The acceleration due to gravity on the surface of the moon is 1.62 m/s2. The moon’s radius is RM –1738 km. (See Example 1.5.)
(a)What is the weight in newtons on the surface of the moon of an object that has a mass of 10 kg?
(b)Using the approach described in Example 1.5, determine the force exerted on the object by the gravity of the moon if the object is located 1738 km above the moon’s surface.
(a)
Write the expression for weight on the surface of the moon.
$$ W=m g_{m} $$
Here, the weight on the surface of the moon is \(W\), the mass of the object is \(m\), and the acceleration due to gravity on the surface of the moon is \(g_{m}\).
Substitute \(10 \mathrm{~kg}\) for \(m\) and \(1.62 \mathrm{~m} / \mathrm{s}^{2}\) for \(g_{m}\).
$$ \begin{aligned} W &=(10)(1.62) \\ &=16.2 \mathrm{~N} \end{aligned} $$
Thus, the weight of the object on the surface of the object is \(16.2 \mathrm{~N}\).
(b)
Write the expression for the acceleration of the object.
$$ a_{m}=\left(\frac{R_{M}}{R_{M}+H}\right)^{2} g_{m} $$
Here, the radius of the moon is \(R_{M}\), the distance of the object from the surface of the moon. Substitute \(1738 \mathrm{~km}\left|\frac{10^{3} \mathrm{~m}}{1 \mathrm{~km}}\right|\) for \(R_{M}, 1738 \mathrm{~km}\left|\frac{10^{3} \mathrm{~m}}{1 \mathrm{~km}}\right|\) for \(H\), and \(1.62 \mathrm{~m} / \mathrm{s}^{2}\) for \(g_{m}\).
$$ \begin{aligned} a_{m} &=\left(\frac{1738 \times 10^{3}}{1738 \times 10^{3}+1738 \times 10^{3}}\right)^{2}(1.62) \\ &=0.405 \mathrm{~m} / \mathrm{s}^{2} \end{aligned} $$
Write the expression for the force exerted on the object.
$$ F=m a_{m} $$
Substitute \(10 \mathrm{~kg}\) for \(m\) and \(0.405 \mathrm{~m} / \mathrm{s}^{2}\) for \(a_{m n}\).
$$ \begin{aligned} F &=(10)(0.405) \\ &=4.05 \mathrm{~N} \end{aligned} $$
Thus, the force exerted on the object by the gravity of the moon is \(4.05 \mathrm{~N}\).