Question

1. (5pts) the weight w of an object of mass m on the surface of the earth is given by (w_e = m(\frac{gm_e}{r_e^2})) where g is a constant of value (6.67\times10^{-11} m^3kg^{-1}s^{-2}), (r_e) is the radius of the earth, and (m_e) is the mass of the earth. if the mass is placed on the surface of a planet of mass (18m_e) with a radius of (r_e/3), what would its weight on the new planet be? give your answer in terms of its weight on earth.

Explanation:

Step1: Write the weight formula for Earth

The weight of an object of mass $m$ on Earth is $W_E=\frac{GM_Em}{R_E^2}$.

Step2: Write the weight formula for the new planet

Let the weight on the new - planet be $W_{new}$. The mass of the new planet $M_{new} = 18M_E$ and the radius of the new planet $R_{new}=\frac{R_E}{3}$. Using the formula $W=\frac{GMm}{R^2}$, we have $W_{new}=\frac{G\times(18M_E)m}{(\frac{R_E}{3})^2}$.

Step3: Simplify the weight formula for the new planet

\[

$$\begin{align*} W_{new}&=\frac{G\times18M_E m}{\frac{R_E^2}{9}}\\ &=18\times9\times\frac{GM_Em}{R_E^2} \end{align*}$$

\]
Since $W_E=\frac{GM_Em}{R_E^2}$, then $W_{new}=162W_E$.

Answer:

The weight of the object on the new planet is 162 times its weight on Earth.