the resultant of two forces 13n and 20 n is f if the first forces is doubled the resultant is also doubled to 2f find the angle betwen the two forces

To find the angle between the two forces, we can use the law of cosines.

Let's consider the original forces as A = 13 N and B = 20 N, and the resultant as F.

When the first force is doubled, it becomes 2A = 26 N, and the resultant becomes 2F.

The law of cosines states that the square of the resultant force is equal to the sum of the squares of the individual forces, minus twice the product of the magnitudes of the forces and the cosine of the angle between them.

So, applying the law of cosines to the original forces (A and B) and the resultant force (F), we have:

F^2 = A^2 + B^2 - 2AB * cos(theta)

And for the doubled forces (2A and B) and the doubled resultant force (2F), we have:

(2F)^2 = (2A)^2 + B^2 - 2(2A)(B) * cos(theta)

Simplifying the equations:

4F^2 = 4A^2 + B^2 - 4AB * cos(theta)

4F^2 = 4A^2 + B^2 - 8AB * cos(theta)

Dividing the second equation by 4 and substituting the first equation, we get:

F^2 = A^2 + B^2 - 2AB * cos(theta)

This implies that:

4A^2 = A^2

Simplifying further, we get:

3A^2 = 2AB * cos(theta)

Dividing by 2A on both sides, we get:

cos(theta) = 3A / (2B)

Now, we substitute the given values A = 13 N and B = 20 N into the equation:

cos(theta) = (3 * 13 N) / (2 * 20 N)

cos(theta) = 39 / 40

To find the angle theta, we need to find the inverse cosine (cos^(-1)) of 39/40.

theta = cos^(-1)(39/40)

Using a calculator or trigonometric tables, we find theta ≈ 14.48 degrees.

Therefore, the angle between the two forces is approximately 14.48 degrees.