The moon revolves about the Earth in a trajectory that is very nearly a circle of radius r = 384,401 km, and requires 27.3 days (23.4 x 10^5 seconds) to make a complete revolution. What is the acceleration of the moon toward the Earth?
centripetal acceleration = V^2/R
You need to express the moon's velocity V in meters per second when you do the calculation, so that the answer will be in m/s^2. The orbital radius r needs to be in meters.
r = 3.844*10^8 m
V = 2 pi r/(23.4*10^5 s)= 1.032*10^3 m/s
Now finish the calculation
To find the acceleration of the moon toward the Earth, we can use the centripetal acceleration formula:
a = (v^2) / r
Where:
a = acceleration
v = velocity
r = radius
First, we need to find the velocity of the moon using the formula:
v = (2πr) / T
Where:
T = period of revolution
Given that the period of revolution is 27.3 days, which is equal to 23.4 x 10^5 seconds:
T = 23.4 x 10^5 seconds
Now, we can calculate the velocity:
v = (2π * 384,401 km) / (23.4 x 10^5 seconds)
= (2π * 384,401 km) / (23.4 x 10^5 seconds)
≈ 1022.25 m/s
Now, we can calculate the acceleration:
a = (1022.25 m/s)^2 / 384,401 km
= (1022.25 m/s)^2 / (384,401,000 m)
≈ 2.691 x 10^-3 m/s^2
Therefore, the acceleration of the moon toward the Earth is approximately 2.691 x 10^-3 m/s^2.
To find the acceleration of the moon toward the Earth, we can use Newton's law of universal gravitation. This law states that the gravitational force between two objects is given by:
F = (G * m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant (approximately 6.67 x 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.
In this case, the moon revolves around the Earth, so m1 is the mass of the Earth and m2 is the mass of the moon. The mass of the Earth (m1) is approximately 5.972 x 10^24 kg, and the mass of the moon (m2) is approximately 7.348 x 10^22 kg.
The radius of the moon's orbit (r) is given as 384,401 km, which is equal to 3.84401 x 10^8 meters.
First, we need to calculate the gravitational force between the Earth and the moon using the equation above. Then we can find the acceleration of the moon by using Newton's second law, which states that force is equal to mass times acceleration:
F = m * a
Rearranging the equation, we have:
a = F / m
Now let's calculate the acceleration:
Step 1: Calculate the gravitational force between the Earth and the moon:
F = (G * m1 * m2) / r^2
F = (6.67 x 10^-11 N m^2/kg^2 * 5.972 x 10^24 kg * 7.348 x 10^22 kg) / (3.84401 x 10^8)^2
Step 2: Calculate the acceleration of the moon:
a = F / m2
a = F / (7.348 x 10^22 kg)
By performing these calculations, we can obtain the value of the acceleration of the moon toward the Earth.