You hold a spherical salad bowl 60 cm in front of your face with the bottom of the bowl facing you. The bowl is made of polished metal with a 35-cm radius of curvature. Where is the of your 5.0-cm tall nose located?

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Identify the type of mirror: The problem describes a spherical bowl made of polished metal, which acts as a concave mirror. The radius of curvature (R) is given as 35 cm.

Use the mirror equation to find the image distance (\(d_i\)): \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length and \(d_o\) is the object distance. The focal length \(f\) is half the radius of curvature, so \(f = \frac{R}{2} = \frac{35}{2} = 17.5\) cm.

Substitute the known values into the mirror equation: The object distance \(d_o\) is 60 cm (the distance from your face to the bowl). Substitute \(f = 17.5\) cm and \(d_o = 60\) cm into the equation: \(\frac{1}{17.5} = \frac{1}{60} + \frac{1}{d_i}\).

Solve for the image distance \(d_i\): Rearrange the equation to solve for \(\frac{1}{d_i}\): \(\frac{1}{d_i} = \frac{1}{17.5} - \frac{1}{60}\). Calculate \(\frac{1}{d_i}\) and then take the reciprocal to find \(d_i\).

Interpret the result: The sign of \(d_i\) will indicate the nature of the image. A positive \(d_i\) means the image is real and located on the same side as the object, while a negative \(d_i\) indicates a virtual image located on the opposite side of the mirror.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Mirror Equation

The mirror equation relates the object distance (d_o), image distance (d_i), and the focal length (f) of a spherical mirror: 1/f = 1/d_o + 1/d_i. For a concave mirror, the focal length is half the radius of curvature. This equation helps determine the position of the image formed by the mirror.

Radius of Curvature

The radius of curvature is the distance from the mirror's surface to its center of curvature. For spherical mirrors, it is twice the focal length. In this problem, the radius of curvature is given as 35 cm, which is crucial for calculating the focal length and subsequently the image position.

Image Formation by Concave Mirrors

Concave mirrors can form real or virtual images depending on the object's position relative to the focal point. If the object is outside the focal length, a real, inverted image is formed. Understanding this concept helps predict the nature and position of the image of the nose in the bowl.