A metallic,spherical Christmas tree ornament has a diameter of 16.00 cm.If Saint Nicholas is by the fireplace,0.720 m away,where will he see his reflection relative to the surface of the ornament?

Question

Quizplus · Accepted Answer

The reflection in a spherical surface can be analyzed using the mirror equation for a spherical mirror, $ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $, where $f$ is the focal length, $d_o$ is the object distance, and $d_i$ is the image distance. For a spherical mirror, the focal length $f$ is half the radius of the sphere. Given the diameter of the ornament is 16.00 cm, the radius $r$ is 8.00 cm, making the focal length $f = \frac{r}{2} = 4.00$ cm. Since the ornament is convex for an observer outside, we consider the focal length as negative for a convex mirror. Saint Nicholas is 0.720 m (or 72.0 cm) away, so $d_o = 72.0$ cm, and we consider this distance as positive. Plugging these into the mirror equation with the correct signs gives:$$ \frac{1}{-4.0} = \frac{1}{72.0} + \frac{1}{d_i} $$Solving for $d_i$ gives the image distance, which will be negative, indicating that the image is virtual and appears on the same side of the mirror as the object (but still appears to be behind the mirror). Solving the equation yields $d_i \approx -3.8$ cm, meaning the image appears 3.8 cm behind the surface of the ornament, from the perspective of Saint Nicholas.

A Metallic,spherical Christmas Tree Ornament Has a Diameter of 16