Determine the focal length of a concave mirror that produces an image 12.2 cm away when an object is placed 49.5 cm in front of it.

Question

BlackTom AI · Accepted Answer

To determine the focal length of the concave mirror, we start with the mirror formula: 1/f = 1/v + 1/u.
We assign sign conventions typical for Cartesian coordinates: distances measured to the left of the mirror are negative.
Given an object distance u = -49.5 cm (object is in front of the mirror, on the left), and the image distance v = -12.2 cm (the real image forms on the same side as the object, i.e., to the left).
Substituting these into the equation gives 1/f = 1/(-12.2) + 1/(-49.5) = -(1/12.2 + 1/49.5).
Calculating the reciprocal values: 1/12.2 ≈ 0.08197 and 1/49.5 ≈ 0.02020. Adding them yields 0.10217, and applying the negative sign from the sign convention gives 1/f ≈ -0.10217.
Thus f ≈ -9.78 cm.
The negative sign indicates a concave mirror under this sign convention. If one reports the focal length by its magnitude, it is about 9.78 cm. Note that some contexts quote the focal length as a positive value for concave mirrors, referring only to its magnitude, while others preserve the sign to indicate the type of mirror.
In summary, applying the mirror equation with the given u and v leads to a focal length magnitude of approximately 9.78 cm, with the sign indicating a concave mirror.

Determine the focal length of a concave mirror that produces an image 12.2 cm away when an object is placed 49.5 cm in front of it. Short answer

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