An object of length 6 cm is placed the principal axis of a concave mirror (f = -40 cm) at 50 cm from its pole. Find the image length if the object is perpendicular to the principal... An object of length 6 cm is placed the principal axis of a concave mirror (f = -40 cm) at 50 cm from its pole. Find the image length if the object is perpendicular to the principal axis.
Understand the Problem
The question involves finding the image length of an object placed in front of a concave mirror. The object is 6 cm long, positioned 50 cm from the mirror's pole, which has a focal length of -40 cm. The task is to determine the length of the image formed, assuming the object is perpendicular to the principal axis.
Answer
The image length is \( 24 \, \text{cm} \).
Answer for screen readers
The image length is ( h_i = 24 , \text{cm} ).
Steps to Solve
- Identify the given values
We have the following values:
- Object length ( h_o = 6 , \text{cm} )
- Object distance ( u = -50 , \text{cm} ) (distance is taken as negative in mirror formula)
- Focal length ( f = -40 , \text{cm} )
- Use the mirror formula to find image distance
The mirror formula is given by: $$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $$
Substituting the known values: $$ \frac{1}{-40} = \frac{1}{v} + \frac{1}{-50} $$
Rearranging gives: $$ \frac{1}{v} = \frac{1}{-40} + \frac{1}{50} $$
- Calculate the image distance ( v )
Finding the common denominator and calculating: $$ \frac{1}{v} = \frac{-5 + 4}{200} = \frac{-1}{200} $$
Thus: $$ v = -200 , \text{cm} $$
- Calculate the magnification
The magnification ( m ) is given by: $$ m = -\frac{v}{u} $$
Substituting in the known values: $$ m = -\frac{-200}{-50} = 4 $$
- Determine the image length
The image length ( h_i ) can be calculated using: $$ h_i = m \cdot h_o $$
Substituting the magnification and object length: $$ h_i = 4 \cdot 6 = 24 , \text{cm} $$
The image length is ( h_i = 24 , \text{cm} ).
More Information
The result indicates that the image formed by the concave mirror is larger than the object and is located on the same side as the object, which is consistent with the properties of convex mirrors.
Tips
- A common mistake is not taking the signs of distances correctly. Remember that object distances are negative for concave mirrors.
- Forgetting to square the magnification when relating it to image dimensions can lead to incorrect image sizes.
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