Find the period time of the moon from the following values,
R=6400km,the distance between the earth and the moon=384000km,g=9.8m/s2
First calculate the speed of the moon in its orbit, V.
V^2/D = g*(R/D)^2
(the value of g at the moon's location)
V^2 = g*R^2/D = 1.045*10^6 m^2/s^2
V = 1022 m/s
V = 2 pi D/P
P (the period) = 2 pi D/V = 2.361*10^6 s
= 27.3 days
Note that D and R had to be in meters, not km, for these calculations
If you need a single equation:
P = 2*pi*D/V = 2*pi*D*sqrt[D/gR^2]
= 2*pi*D^3/2/[R*g^1/2)]
g of earth at moon = 9.8 (6400/384000)^2
=.00272 m/s^2
so
.00272 = w^2 r = w^2 (3.84*10^8) (note r in meters)
w^2 = 7.1 * 10^-4 * 10^-8
w = 2.66 * 10^-6
w = 2 pi/T
so
T = 2.35 * 10^6 seconds
/3600 s/hr /24 hr/day = 27.3 days
To find the period time of the moon, we need to use the formula:
T = 2π√(R³/gM)
Where:
T is the period time of the moon.
R is the radius of the Earth (6400 km or 6,400,000 meters).
g is the acceleration due to gravity (9.8 m/s²).
M is the mass of the Earth (5.97 x 10^24 kg).
We know the radius of the Earth (R = 6,400,000 meters) and the value of g (9.8 m/s²). We can find the mass of the Earth (M) using the formula:
g = (G * M) / R²
Where:
G is the gravitational constant (approximately 6.674 × 10^-11 N m²/kg²).
Rearranging the formula, we have:
M = (g * R²) / G
Let's calculate the mass of the Earth:
M = (9.8 * (6,400,000)²) / (6.674 × 10^-11)
M ≈ 5.972 × 10^24 kg
Now, we can substitute the values of R, g, and M into the period formula:
T = 2π√((6,400,000)³ / (9.8 * (5.972 x 10^24)))
Calculating the expression inside the square root:
(6,400,000)³ / (9.8 * (5.972 x 10^24)) ≈ 7.387 x 10^8
T = 2π√(7.387 x 10^8)
T ≈ 2π * 2.718 ≈ 17.09 days
Therefore, the period time of the moon is approximately 17.09 days.