By observing that the centripetal acceleration of the Moon around the Earth is \( a_c = 2.7 \times 10^{-3} \text{m/s}^2 \), we can determine the gravitational constant \( G \) in cubic meters per kilogram per square second. Assuming the Earth has a mass of \( M_E = 5.99 \times 10^{24} \text{kg} \), and the mean distance between the centers of the Earth and Moon is \( r_m = 3.88 \times 10^8 \text{m} \), calculate \( G \).
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To calculate the gravitational constant \( G \) using the given values of the centripetal acceleration of the Moon, Earth's mass, and the mean distance between the Earth and the Moon, we will use Newton's law of universal gravitation and the equation for centripetal acceleration.
Newton's law of universal gravitation states that the force of gravity \( F_g \) between two masses \( M_1 \) and \( M_2 \) separated by a distance \( r \) is given by \[ F_g = G \cdot \frac{M_1 M_2}{r^2} \tag{1} \]
The centripetal force that keeps the Moon in orbit is due to this gravitational attraction, and the equation for centripetal force \( F_c \) is \[ F_c = \frac{m v^2}{r} = m \cdot a_c \tag{2} \]
where - \( m \) is the mass of the orbiting object (the Moon in this case), - \( v \) is the orbital velocity of the object, - \( r \) is the radius of the circular path (the distance between the Earth and Moon), - \( a_c \) is the centripetal acceleration.
In a stable orbit, the gravitational force is equal to the centripetal force needed to keep the Moon in its orbit; therefore \[ F_g = F_c \] \[ G \cdot \frac{M_E m}{r_m^2} = m \cdot a_c \] \[ G \cdot \frac{M_E}{r_m^2} = a_c \]
We can solve for \( G \) by rearranging this equation: \[ G = \frac{a_c r_m^2}{M_E} \] Plugging in the values given: \[ G = \frac{2.7 \times 10^{-3} \text{m/s}^2 \cdot (3.88 \times 10^8 \text{m})^2}{5.99 \times 10^{24} \text{kg}} \] \[ G = \frac{2.7 \times 10^{-3} \text{m/s}^2 \cdot 1.50464 \times 10^{17} \text{m}^2}{5.99 \times 10^{24} \text{kg}} \] \[ G \approx \frac{4.063128 \times 10^{14} \text{m}^3/\text{s}^2}{5.99 \times 10^{24} \text{kg}} \] \[ G \approx 6.78 \times 10^{-11} \text{m}^3/\text{kg} \cdot \text{s}^2 \]
The calculated value of \( G \) is approximately \( 6.78 \times 10^{-11} \text{m}^3/\text{kg} \cdot \text{s}^2 \).