2 masses m and 4m are placed at 12meter distance. where a mass of 10 kg can be placed between them so that the net gravitation force on it become zero.
To find the position where a mass of 10 kg can be placed such that the net gravitational force on it becomes zero, we can use the concept of gravitational force and Newton's Law of Universal Gravitation.
Let's denote the distance between the mass m (located at point A) and the mass 4m (located at point B) as r = 12 meters.
According to Newton's Law of Universal Gravitation, the gravitational force between two masses m1 and m2 is given by:
F = G * (m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses, and r is the distance between them.
In this case, we want to find the position where the net gravitational force on a mass of 10 kg (located at point C) becomes zero. Let's denote the distance between the mass m (located at point A) and the mass of 10 kg (located at point C) as x, and the distance between the mass 4m (located at point B) and the mass of 10 kg (located at point C) as 12 - x.
Now, let's calculate the gravitational forces on the mass of 10 kg due to masses m and 4m respectively.
Gravitational force due to mass m: F1 = G * ((10 kg) * m) / x^2
Gravitational force due to mass 4m: F2 = G * ((10 kg) * (4m)) / (12 - x)^2
Since we want the net gravitational force on the mass of 10 kg to be zero, we can set F1 equal to F2:
F1 = F2
G * ((10 kg) * m) / x^2 = G * ((10 kg) * (4m)) / (12 - x)^2
Canceling out G and (10 kg) from both sides:
(m) / x^2 = (4m) / (12 - x)^2
Cross multiplying the equation:
m * (12 - x)^2 = (4m) * x^2
Expanding the squares:
m * (144 - 24x + x^2) = 4m * x^2
Distributing the terms:
m * 144 - m * 24x + m * x^2 = 4m * x^2
Rearranging the terms:
m * x^2 - 4m * x^2 + m * 24x - m * 144 = 0
Combining like terms:
-3m * x^2 + 24m * x - 144m = 0
Now, we can solve this quadratic equation for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a = -3m, b = 24m, and c = -144m.
Substituting the values:
x = (-24m ± √((24m)^2 - 4 * (-3m) * (-144m))) / (2 * (-3m))
Simplifying the equation:
x = (-24m ± √(576m^2 - 1728m^2)) / (-6m)
x = (-24m ± √(-1152m^2)) / (-6m)
x = (-24m ± √(1152m^2) * i) / (-6m)
x = (-24m ± 48m * i) / (-6m)
x = 4 ± 8i
Here, i represents the imaginary unit. Since we are dealing with physical distances, we can ignore the imaginary solution.
Therefore, the possible position where a mass of 10 kg can be placed such that the net gravitational force on it becomes zero is at a distance of 4 meters from the mass m (point A) or a distance of 8 meters from the mass 4m (point B).
So, the mass of 10 kg can be placed either 4 meters from mass m (point A) or 8 meters from mass 4m (point B) to experience a net gravitational force of zero.