A 10 mm Long Awl Pin is Placed Vertically in Front of a Concave Mirror. A 5 mm Long Image of the Awl Pin is Formed at 30 Cm in Front of the Mirror. The Focal Length of This Mirror is?

(a) 30 cm

(b) 20 cm

(c) 40 cm

(d) 60 cm

A 10 mm long awl pin is placed vertically in front of a concave mirror. A 5 mm long image of the awl pin is formed at 30 cm in front of the mirror. The focal length of this mirror is 20 cm.

Step 1: Given data

Since the image is real and is formed in front of the mirror,

v = -30 cm

Height of the image h_i = 5 mm = 0.5 cm

Height of the object h₀ = 10 mm = 1 cm

Step 2: Now we have to calculate the Image Distance

Since the magnification for a concave mirror is specified as,

m = h_i/h₀ = -v/u

substituting the values in above equation, we get,

0.5/1 = – (-30)/u

On rearranging we get

u = 30/0.5 = 60

Since, the object is in front of the mirror,

u = -60 cm

Step 3: Now we have to Calculate the Focal Length

According to the mirror formula, the focal length is given by

1/f = 1/v + 1/u

Now substituting the values in above equation we get,

1/f = 1/-30 + 1/-60

= -(2 + 1)/ 60

On simplifying we get

= -3/60

= -1/20

f = -20 cm

Focal length cannot be negative so f = 20cm