Why the uv graph of concave mirror is parobilic and1/u and1/v graph is a linear graph???
The uv graph is actually a hyperbola
1/v + 1/u = 1/f
u + v = uv/f
uv = f(u+ v)
We know that equation of the type xy = a(x + y) is a hyperbola asymptotic to x = a, y = a
Therefore the equation
uv = f(u + v) is also a hyperbola asymptotic to u = f, v = f
Plot u on x axis and v on y axis you will get a curve of the shape shown in figure.
---------CONSIDER THE BRANCH OF HYPERBOLA IN THE FIRST QUADRANT-------
A line drawn at angle 45o (u = v) from the origin intersects the hyperbola at point C.
Find the intersection point of v = u and uv = f(u + v). Let the intersection point be C
u.u = f (u + u)
u = 2f
v = 2f
The coordinates of point C are (2f, 2f). The focal length of mirror can be calculated by measuring the coordinates of point C on your graph.
------------
-----------
SECOND METHOD (USING STRAIGHT LINE)
1/v + 1/u = 1/f
or
1/v = -1/u + 1/f
Comparing above equation with the equation of line y = mx + c
1/v is plotted on y axis
1/u is plotted on x axis
Slope of line m is -1
Constant term will be 1/f
Measure the constant term in your graph. The reciprocal of which will give the focal length.
