Problem 138 - Fundamentals of Physics

Question

A grasshopper hops to a point on the central axis of a spherical mirror. The absolute magnitude of the mirror's focal length is \(40.0 \mathrm{~cm}\), and the lateral magnification of the image produced by the mirror is \(+0.200 .\) (a) Is the mirror convex or concave? (b) How far from the mirror is the grasshopper?

Step-by-Step Solution

Answer

(a) The mirror is convex. (b) The grasshopper is 160.0 cm from the mirror.

Step 1: Identify Mirror Type

The magnification, given by the formula \( m = \frac{-d_i}{d_o} \), is positive. This indicates an upright image, which is characteristic of a virtual image produced by a convex mirror. If it were a concave mirror, the image would be real and inverted unless the object is within the focal length.

Step 2: Use Magnification Formula

Given the magnification \( m = +0.200 \) and knowing for a convex mirror the magnification formula is \( m = \frac{-d_i}{d_o} \), we conclude that \(-d_i = 0.200 \, d_o\). This means \(d_i = -0.200 \, d_o\), indicating a virtual image.

Step 3: Apply Mirror Formula

The mirror formula is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). For a convex mirror, the focal length \( f = -40.0 \, \mathrm{cm} \) and \( d_i = -0.200 \, d_o \). Substitute these into the equation: \( \frac{1}{-40.0} = \frac{1}{d_o} + \frac{1}{-0.200 \, d_o} \).

Step 4: Solve for Object Distance

Combining terms, we get \( \frac{1}{-40.0} = \frac{1}{d_o} - \frac{5}{d_o} \), simplifying to \( \frac{1}{-40.0} = -\frac{4}{d_o} \). Solving for \( d_o \), we find \( d_o = 4 \times 40.0 \), giving \( d_o = 160.0 \, \mathrm{cm} \).

Key Concepts

Spherical Mirror Theory

Spherical mirrors are found in two main types: **concave** and **convex** mirrors. Both have unique characteristics that determine their use in various applications. A concave mirror is curved inward, like the inside of a bowl, and can focus incoming light to a point. This type of mirror forms real and inverted images if the object is placed outside the focal length. However, if the object is within the focal length, the image appears virtual and upright.
On the other hand, a **convex mirror** is curved outward, like the back of a spoon. Convex mirrors diverge light rays, which means they always form a virtual image that is upright and diminished in size, regardless of the object's position. This property makes them useful for purposes like vehicle side mirrors, where a broader field of view is beneficial.
In the context of the exercise, the sign of the magnification and the nature of the image helps determine the type of mirror. A positive magnification indicates that the image is upright, leading to the conclusion that the mirror in question is convex.

Lens Formula

The lens formula (also known as the mirror formula when applied to mirrors) is a crucial equation in optics. It defines the relationship between the **focal length (f)** of a mirror or lens, the distance of the object from the mirror (denoted as **d_o**), and the distance of the image from the mirror (referred to as **d_i**). The formula is written as: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]This equation is vital in determining unknown variables when two of these quantities are known. For convex mirrors, since they produce virtual images, the image distance (**d_i**) is considered negative. Similarly, the focal length (**f**) is also negative as a convention. By substituting these into the formula, one can solve for unknown values such as the object distance, when the focal length and magnification are known, as shown in the original problem exercise.
Understanding this formula allows for solving a wide range of physics problems involving spherical mirrors, enabling predictions of how images will form under different conditions.

Magnification in Optics

Magnification is a measure of how much larger or smaller an image is compared to the object itself. It is represented by the variable **m** and is calculated as:\[ m = \frac{-d_i}{d_o} \]The sign of **m** provides insight into the nature of the image. A positive magnification means the image is upright relative to the object. Conversely, a negative magnification indicates an inverted image. In concave mirrors, depending on the object's position relative to the focal length, the image can be real or virtual, thereby affecting the sign of **m**. In contrast, convex mirrors always yield a positive magnification as they form virtual images that are smaller and upright compared to the actual object.
In the given exercise, the magnification value of **+0.200** indicates that the image is upright and smaller than the object. Such a positive value affirms the virtual nature of the image, aligning with the properties of a convex mirror and confirming the initial assessment of the mirror type.