Question

The Moon causes tides because the gravitational force it exerts differs between the side of the Earth nearer to it and the side farther from it. Find the difference in the accelerations toward the Moon of objects on the nearer and farther sides of the Earth.

Short answer

Based on the provided step-by-step solution, the difference in the accelerations towards the Moon of objects on the nearer and farther sides of the Earth is approximately \(1.108 × 10^{-6}\ m/s^2\).

Step-by-step solution

Constants and Variables

First, let's list down the constants and variables we have: 1. Mass of the Earth: \(M_E = 5.972 × 10^{24}\ kg\) 2. Mass of the Moon: \(M_M = 7.342 × 10^{22}\ kg\) 3. Mean distance between the Earth and the Moon: \(r = 3.844 × 10^8\ m\) 4. Radius of the Earth: \(R_E = 6.371 × 10^6\ m\) 5. Gravitational constant: \(G = 6.674 × 10^{-11} \ Nm^2/kg^2\) We have to find the gravitational acceleration of objects on the two sides of the Earth.

Gravitational Acceleration Formula

To find the gravitational acceleration, we use the formula: \(g = \frac{GM}{r^2}\) where \(G\) is the gravitational constant, \(M\) is the mass of one object, and \(r\) is the distance between the centers of mass of the two objects.

Gravitational Acceleration on the Nearer Side

To find the acceleration of an object at the Earth's surface nearer to the Moon, we need to use the gravitational acceleration formula for the Moon, keeping in mind the distance between the Moon and the object is \(r-R_E\): \(g_{nearer} = \frac{GM_M}{(r-R_E)^2}\)

Gravitational Acceleration on the Farther Side

And for an object at the Earth's surface farther from the Moon, the distance between the Moon and the object is \(r+R_E\): \(g_{farther} = \frac{GM_M}{(r+R_E)^2}\)

Finding the Difference in Accelerations

Now, we need to find the difference between these two accelerations: \(\Delta g = g_{nearer} - g_{farther} = \frac{GM_M}{(r-R_E)^2} - \frac{GM_M}{(r+R_E)^2}\)

Plugging in the Constants

Now, plug in the constants and evaluate the equation: \(\Delta g = \frac{6.674 × 10^{-11} \ Nm^2/kg^2 × 7.342 × 10^{22}\ kg}{(3.844 × 10^8\ m - 6.371 × 10^6\ m)^2} - \frac{6.674 × 10^{-11} \ Nm^2/kg^2 × 7.342 × 10^{22}\ kg}{(3.844 × 10^8\ m + 6.371 × 10^6\ m)^2}\) After calculating, we get: \(\Delta g \approx 1.108 × 10^{-6}\ m/s^2\) Hence, the difference in the accelerations towards the Moon of objects on the nearer and farther sides of the Earth is approximately \(1.108 × 10^{-6}\ m/s^2\).