Define the focal length of a spherical mirror. A concave mirror produces 3 times a real image of an object placed at a distance of 10 cm in front of it. Find the radius of curvature of the mirror.

The focal length of a spherical mirror is defined as the distance between the focal point and the center of the mirror. For a concave mirror, the focal length is positive.

Given that the concave mirror produces a real image of an object that is three times the size, and the object is placed at a distance of 10 cm in front of the mirror, we can use the mirror formula to find the focal length and then calculate the radius of curvature.

The mirror formula is given by:

1/f = 1/v - 1/u

where:
f = focal length of the mirror
v = image distance (distance of the image from the mirror)
u = object distance (distance of the object from the mirror)

Since the image is real and three times the size, we can assume that the magnification (M) is -3.

M = -v/u = -3

Given that u = -10 cm (negative sign indicates that the object is placed in front of the mirror) and M = -3, we can solve for v:

-3 = -v/(-10)
v = 3 * 10
v = 30 cm

Substituting the values of v and u into the mirror formula, we can solve for f:

1/f = 1/30 - 1/(-10)
1/f = (1 - 3)/30
1/f = -2/30
1/f = -1/15

Taking the reciprocal of both sides:

f = -15 cm

Since the focal length (f) is the distance between the focal point and the center of the mirror, and for a concave mirror the focal length is positive, we take the absolute value of f:

f = 15 cm

The radius of curvature (R) of the mirror is twice the focal length:

R = 2 * f
R = 2 * 15
R = 30 cm

Therefore, the radius of curvature of the concave mirror is 30 cm.