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

The mirror is convex and the grasshopper is 32 cm away from the mirror.

Step 1: Understanding the Mirror Type

The lateral magnification (\(m\)) is positive, indicating an upright image. Therefore, the mirror is convex, because convex mirrors always produce virtual, upright, and diminished images.

Step 2: Using the Magnification Formula

The formula for magnification is \(m = -\frac{d_i}{d_o}\). Given \(m = +0.200\), it implies that the image distance \(d_i\) is negative because the image is virtual. Hence, we have \( +0.200 = -\frac{d_i}{d_o}\).

Step 3: Finding the Mirror Formula

The mirror formula is \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\). Since \(f = -40.0\, \text{cm}\) (negative for convex mirrors), we can substitute this into the equation.

Step 4: Determining Image Distance

As derived from the magnification formula in Step 2, we know \(d_i = -0.200 \cdot d_o\). Substitute \(d_i\) into the mirror equation in Step 3: \(\frac{1}{f} = \frac{1}{d_o} - \frac{0.200}{d_o}\).

Step 5: Simplifying the Equation

The equation becomes \(\frac{1}{d_o} - \frac{0.200}{d_o} = \frac{1-0.200}{d_o} = \frac{0.800}{d_o}\). So, \(\frac{1}{-40} = \frac{0.800}{d_o}\).

Step 6: Finding Distance from the Mirror

Solving for \(d_o\), we get \(d_o = 0.800 \times 40 = 32.0\, \text{cm}\). Therefore, the grasshopper is 32 cm away from the mirror.

Key Concepts

Convex Mirrors

Convex mirrors are uniquely suited for a variety of applications due to their special properties. These mirrors are curved outward, like the exterior of a sphere. They have the remarkable ability to form images that are always virtual, upright, and reduced in size.
This property makes them particularly useful in situations where a broader field of view is needed, such as in vehicle side mirrors or surveillance.
- **Curved Surface**: The outward bulge allows for the collection and reflection of light over a wide area. - **Virtual Images**: Unlike other mirror types, convex mirrors always produce images that seem to be coming from a location inside the mirror. - **Upright and Reduced**: The images formed are reduced in size and upright compared to the object. They help in understanding spatial proportions without distortion.
All these characteristics stem from the fact that the focal point and center of curvature are imaginary, lying behind the mirror's surface."

Lateral Magnification

Lateral magnification (\(m\)) in mirrors deals with the size and orientation of the image relative to the object. When we talk about lateral magnification being positive or negative, we're also talking about whether the image is upright or inverted.
The formula for lateral magnification is given by:\[m = -\frac{d_i}{d_o}\\]- **Upright Image**: A positive magnification implies an upright image, as seen in convex mirrors.- **Value Representation**: Here, with \(m = +0.200\), the image is one-fifth the size of the object, maintaining the original orientation.- **Real-life Implications**: For every 1 cm of object height, the image is 0.2 cm in size. This reduction helps in fitting larger scenes within a smaller view, assisting in making safe judgments in real-world scenarios."

Mirror Formula

The mirror formula is a fundamental recipe that relates the object distance \(d_o\), the image distance \(d_i\), and the focal length \(f\) of a spherical mirror.The formula is expressed as:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\\]This equation holds true for both concave and convex mirrors by accounting for their geometric properties and sign conventions. Here's how it works:
- **Determining Image and Focal Lengths**: By knowing any two of these variables, you can solve for the third.- **Application in Convex Mirrors**: The focal length (\(f\)) is negative, as these mirrors are divergent and the focal point is hypothetical.- **Understanding Interactions**: The mirror formula guides us in predicting how rays will converge or diverge, forming distinct images based on the principal axis.
It provides crucial support in mathematical predictions and practical applications for optics."

Virtual Images

Virtual images are a distinctive aspect of spherical mirrors, particularly convex mirrors. Unlike real images, these cannot be projected onto a screen as they appear to be located behind the mirror. - **Formation in Convex Mirrors**: In these mirrors, the reflected light rays appear to diverge from a single point. The brain perceives these rays as originating from the mirror’s own surface, forming a virtual image. - **Characteristics**: Virtual images are always upright and reduced in size. They offer clarity in vision by providing an unobstructed, wide-angle view. - **Uses in Daily Life**: They are crucial in applications such as rearview mirrors of cars, where reduced images help drivers gain a broad perspective of the traffic situation behind. Being virtual doesn’t diminish their importance. They are a cornerstone in safety and precision in everyday technologies."