Problem 127 - Fundamentals of Physics

Question

A concave mirror has a radius of curvature of \(24 \mathrm{~cm}\). How far is an object from the mirror if the image formed is (a) virtual and \(3.0\) times the size of the object, (b) real and \(3.0\) times the size of the object, and (c) real and \(1 / 3\) the size of the object?

Step-by-Step Solution

Answer

(a) 8 cm, (b) 16 cm, (c) 4 cm.

Step 1: Understand the Mirror Formula

The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \) of a mirror as follows: \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \).

Step 2: Calculate the Focal Length

For a concave mirror, the focal length \( f \) is half of the radius of curvature. Given a radius of curvature of \( 24 \mathrm{~cm} \), the focal length is \( f = \frac{24}{2} = 12 \mathrm{~cm} \).

Step 3: Consider Magnification for (a) Virtual Image

The magnification \( m \) is given by \( m = \frac{v}{u} \). For a virtual image that is 3 times the size of the object, \( m = -3 \). Use this to find the object distance \( u \).

Step 4: Calculate Object Distance for (a)

With \( m = -3 \), we have \( v = -3u \). Substitute into the mirror formula: \( \frac{1}{12} = \frac{1}{u} + \frac{1}{-3u} \), solve for \( u \) resulting in \( u = 8 \mathrm{~cm} \).

Step 5: Consider Magnification for (b) Real Image

For a real image that is 3 times the size of the object, \( m = 3 \). Hence, \( v = 3u \). Use this in the mirror formula.

Step 6: Calculate Object Distance for (b)

Substitute \( v = 3u \) into the mirror formula: \( \frac{1}{12} = \frac{1}{u} + \frac{1}{3u} \), solving for \( u \) results in \( u = 16 \mathrm{~cm} \).

Step 7: Consider Magnification for (c) Real Image

For a real image that is \( \frac{1}{3} \) the size of the object, \( m = \frac{1}{3} \). Hence, \( v = \frac{u}{3} \). Use this in the mirror formula.

Step 8: Calculate Object Distance for (c)

Substitute \( v = \frac{u}{3} \) into the mirror formula: \( \frac{1}{12} = \frac{1}{u} + \frac{3}{u} \), solving for \( u \) results in \( u = 4 \mathrm{~cm} \).

Key Concepts

Mirror Formula

The mirror formula is a key concept in understanding how an object and its image relate to a mirror's properties. This formula is expressed mathematically as \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \), where:

• \( f \) is the focal length of the mirror.
• \( u \) is the object distance, i.e., how far the object is from the mirror.
• \( v \) is the image distance, or how far the image appears from the mirror.

Using this equation allows us to determine one quantity if two are known. For example, knowing the focal length and object distance, we can calculate the image distance. The signs of \( u \) and \( v \) are important, based on the convention that distances are positive if they are in front of the mirror and negative if behind the mirror.

Focal Length

The focal length of a mirror, symbolized by \( f \), is the distance from the mirror to the focal point, where light converges or appears to diverge. For a concave mirror, this point is in front of the mirror. The focal length is directly influenced by the radius of curvature \( R \), which is the radius of the sphere of which the mirror forms a part. It is exactly half the radius:

• \( f = \frac{R}{2} \)

For instance, if the radius of curvature is \( 24 \, \text{cm} \), then \( f = \frac{24}{2} = 12 \, \text{cm} \). Knowing this helps us apply the mirror formula effectively to solve problems related to mirror imaging.

Magnification

Magnification gives us an idea of how much larger or smaller the image is compared to the original object. It is represented by \( m \), and can be calculated with the formula \( m = \frac{v}{u} \), where both \( v \) and \( u \) relate to their respective distances.

• If \( m \) is positive, the image is upright and virtual.
• If \( m \) is negative, the image is inverted and real.

In the problem, if a virtual image is three times as large as the object, then \( m = -3 \), indicating image size and orientation. For real images of different sizes, use \( m = 3 \) for larger and \( m = \frac{1}{3} \) for smaller real images.

Radius of Curvature

The radius of curvature \( R \) of a mirror is vital to understanding its reflecting properties. It represents the radius of the entire spherical surface of which the mirror is a segment.

• A small radius indicates a tightly curved surface, leading to a shorter focal length.
• A larger radius means a less curved mirror, leading to a longer focal length.

As such, \( R \) directly influences the focal length by the relationship \( f = \frac{R}{2} \). Knowing \( R \) allows calculations regarding how light will reflect and focus, essential for determining distances in mirror equations.