The Moon causes tides because the gravitational force it exerts differs between the side of the Earth nearest the Moon and that farthest from the Moon. Find the difference in the accelerations toward the Moon of objects on the nearest and farthest sides of the Earth. (The mass of the Moon is Mm = 7.36 1022 kg, the radius of the Earth is RE = 6.37 106 m and the distance between the Earth and the Moon is d = 3.82 108 m.)
Moon orbit radius r = 3.84•10^8 m
Earth radius R = 6.37•10^6 m
F1 = G •m₁•m₂/(r+R)²
m₁ = mass of the object
m₂ = moon's mass
r+R = 3.84•10^8 + (m)
far side
F2 = G• m₁•m₂/(r-R)²
m₁ = mass of the oject
m₂ = moon's mass
r-R = 3.84•10^8 - 6.37•10^6 (m)
∆F = G• m₁•m₂/(r-R)² – G• m₁•m₂/(r+R)²
∆F = G• m₁•m₂[ 1/(r-R)² – 1/(r+R)² ]
∆F/m₁ = ∆a = G•m₂[ 1/(r-R)² – 1/(r+R)² ]
To calculate the difference in accelerations toward the Moon on the nearest and farthest sides of the Earth, we can use Newton's law of universal gravitation and the concept of centripetal acceleration.
Step 1: Calculate the gravitational force on the nearest and farthest sides of the Earth.
We can use the formula for the gravitational force:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (6.674 × 10^-11 N(m/kg)^2)
m1 and m2 are the masses of the two objects (in this case, the Moon and the Earth)
r is the distance between the centers of the two objects
For the nearest side of the Earth:
F_nearest = G * (Mm * ME) / (r_nearest)^2
where Mm is the mass of the Moon, ME is the mass of the Earth, and r_nearest is the distance between the Earth's center and the nearest side.
For the farthest side of the Earth:
F_farthest = G * (Mm * ME) / (r_farthest)^2
where r_farthest is the distance between the Earth's center and the farthest side.
Step 2: Calculate the acceleration on the nearest and farthest sides of the Earth.
We can use the formula for centripetal acceleration:
a = v^2 / r
where:
a is the centripetal acceleration
v is the velocity of the object
r is the distance from the center of the object to the axis of rotation
For an object in orbit around the Earth, the centripetal acceleration is provided by the gravitational force:
a = F / m
where m is the mass of the object.
For the nearest side of the Earth:
a_nearest = F_nearest / ME
For the farthest side of the Earth:
a_farthest = F_farthest / ME
Step 3: Calculate the difference in accelerations.
Difference = a_nearest - a_farthest
Let's plug in the values:
- Gravitational constant (G) = 6.674 × 10^-11 N(m/kg)^2
- Mass of the Moon (Mm) = 7.36 × 10^22 kg
- Mass of the Earth (ME) = 5.972 × 10^24 kg
- Distance between the Earth and the Moon (d) = 3.82 × 10^8 m
- Radius of the Earth (RE) = 6.37 × 10^6 m
r_nearest = RE + d
r_farthest = RE + d
After substituting the values and solving the equations, you should be able to find the difference in accelerations between the nearest and farthest sides of the Earth.