The weight of an apple near the surface of the earth is 1N. What is the weight of Earth in the gravitational field of the apple?
They are asking for the force (weight) on the Earth due to the presence of the apple. It is also 1 N, but in the opposite direction (towards the apple).
This follows from Newton's third law of motion, but can also be derived from his law of gravity.
To find the weight of Earth in the gravitational field of the apple, we can use Newton's law of universal gravitation:
F = G * (m₁ * m₂) / r²
where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 x 10^-11 N*m²/kg²)
m₁ is the mass of the apple
m₂ is the mass of the Earth
r is the distance between the apple and Earth's center
From the information given, we know the weight of the apple near the surface of the Earth is 1N. Weight is defined as the force of gravity acting on an object, which can be calculated using the following formula:
Weight = mass * acceleration due to gravity (g)
Since weight is given as 1N, we can replace it with mass * g:
1N = m₁ * g
Now, we need the mass of the apple. But since weight is the force being exerted on the apple due to Earth's gravity, we can also say it is equal to:
1N = m₁ * (m₂ * g) / r²
Since g is the acceleration due to gravity and is the same on both sides of the equation, we can cancel it out:
1 = (m₁ * m₂) / r²
If r is the average distance between the apple and Earth's center, we can assume it to be approximately equal to the radius of Earth, which is about 6,371 kilometers (6,371,000 meters).
Plugging in the numbers, we get:
1 = (m₁ * m₂) / (6,371,000²)
Now, we need to solve for m₂ (the mass of Earth). Rearranging the equation:
m₂ = (1 * (6,371,000²)) / m₁
Therefore, the weight of Earth in the gravitational field of the apple is ((1 * (6,371,000²)) / m₁) kilograms.
To determine the weight of Earth in the gravitational field of the apple, we need to make a few assumptions and calculations.
First, we need to know the mass of the apple. Weight is directly proportional to mass, so we can assume that the weight of the apple is equal to the force of gravity acting on it, which is 1 Newton.
Using Newton's second law of motion (F = m * a), we can determine that the force acting on the apple (1N) is the mass of the apple (m) multiplied by the acceleration due to gravity (9.8 m/s^2 on the surface of Earth).
1N = m * 9.8 m/s^2
Solving for the mass (m), we get:
m = (1N) / (9.8 m/s^2)
m ≈ 0.102 kilograms
Now, to find the weight of Earth in the gravitational field of the apple, we need to use Newton's law of universal gravitation. The formula states that the force of gravity (F) between two objects is equal to the product of their masses (m1 and m2) divided by the square of the distance between their centers (r) and multiplied by the gravitational constant (G).
F = (G * m1 * m2) / r^2
Assuming the mass of the apple is negligible compared to the mass of Earth, we can consider only the mass of Earth (m1) in this equation.
The distance between Earth's center and the apple's center is the radius of Earth, which is approximately 6,371 kilometers (6,371,000 meters).
Now, to calculate the weight of Earth in the apple's gravitational field, we need to rearrange the equation to solve for m2 (mass of Earth):
m2 = (F * r^2) / (G * m1)
Substituting the values:
m2 = (1N * (6,371,000 m)^2) / (6.67430 × 10^-11 N(m/kg)^2 * 0.102 kg)
m2 ≈ 5.985 × 10^24 kilograms
So, the weight of Earth in the gravitational field of the apple is approximately 5.985 × 10^24 kilograms.