7. The Moon, which revolves around Earth with a period of
about 27.3 d in a nearly circular orbit, has a centripetal
acceleration of magnitude 2.7 103 m/s2. What is the
average distance from Earth to the Moon?
To solve this problem, we can use the formula for centripetal acceleration:
a = v^2 / r
Where:
a = centripetal acceleration (2.7 * 10^3 m/s^2)
v = velocity of the Moon
r = distance from Earth to the Moon
The velocity of the Moon can be calculated using the formula for the speed of an object in circular motion:
v = 2 * π * r / T
Where:
v = velocity
r = distance from Earth to the Moon
T = period of rotation (27.3 days = 2358720 seconds)
Substitute the expression for v into the equation for centripetal acceleration:
2.7 * 10^3 m/s^2 = (2 * π * r / 2358720)^2 / r
Simplify the equation:
2.7 * 10^3 = (4 * π^2 * r) / (2358720^2)
Multiply both sides by 2358720^2:
2.7 * 10^3 * 2358720^2 = 4 * π^2 * r
r = (2.7 * 10^3 * 2358720^2) / (4 * π^2)
r = 384,400 km
Therefore, the average distance from Earth to the Moon is approximately 384,400 km.