Problem 8 - Applied Physics

Question

Given \(h_{i}=3.50 \mathrm{~cm}, h_{o}=2.50 \mathrm{~cm}\), and \(s_{i}=15.5 \mathrm{~cm}\), find \(s_{o}\).

Step-by-Step Solution

Answer

The object distance \( s_o \) is approximately 11.07 cm.

Step 1: Understand the Magnification Formula

The magnification formula relates the heights and distances of images and objects in a lens or mirror setup. It is given by the equation \( m = \frac{h_i}{h_o} = \frac{s_i}{s_o} \), where \( m \) is the magnification, \( h_i \) is the height of the image, \( h_o \) is the height of the object, \( s_i \) is the distance of the image, and \( s_o \) is the distance of the object.

Step 2: Calculate the Magnification

Using the given heights, calculate the magnification: \[ m = \frac{h_i}{h_o} = \frac{3.50}{2.50} = 1.4 \]This means the magnification is 1.4.

Step 3: Rearrange the Formula to Find Object Distance

Rearrange the formula \( m = \frac{s_i}{s_o} \) to solve for \( s_o \):\[ s_o = \frac{s_i}{m} \]

Step 4: Substitute the Known Values

Substitute the values we have into the equation:\[ s_o = \frac{15.5}{1.4} \]

Step 5: Calculate the Object Distance

Perform the division to find \( s_o \):\[ s_o = \frac{15.5}{1.4} \approx 11.07 \]So, the object distance \( s_o \) is approximately 11.07 cm.

Key Concepts

Magnification Formula

The magnification formula is a fundamental concept in optics that helps describe how images are scaled in size compared to their original objects. Imagine looking through a magnifying glass. This formula tells us how much bigger or smaller the image appears than the actual object. It is expressed as:

• \( m = \frac{h_i}{h_o} = \frac{s_i}{s_o} \)

Here, \( m \) stands for magnification.

\( h_i \) is the height of the image seen.

\( h_o \) is the actual height of the object.

\( s_i \) is the distance from the image to the lens or mirror.

\( s_o \) is the distance from the object to the lens or mirror.

By using this formula, understanding how an object changes its appearance through lenses or mirrors becomes much more straightforward.

Lens and Mirror Equations

In the realm of optics, lenses and mirrors are crucial for shaping how we see images. Their behavior is governed by the lens and mirror equations, which connect the object distance, image distance, and focal length. While we did not explicitly use the lens equation in our exercise, understanding it gives more insight.The basic forms of these equations are:

• For lenses: \( \frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i} \)
• For mirrors: \( \frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i} \)

In these equations:

\( f \) is the focal length of the lens or mirror.

These equations help predict where an image will form and how large it will be, aiding in designing optical instruments and understanding human vision.

Image Distance Calculation

Calculating the image distance, denoted as \( s_i \), is an essential aspect of working with optical systems. This parameter tells us how far away the image appears from the lens or mirror, which is critical for focusing.In our example exercise, the image distance was given as \( s_i = 15.5 \text{ cm} \). Image distance calculation often involves:

• Using the magnification equation: Rearranging \( m = \frac{s_i}{s_o} \) to rely on known values for one variable.
• In complex setups, leveraging the lens or mirror equation where the focal length is also known.

Understanding image distance helps in focusing cameras or telescopes and adjusting lenses correctly.

Object Distance Determination

Determining the object distance, symbolized as \( s_o \), is a key step in many optical calculations. It describes how far the original object is from the lens or mirror.Here’s how we determine it:

• First, calculate the magnification \( m \) using \( \frac{h_i}{h_o} \), which tells how the image size relates to the object size.
• Then, use the rearranged formula \( s_o = \frac{s_i}{m} \) to find the object distance.

In our exercise, substituting in the values gives \( s_o = \frac{15.5}{1.4} \approx 11.07 \text{ cm}\). This practical method allows us to gather vital information about the placement of objects in relation to lenses or mirrors.