Analyzing the Setup
Imagine you are in a physics lab, standing in front of an optical bench. You have a concave mirror with a known focal length of f=24 cm. Your task is to place an object pin at various distances (u) and locate the sharpest real image (v) using an image pin.
The optical bench is 1.5 m (or 150 cm) long, which gives you plenty of room to move the pins around. However, human eyes and instruments aren't perfect. The problem states that the maximum permissible error in locating the image is 0.2 cm. This means if your theoretical calculation for v differs from your experimental reading by more than 0.2 cm, your reading is fundamentally flawed and incorrectly recorded.
The Master Equation
To verify the student's readings, we must rely on the trusty mirror formula:
For a concave mirror forming a real image, the object distance u, image distance v, and focal length f are all negative according to the Cartesian sign convention. If we plug in the magnitudes, the equation simplifies beautifully to:
We will use this master equation to calculate the exact theoretical value of v for each given u, and then compare it with the student's recorded v.
Verifying the Data Sets
Let's test the options one by one.
Option (a): u=42 cm
Plugging this into our formula:
v=42−2442×24=181008=56 cm
The student recorded
56 cm. This is a perfect match! The error is
0 cm, which is well within the
0.2 cm limit.
Option (b): u=48 cm
v=48−2448×24=241152=48 cm
The student recorded
48 cm. Another perfect match! (Fun fact: This is the center of curvature, where
u=v=2f).
Option (c): u=66 cm
v=66−2466×24=421584≈37.71 cm
The student recorded
33 cm. The absolute error here is
∣37.71−33∣=4.71 cm. This is massively larger than the allowed
0.2 cm error. This data set is definitely incorrectly recorded.
Option (d): u=78 cm
v=78−2478×24=541872≈34.66 cm
The student recorded
39 cm. The absolute error is
∣34.66−39∣=4.34 cm, which again far exceeds the
0.2 cm threshold. This reading is also flawed.
Final Conclusion
By systematically applying the mirror formula and respecting the boundaries of experimental error, we have successfully identified the anomalies. The data sets in options (c) and (d) are physically impossible under the given constraints and are therefore incorrectly recorded.
