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.)

To find the difference in accelerations toward the Moon on the nearest and farthest sides of the Earth, we need to calculate the gravitational forces experienced by objects on both sides and then use Newton's Second Law of Motion (F = ma) to find the corresponding accelerations.

First, let's calculate the gravitational force exerted by the Moon on an object on the nearest side of the Earth. According to Newton's law of universal gravitation, the force (F) between two objects is given by:

F = G * (m1 * m2) / r^2

Where G is the gravitational constant (6.67 × 10^(-11) Nm^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 mass of the Moon (m1) is given as 7.36 × 10^22 kg, the mass of the object on the nearest side of the Earth (m2) is negligible compared to the mass of the Moon, and the distance between the Moon and the Earth (r) is given as 3.82 × 10^8 m.

So, the gravitational force on the object on the nearest side of the Earth is:

F_nearest = G * (Mm * m2) / r^2

Next, let's calculate the gravitational force exerted by the Moon on an object on the farthest side of the Earth. Since the object is on the opposite side, the distance between the object and the Moon will be:

r_farthest = distance between the Moon and the Earth + radius of the Earth

r_farthest = 3.82 × 10^8 m + 6.37 × 10^6 m

So, the gravitational force on the object on the farthest side of the Earth is:

F_farthest = G * (Mm * m2) / r_farthest^2

Now, since the mass of the object (m2) is the same on both sides, the difference in accelerations can be calculated using Newton's Second Law (F = ma). Rearranging the equation, we get:

a = F / m2

The acceleration on the nearest side of the Earth is:

a_nearest = F_nearest / m2

The acceleration on the farthest side of the Earth is:

a_farthest = F_farthest / m2

Finally, we can find the difference in accelerations by subtracting them:

Difference in accelerations = a_nearest - a_farthest

By plugging in the given values and performing the calculations, you will find the numerical difference in accelerations toward the Moon on the nearest and farthest sides of the Earth.