Question

A student performed the experiment of determination of focal length of a concave mirror by u−v method using an optical bench of length 1.5 meter. The focal length of the mirror used is 24 cm. The maximum error in the location of the image can be 0.2 cm. The 5 sets of (u, v) values recorded by the student (in cm) are : (42,56),(48,48),(60,40),(66,33), and (78,39). The data set(s) that cannot come from experiment and is (are) incorrectly recorded is (are).
This question has multiple correct options.
A. (42, 56)
B. (48, 48)
C. (66, 33)
D. (78, 39)

Answer 1

Hint: It is said that a student performs an experiment for finding the focal length of a concave mirror using u –v method. The focal length of the mirror and five sets of data of u and v collected by the student is given to us, and also the maximum possible error in the image distance. By using the lens formula we can formulate an equation to find the image distance and check for all the given values. Thus we can find the correct solution.
Formula used:
\[\dfrac{1}{f}=\dfrac{1}{u}+\dfrac{1}{v}\]

Complete answer:
In the question it is said that a student performs an experiment to determine the focal length of a concave mirror by u – v method.
The length of the optical bench used in the experiment is given as 1.5 meter.
We are given the focal length of the mirror as 24 cm and the maximum error in the position of the image as 0.2 cm.
The 5 sets of object distance and image distance noted by the student is given to us in the form (u, v).
By lens formula we have an equation that relates the focal length, object distance and image distance given as,
\[\dfrac{1}{f}=\dfrac{1}{u}+\dfrac{1}{v}\], were ‘f’ is the focal length of the mirror, ‘u’ is the object distance and ‘v’ is the image distance.
From this equation we can calculate the image distance as,
$\Rightarrow v=\dfrac{uf}{u-f}$
The first data recorded by the student is (42, 56).
Here $u=42cm$, $v=56cm$ and $f=24cm$
Now by using the lens formula we can find the image distance from the known object distance and focal length, and check if it is the same recorded by the student.
$\Rightarrow v=\dfrac{42\times 24}{42-24}$
$\Rightarrow v=\dfrac{1008}{18}$
$\Rightarrow v=56cm$
Therefore the image distance when the object distance is 42 is 56. Hence this data is correct.
The second data recorded is (48, 48).
Here $u=48cm$, $v=48cm$ and we have $f=24cm$.
Therefore we can check this,
$\Rightarrow v=\dfrac{48\times 24}{48-24}$
$\Rightarrow v=\dfrac{1152}{24}$
$\Rightarrow v=48cm$
Therefore this recording is also correct.
Now the third data set is (60, 40).
Here $u=60cm$, $v=40cm$ and we have $f=24cm$
Therefore we can write,
$\Rightarrow v=\dfrac{60\times 24}{60-24}$
$\Rightarrow v=\dfrac{1440}{36}$
$\Rightarrow v=40cm$
This recording is also correct.
The fourth recording is (66, 33).
Here we have $u=66cm$, $v=33cm$ and $f=24cm$.
Therefore,
$\Rightarrow v=\dfrac{66\times 24}{66-24}$
$\Rightarrow v=\dfrac{1584}{42}$
$\Rightarrow v=37.71cm$
Here we get the image distance as 37.71 cm. but in the recorded data the image distance is 33 cm. In the question it is said that a maximum error of 0.2 cm is possible.
The error here is,
$Error=37.71-33$
$\Rightarrow Error=4.71cm$
An absolute error of 4.71 cm is not allowed. Hence this data is wrong.
Therefore we can say that option C is wrong.
Now we have the last data set as (78, 39).
Here we have $u=78cm$, $v=39cm$ and $f=24cm$.
Therefore,
$\Rightarrow v=\dfrac{78\times 24}{78-24}$
$\Rightarrow v=\dfrac{1872}{54}$
$\Rightarrow v=34.66cm$
The recorded image distance is 39 cm and the calculated image distance is 34.66 cm.
By calculating the absolute error, we will get,
$Error=34.66-39$
$\Rightarrow Error=4.34cm$
Since the maximum possible error is 0.2 cm this data set is also incorrect.
Therefore option D is also wrong.

Hence the correct answers are option C and option D.

Note:
Mirrors are of two types; curved mirrors and plane mirrors. The image formed using a plane mirror cannot be seen on a screen and hence they are virtual images. When a curved mirror is part of a sphere, then that mirror is known as a spherical mirror. The two types of spherical mirrors are convex mirror and concave mirror.
A concave mirror is a converging mirror that is curved inward in the middle. These mirrors are used to focus the reflected light.
A convex mirror is called a diverging mirror because it diverges the light that strikes on the reflecting surface of the mirror.