An image is produced by a concave mirror of a focal length of 8.7 cm. The object is 13.2 cm tall and at a distance of 19.3 cm from the mirror. 
(a) Find the location and height of the image. 
(b) Find the height of the image produced by the mirror if the object is twice as far from the mirror. 
Ans. [(a) -15.84 cm, 10.83 cm (b) 5.42 cm]


Note: In this numerical, all the sign conventions are according to New Cartesian Sign Conventions.
For an easy and comprehensive understanding of the sign convention of spherical mirrors problems please Click on the below link


Given Data:


Focal Length = f  = - 8.70 cm

Height of the object: h₀ = 30 cm

Distance of the object from the mirror = p = -19.3 cm

Note: You can use notations u and v instead of p and q depending upon the notation used in your book.



To Find:


(a) 
Distance of the image from the mirror = q  = ?
Image height = hᵢ = ? 

(b) 
Image height = hᵢ = ? (when the object is twice far from the mirror)


Solution:


(a)
We have the mirror formula as 

`\frac{1}{f}` = `\frac{1}{p}` + `\frac{1}{q}`

or

`\frac{1}{q}` = `\frac{1}{f}` - `\frac{1}{p}`

Now by putting the values

`\frac{1}{q}` = `\frac{1}{-8.7 cm}` - `\frac{1}{-19.3 cm}`

`\frac{1}{q}` = -`\frac{10}{87}` cm + `\frac{10}{193}` cm

`\frac{1}{q}` = `\frac{-1930+870}{(87)(193)}` cm

`\frac{1}{q}` = `\frac{-1,060}{16,791}` cm

by reversing the fraction we get

`\frac{q}{1}` = -`\frac{16,791}{1,060}` cm

by simplifying we get

q = - 15.83 cm ---------------Ans. 1(a)



Now using the magnification formula

magnification = `\frac{hᵢ }{h₀}` = `\frac{q}{p}`

or

`\frac{hᵢ }{h₀}` = `\frac{q}{p}`

by putting values

`\frac{hᵢ}{13.2 cm}` =`\frac{-15.84 cm}{-19.3 cm}`

hᵢ = `\frac{15.84 cm}{19.3 cm}` 𝐱 13.2 cm

by simplifying we get

hᵢ = 10.83 cm ---------------Ans. 2(a)



(b)
when the object is twice far from the mirror i.e
P = -19.3 cm  𝐱 2 = - 38.6 cm

Using the magnification formula

`\frac{hᵢ }{h₀}` = `\frac{q}{p}`

by putting values

`\frac{hᵢ}{13.2 cm}` =`\frac{-15.84 cm}{-38.6 cm}`

hᵢ = `\frac{15.84 cm}{38.6 cm}` 𝐱 13.2 cm

by simplifying we get

hᵢ = 5.42 cm



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