In a subway station, a convex mirror allows the attendant to...

In a subway station, a convex mirror allows the attendant to view activity on the platform. A woman $1.64 \mathrm{m}$ tall is standing $4.5 \mathrm{m}$ from the mirror. The image formed of the woman is $0.500 \mathrm{m}$ tall. (a) What is the radius of curvature of the mirror? (b) The mirror is $0.500 \mathrm{m}$ in diameter. If the woman's shoes appear at the bottom of the mirror, does her head appear at the top-in other words, does the image of the woman fill the mirror from top to bottom? Explain.

Key Concepts

Magnification

Magnification, defined as the ratio of the image height to the object height (m = hi/ho), provides insight into the relative sizes of the image and the object. In mirror problems, it helps determine how changes in object distance affect the size of the produced image, and it also indicates the image's orientation with respect to the object.

Image Coverage on Mirror Surface

This concept involves understanding whether the entirety of an image produced by a mirror fills the available reflective surface. It requires analyzing the geometry of the image formation to determine if an object's features (such as the top and bottom) are fully reflected within the mirror's physical boundaries, which is important when assessing the effectiveness of mirror placement in practical applications.

Convex Mirrors

Convex mirrors are curved outward and always produce virtual, upright, and diminished images, regardless of the object's position. Their geometry offers a wide field of view, making them ideal for applications like security and rear-view mirrors. In solving problems, understanding that convex mirrors diverge light rays is key to determining image location and size.

Mirror Equation

The mirror equation (1/f = 1/do + 1/di) is central to geometrical optics, linking the focal length of a mirror with the distances of the object and the image from the mirror. It enables us to calculate unknown distances and is applicable to both concave and convex mirrors, with appropriate sign conventions.

Radius of Curvature

The radius of curvature of a mirror is twice its focal length and is a measure of the mirror’s bending. It determines how strongly the mirror converges or diverges light and directly influences the properties of image formation. Calculating the radius of curvature is essential in designing optical systems and analyzing their performance.

Transcript

00:01 Okay, so there's some information that's given to us in this problem.

00:05 The height of the woman is 1 .64 meters.

00:08 She is positioned 4 .5 meters from the mirror.

00:11 The height of her image is 0 .5 meters.

00:15 And the diameter of the mirror is 0 .5 meters.

00:19 So we want to figure out where, i'm sorry, with a radius of curvature is for this mirror.

00:28 And then we want to know if the woman's entire body is visible in the mirror.

00:35 So to get started, in order to get the radius, we need the focal length.

00:40 So we're going to need to solve for the focal length first using the magnification relation.

00:46 And that is negative q over p equal to the height of the image over the height of the object.

00:55 We can plug in our values and solve for q.

00:59 We're doing this because we need to know the distance of the image in order to solve for the focal length.

01:05 We're given the placement of the position, i'm sorry, the placement of the object, the height of the image, and the height of the object.

01:16 And then we can rearrange it to solve for q.

01:27 And when we calculate that out, we find that the image is located 1 .37 meters behind the mirror.

01:36 So negative 1 .37 meters...

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