the force of attraction between two equal spheres is 8N. calculate the sum of their masses if their circumference distance is 10cm and their radius is 1cm each. (take G=6.7*10^-11)
The force of attraction between two spherical objects is given by the formula:
F = (G * m1 * m2) / r^2
Where:
F is the force of attraction
G is the gravitational constant (6.7 * 10^-11 N * (m/kg)^2)
m1 and m2 are the masses of the two spheres
r is the distance between the centers of the two spheres
We are given that the force of attraction is 8N, the distance between the spheres is 10cm, and the radius of each sphere is 1cm.
First, we need to find the masses of the spheres. We can use the formula for the circumference of a sphere to find the radius:
C = 2πr
10cm = 2πr
10cm = 2 * 3.14 * 1cm
r = 5/π = 1.59cm
Now, we can plug in the values into the equation for the force of attraction and solve for the mass:
8N = (6.7 * 10^-11 N * (m/kg)^2 * m1 * m2) / (1.59cm)^2
Simplifying:
8N = (6.7 * 10^-11 N * (m/kg)^2 * m1 * m2) / 2.53cm^2
Rearranging:
m1 * m2 = (8N * 2.53cm^2) / (6.7 * 10^-11 N * (m/kg)^2)
m1 * m2 = 3.01 * 10^21 kg^2
The sum of their masses is given by m1 + m2:
m1 + m2 = √(m1 * m2)
m1 + m2 = √(3.01 * 10^21) kg
m1 + m2 = 5.48 * 10^10 kg
Therefore, the sum of the masses of the two spheres is approximately 5.48 * 10^10 kg.
To find the sum of their masses, we can use Newton's Law of Universal Gravitation:
F = (G * m1 * m2) / r^2
where:
F = force of attraction
G = gravitational constant (6.7 * 10^-11 N(m/kg)^2)
m1, m2 = masses of the spheres
r = distance between the centers of the spheres
In this case, F = 8N and r = 10cm = 0.1m.
First, let's find the individual masses of the spheres using the information given.
Circumference = 2 * π * radius
10cm = 2 * π * 1cm
10cm = 2π cm
Since the radius is 1cm, the circumference is equal to 2π.
Therefore, the distance between the centers of the spheres is equal to the sum of their radii plus the circumference:
r = (2 * radius) + circumference
0.1m = (2 * 0.01m) + 2π
Now, let's rearrange the equation to solve for the individual masses:
F = (G * m1 * m2) / r^2
Using the given values, we have:
8N = (6.7 * 10^-11 N(m/kg)^2) * m1 * m2 / (0.1m)^2
Simplifying, we get:
8N = (6.7 * 10^-11 N(m/kg)^2) * m1 * m2 / 0.01m^2
Cross-multiplying, we get:
1 = (6.7 * 10^-11 N(m/kg)^2) * m1 * m2 / 0.01m^2
Next, we can simplify:
1 = (6.7 * 10^-11 N(m/kg)^2) * m1 * m2 / 0.0001m^2
To isolate the product of the masses, we multiply both sides of the equation by 0.0001m^2:
0.0001m^2 = (6.7 * 10^-11 N(m/kg)^2) * m1 * m2
Next, we divide both sides of the equation by (6.7 * 10^-11 N(m/kg)^2):
m1 * m2 = 0.0001m^2 / (6.7 * 10^-11 N(m/kg)^2)
m1 * m2 = 1.49253731... * 10^(-6)kg^2
Now, let's find the sum of the masses by taking the square root of both sides:
√(m1 * m2) = √(1.49253731... * 10^(-6)kg^2)
m1 + m2 = √(1.49253731... * 10^(-6)kg^2)
Calculating the square root, we find:
m1 + m2 = 0.00122392... kg
Therefore, the sum of the masses of the two spheres is approximately 0.0012 kg.