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 metre. 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 the 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

Use the formula of the mirror $\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}$. Substitute the values of u and v from each of the data sets given in the options. Do not forget to substitute the values of u and v according to the sign convection. f for a concave mirror is always negative.
Formula used:
$\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}$

Complete answer:
It is given that a student performs an experiment where he/she determines the focal length of a concave mirror by u-v method using an optical bench.
In the u-v method of determining the focal length of a given mirror, the object is placed at a known location in the line of the mirror. Then the image is found by focusing it on a white screen. This is the location of the image of the object.
Once the position of the image and the object are known then the focal length of the mirror is calculated with the mirror formula.
i.e. $\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}$
$\Rightarrow \dfrac{1}{f}=\dfrac{u+v}{vu}$
$\Rightarrow f=\dfrac{uv}{u+v}$ …. (i).
Here, the values of u and v are according to the sign convection.
For a concave mirror, u is always negative for a real object. v is negative when the magnitude of u is greater than or equal to the double of the focal length. f is always negative for a concave mirror.
Therefore, in this case, f=-24.
Since u is greater than 2f for all observations, v is negative in all.
Let us find the focal length using equation (i) for each data set given in the option. It is given that there can be a maximum error of 0.2cm.
Therefore, if the value of f lies between -24.2cm and -23.8 cm then the data set is correct.
Option A: Here, u=-42cm, v=-56cm
$\Rightarrow f=\dfrac{(-42)(-56)}{(-42)+(-56)}=-24cm$.
This means that this data set is correct.
Option B: Here, u=-48cm, v=-48cm
$\Rightarrow f=\dfrac{(-48)(-48)}{(-48)+(-48)}=-24cm$.
This means that this data set is correct.
Option C: Here, u=-66cm, v=-33cm
$\Rightarrow f=\dfrac{(-66)(-33)}{(-66)+(-33)}=-22cm$.
This means that this data set is incorrect.
Option D: Here, u=-78cm, v=-39cm
$\Rightarrow f=\dfrac{(-78)(-39)}{(-78)+(-39)}=-26cm$.
This means that this data set is incorrect.

Hence, the correct options are C and D.

Note:
Some students may forget to use the sign convection in the given problem. They may get the correct answers without applying the sign convection. However, it will not always happen.
According to the sign convection, the direction of the incident light rays is taken as position direction and the opposite direction to this is negative. All the positions are measured from the pole of the mirror.