The force of attraction F, between two bodies, varies directly as the product of their masses, \(m_1\) and m\(_2\) and inversely as the square of the distance, d, between them. Given that F = 20N, when m\(_1\) = 25kg, m\(_2\) = 10kg and d = 5m, find:
(I) To find the expression for F in terms of m1, m2, and d:
We can use the formula for direct variation and inverse variation:
F ∝ (m1 m2) / d2
where k is a constant of proportionality. To find the value of k, we can use the given values:
20 = k (25 x 10) / (52)
Simplifying this equation gives:
k = 4
Therefore, the expression for F in terms of m1, m2, and d is:
F = 4(m1 m2) / d2
(ii) To find the distance, d, for F = 30N, m1 = 7.5kg, and m2 = 4kg:
We can use the expression we found in part (i):
30 = 4(7.5 x 4) / d2
Simplifying this equation gives:
d2 = (4 x 7.5 x 4) / 30
d2 = 4
Taking the square root of both sides gives:
d = 2 meters
Therefore, the distance between the two bodies is 2 meters.
(b) In the diagram:
We can see that there are two right triangles that share a common side, which is x. The two triangles have sides of length 3 and 4, and sides of length 5 and x.
Since these are right triangles, we can use the Pythagorean theorem:
32 + 42 = 52 + x2
Simplifying this equation gives:
9 + 16 = 25 + x2
25 = x2
Taking the square root of both sides gives:
x = 5
Therefore, the value of x in the diagram is 5.
