when two forces of magnitude p and q are perpendicular to each other their resultant is of magnitude 'r' whaen they are at an angle 180 degree to each other their resultant is of magnitude 'r under root 2' find the ratio of p and q.
11/3
To find the ratio of p and q, we can use the concept of vector addition and trigonometry.
Let's consider the case where the two perpendicular forces of magnitude p and q have a resultant magnitude of r:
Using vector addition, we can represent the forces as vectors P and Q, and their resultant as vector R.
The magnitude of the resultant vector R can be calculated using the Pythagorean theorem:
r^2 = p^2 + q^2
Now, let's consider the case where the forces are at an angle of 180 degrees to each other:
The magnitude of the resultant vector in this case is given as square root of 2 times r (sqrt(2) * r).
Using vector addition again, we can represent the forces as vectors -P and Q (since they are at 180 degrees), and their resultant as vector R.
The magnitude of the resultant vector R can be calculated as follows:
(sqrt(2) * r)^2 = p^2 + q^2
Simplifying this equation, we get:
2 * r^2 = p^2 + q^2
Now, we can equate the expressions for r^2:
p^2 + q^2 = 2 * r^2
We know from the first case that:
r^2 = p^2 + q^2
Substituting this into the equation above, we have:
p^2 + q^2 = 2 * p^2 + 2 * q^2
Simplifying further, we get:
0 = p^2 + q^2
From this equation, we can deduce that the ratio of p and q is equal to 1:1, meaning p and q are equal.
Therefore, the ratio of p and q is 1:1.