when two forces of magnitude p and q are perpendicular to each other their resultant is of magnitude 'r'. When they are at an angle of 180 degree to each other their resultant is 'r root 2'find the ratio of p and q.
To find the ratio of p and q, let's consider the given information:
1) Two forces of magnitude p and q are perpendicular to each other, and their resultant is of magnitude 'r'.
2) The same forces, when they are at an angle of 180 degrees to each other, have a resultant of magnitude 'r√2'.
Let's assume that p represents the magnitude of the force in the x-direction, and q represents the magnitude of the force in the y-direction.
When the forces are perpendicular to each other, we can use the Pythagorean theorem to find the resultant magnitude 'r'. According to Pythagoras, the sum of the squares of the magnitudes of p and q is equal to the square of the magnitude of the resultant 'r':
p^2 + q^2 = r^2 ... (Equation 1)
When the forces are at an angle of 180 degrees to each other, we can use the law of cosines to find the resultant magnitude 'r√2'. According to the law of cosines, the square of the magnitude of the resultant 'r√2' is equal to the sum of the squares of the magnitudes of p and q, minus the product of their magnitudes multiplied by the cosine of the angle between them (180 degrees in this case):
(p + q)^2 = (r√2)^2 = r^2 * 2
Simplifying: p^2 + q^2 + 2pq = r^2 * 2 ... (Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with two variables (p and q). We can solve these equations simultaneously to find the ratio of p and q.
First, divide Equation 2 by Equation 1:
(p^2 + q^2 + 2pq) / (p^2 + q^2) = (r^2 * 2) / (r^2)
Simplifying: 1 + (2pq / (p^2 + q^2)) = 2
Next, subtract 1 from both sides of the equation:
2pq / (p^2 + q^2) = 1
Cross-multiply:
2pq = p^2 + q^2
Rearranging terms:
p^2 - 2pq + q^2 = 0
This equation is a quadratic equation in the form of (p - q)^2 = 0. Therefore, p = q.
So, the ratio of p and q is 1:1 or simply p = q.