A particle weighs 120 N on the surface of the earth. At what height above the earth's surface will its weight be 30 N? Radius of the earth = 6,400 km.
6400km is the answer
To find the height above the Earth's surface at which the particle's weight will be 30 N, we need to use the concept of gravitational force and the inverse square relationship of gravity with respect to distance.
The first step is to find the mass of the particle since weight is the force of gravity acting on an object's mass. We can use the equation:
Weight = mass * gravitational acceleration
The gravitational acceleration on the surface of the Earth is usually denoted as "g" and has an approximate value of 9.8 m/s^2. Converting the given weight from Newtons to kg:
120 N = mass * 9.8 m/s^2
From this equation, we can solve for the mass:
mass = 120 N / 9.8 m/s^2 = 12.24 kg
Now, let's consider the decrease in weight as the particle moves away from the Earth's surface. The force of gravity between two objects decreases with the square of the distance between their centers. This is expressed by the equation:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (approximated as 6.67430 × 10^-11 m^3 kg^-1 s^-2),
m1 and m2 are the masses of the two objects, and
r is the distance between their centers.
In this case, m1 refers to the mass of the particle and m2 refers to the mass of the Earth.
To find the height above the Earth's surface at which the weight of the particle is 30 N, we need to set the gravitational force equal to 30 N and solve for r. Rearranging the equation:
30 N = (G * m1 * m2) / r^2
We already know the value of G, the mass of the particle (m1), and the radius of the Earth (r). However, we need to convert the radius of the Earth from kilometers to meters:
Radius of the Earth = 6,400 km = 6,400,000 m
Substituting the known values into the equation:
30 N = (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 12.24 kg * m2) / (r + 6,400,000 m)^2
Simplifying the equation and solving for r:
r = √[(6.67430 × 10^-11 m^3 kg^-1 s^-2 * 12.24 kg * m2 * (r + 6,400,000 m)^2) / 30 N]
Simplifying further,
[(r + 6,400,000 m)^2] / r = (30 N * r) / (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 12.24 kg * m2)
r = [(30 N * r) / (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 12.24 kg * m2)] - 6,400,000 m
This is a complicated equation to solve manually, but you can use numerical methods or software to find the solution. The value of r will represent the height above the Earth's surface at which the particle's weight is 30 N.