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Step-by-Step Solution
Step 1: Identify the given information
• Initial object distance, $u = 2.4\,\text{m} = 240\,\text{cm}$.
• Initial image distance, $v = 12\,\text{cm}$.
• Thickness of glass plate, $t = 1\,\text{cm}$.
• Refractive index of glass plate, $\mu = 1.50$.
• We wish to find the new object distance $u'$ when the glass plate is introduced between the lens and the film.
Step 2: Calculate the focal length of the lens
Using the lens formula in the chosen sign convention:
$$
\frac{1}{f} = \frac{1}{v} - \frac{1}{u}.
$$
Substituting $v = 12\,\text{cm}$ and $u = 240\,\text{cm}$:
$$
\frac{1}{f}
= \frac{1}{12} + \frac{1}{240}
= \frac{20}{240} + \frac{1}{240}
= \frac{21}{240}.
$$
Hence,
$$
f = \frac{240}{21}\,\text{cm}.
$$
Step 3: Determine the shift caused by the glass plate
When a glass plate of thickness $t$ and refractive index $\mu$ is introduced in the path of the image, the image shifts by
$$
\Delta = t \biggl( 1 - \frac{1}{\mu} \biggr).
$$
Here, $t = 1\,\text{cm}$ and $\mu = 1.50$. Thus,
$$
\Delta
= 1\,(\!1 - \frac{1}{1.50})
= 1 \left(1 - \frac{2}{3}\right)
= 1 \times \frac{1}{3}
= \frac{1}{3}\,\text{cm}.
$$
Step 4: Adjust the new image distance
The film remains in the same place, so effectively the lens needs to form the image at a distance $v' = v - \Delta$. Thus,
$$
v'
= 12\,\text{cm}
- \frac{1}{3}\,\text{cm}
= \frac{35}{3}\,\text{cm}.
$$
Step 5: Use the lens formula again for the new arrangement
We keep the same focal length $f = \frac{240}{21}\,\text{cm}$, but now the new image distance is $v' = \frac{35}{3}\,\text{cm}$. From the lens formula,
$$
\frac{1}{f}
= \frac{1}{v'} - \frac{1}{u'}.
$$
Hence,
$$
\frac{21}{240}
= \frac{3}{35} - \frac{1}{u'}.
$$
Rearrange to find $1/u'$ and then $u'$:
$$
\frac{1}{u'}
= \frac{3}{35} - \frac{21}{240}.
$$
Evaluating carefully (as in the provided steps) gives:
$$
u' = -\,560\,\text{cm} = -\,5.6\,\text{m}.
$$
The negative sign in this convention indicates the object is on the opposite side relative to the direction of image formation. In practical terms, the object should be placed $5.6\,\text{m}$ from the lens to maintain sharp focus on the film.
Step 6: State the final answer
Therefore, the object must be shifted to a distance of $5.6\,\text{m}$ from the lens to be in sharp focus on the film after introducing the glass plate.
