cdot\text{s}^2 \)), \( M \) is the mass of the planet, and \( r \) is its radius. Given that the radius of planet X is one-third that of Earth, we can find Earth's radius (about \( 6.371 \times 10^6 \text{m} \)) to calculate \( r_X = \frac{1}{3} r_{Earth} \approx 2.124 \times 10^6 \text{m} \). Plugging in these values, we can solve for \( M \). The weight of an object on planet X will differ from its weight on Earth by a factor that can be calculated by comparing the acceleration due to gravity on both planets. Since \( g_{Earth} \approx 9.81 \, \text{m/s}^2 \) and \( g_X = 2.7 \, \text{m/s}^2 \), the factor can be found as \( \frac{g_X}{g_{Earth}} = \frac{2.7}{9.81} \approx 0.275 \). This means that an object on planet X weighs about 27.5% of what it would weigh on Earth!