Two stars of mass M and 2M, a distance 3x apart rotates in circles about the common center mass "0" The gravitational force acting on the stars is KGM^2/x^2, what is value of K?
G(M)(2M)/(3x)^2 = KGM^2/x^2
2GM^2/9x^2 = KGM^2/x^2
2/9 = K
thanku you soo much
To find the value of K, we can use the concept of centripetal force in circular motion.
Given:
Mass of one star, m1 = M
Mass of the other star, m2 = 2M
Distance between the stars, r = 3x
The gravitational force acting on each star is given by the formula:
F = GMm/r^2
In circular motion, the centripetal force required to keep an object in circular motion is given by:
Fc = mv^2/r
Setting these two forces equal to each other, we have:
GMm/r^2 = mv^2/r
Now, expressing velocity (v) in terms of the period of rotation (T) and the radius of the circular path (r), we have:
v = 2πr/T
Substituting v in the equation above, we get:
GMm/r^2 = m(2πr/T)^2/r
GM/r^2 = 4π^2r/T^2
GM = 4π^2r^3/T^2
Now, substituting the given values:
M = m1 = M
2M = m2
r = 3x
We have:
G(M)(2M) = 4π^2(3x)^3/T^2
Simplifying the equation further, we get:
2GM^2 = 4π^2(27x^3)/T^2
GM^2 = 2π^2(27x^3)/T^2
Now, to find the value of K, we recall that the gravitational force acting on the stars is given by:
F = KGM^2/x^2
Comparing this with GM^2 = 2π^2(27x^3)/T^2, we can deduce the value of K:
K = 2π^2(27x^3)/T^2
Therefore, the value of K is 2π^2(27x^3)/T^2.
To find the value of K, we can use the gravitational force equation and the given information.
The gravitational force acting between two objects can be calculated using the formula:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force between two objects,
G is the gravitational constant (approximately 6.674 × 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 have two stars with masses M and 2M, respectively. The distance between their centers is given as 3x.
Let's substitute the values into the gravitational force formula:
K * G * M * (2M) / (3x)^2 = K * G * M^2 / (9x^2)
The gravitational force acting on the stars is given as K * G * M^2 / x^2. Comparing this with the above equation, we can determine that K = 9.
So, the value of K is 9.