Some people believe that the Moon controls their activities. The Moon moves from being directly on the opposite side of Earth from you to be being directly overhead. Assume that the Earth-Moon (center-to-center) distance is 3.82 multiplied by 108 m and Earth's radius is 6.37 multiplied by 106.
(a) By what percent does the Moon's gravitational pull on you increase?
(b) By what percent does your weight (as measured on a scale) decrease?
To answer these questions, we can apply the concept of gravitational force.
(a) To calculate the percent increase in the Moon's gravitational pull on you, we need to compare the gravitational force at the point when the Moon is directly opposite to when it is directly overhead.
The gravitational force between two objects is given by the equation:
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 the centers of the two objects.
In this case, we can assume that your mass remains constant, so we only need to compare the change in distance.
When the Moon is directly opposite, the distance between the centers of the Earth and the Moon is: r1 = 3.82 x 10^8 m + 6.37 x 10^6 m (radius of the Earth)
When the Moon is directly overhead, the distance between the centers of the Earth and the Moon is: r2 = 3.82 x 10^8 m - 6.37 x 10^6 m (radius of the Earth)
To calculate the percent increase, we can use the following formula:
Percent increase = (r2 - r1) / r1 * 100
Calculating this, we find:
Percent increase = [(3.82 x 10^8 - 6.37 x 10^6) - (3.82 x 10^8 + 6.37 x 10^6)] / (3.82 x 10^8 + 6.37 x 10^6) * 100
(b) To calculate the percent decrease in your weight, we need to relate the gravitational force to your weight.
Weight (W) is given by the equation:
W = m * g
Where:
W is the weight,
m is the mass, and
g is the acceleration due to gravity.
The acceleration due to gravity on Earth is approximately 9.8 m/s^2.
To calculate the percent decrease, we can use the formula:
Percent decrease = (W2 - W1) / W1 * 100
However, it's important to note that your mass remains the same in this calculation.
Using these formulas, you can calculate the percent increase in the Moon's gravitational pull on you and the percent decrease in your weight.
To calculate the percent change in the Moon's gravitational pull on you and the percent change in your weight when the Moon moves from being directly on the opposite side of Earth to being directly overhead, you'll need to understand the concept of gravitational force and the equation for calculating it.
The gravitational force between two objects can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between the two objects.
- G is the gravitational constant (approximately 6.67430 x 10^-11 N m^2/kg^2).
- m1 and m2 are the masses of the two objects.
- r is the distance between the centers of the two objects.
Now, let's break down the problem and find the answers step by step.
(a) By what percent does the Moon's gravitational pull on you increase?
When the Moon moves from being directly on the opposite side of Earth to being directly overhead, the distance between you and the Moon decreases. To calculate the percent increase in gravitational pull, we need to compare the gravitational forces at the two distances.
Let's label the initial distance (when the Moon is on the opposite side) as r1, and the final distance (when the Moon is overhead) as r2.
Given:
- r1 = 3.82 * 10^8 m (Earth-Moon distance when the Moon is on the opposite side)
- r2 = Earth's radius + r1 = 6.37 * 10^6 m + r1 (Earth's radius plus the Earth-Moon distance when the Moon is on the opposite side)
To calculate the percentage increase in gravitational pull, we need to find the ratio of the two forces and convert it to a percentage:
Percentage increase = ((F2 - F1) / F1) * 100
Where F1 is the initial gravitational force and F2 is the final gravitational force.
To calculate F1 and F2, we need to know the masses of the Moon (mMoon) and you (mYou). For simplicity, assume your mass is negligible compared to the Moon's mass.
Given:
- mMoon = mass of the Moon
Using Newton's law of universal gravitation, we can calculate F1 and F2 using the respective distances.
F1 = G * (mMoon * mYou) / r1^2
F2 = G * (mMoon * mYou) / r2^2
Substituting the given values and solving for F1 and F2:
F1 = G * (mMoon * mYou) / (3.82 * 10^8)^2
F2 = G * (mMoon * mYou) / (6.37 * 10^6 + 3.82 * 10^8)^2
Once you have F1 and F2, you can substitute them into the percentage increase equation to calculate the percent change in the Moon's gravitational pull on you.
(b) By what percent does your weight (as measured on a scale) decrease?
Your weight on a scale is the force you exert on the scale due to the gravitational pull of the Earth. When the Moon moves directly overhead, its gravitational force is partially counteracting the Earth's gravitational force, leading to a decrease in your weight.
To calculate the percentage decrease in your weight, we'll use the same formula as in part (a):
Percentage decrease = ((W1 - W2) / W1) * 100
Where W1 is your initial weight (when the Moon is on the opposite side), and W2 is your final weight (when the Moon is overhead).
To calculate W1 and W2, we need to know your mass (mYou). Then we can use the equation:
W1 = mYou * g
W2 = mYou * (g - a)
Where g is the acceleration due to gravity on Earth (approximately 9.8 m/s^2) and a is the acceleration due to the Moon's gravity.
To calculate a, we can use the equation:
a = G * mMoon / r2^2
Once you have W1 and W2, substitute them into the percentage decrease equation to calculate the percent change in your weight.
Remember to plug in the correct values for the masses and distances in the calculations, as they were not provided in the question.
The distance of the observer to the center of the moon changes from d1 = (3.82 +0.064)*10^8 to d2 = (3.82 -0.064)*10^8 m
The gravitational pull changes by a factor (d1/d2)^2 = (3.884/3.756)^2 = 1.069, which is a 6.09% increase with the moon overhead.
Bigger changes occur during a month
because the moon's orbit is not circular.
For part (b) compare the change in the moon's pull to the pull of the Earth (the weight).