Problem 35 - Physics

Question

A concave mirror has a focal length of \(42 \mathrm{~cm}\). The image formed by this mirror is \(97 \mathrm{~cm}\) in front of the mirror. What is the object distance?

Step-by-Step Solution

Answer

The object distance is approximately 29.30 cm.

Step 1: Understanding the Mirror Equation

The mirror equation relates the object distance \(d_o\), image distance \(d_i\), and the focal length \(f\) of a mirror: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] Here, \(f = 42 \text{ cm}\), and \(d_i = -97 \text{ cm}\) (negative for real images formed by concave mirrors). We need to find \(d_o\).

Step 2: Rearranging the Mirror Equation

The mirror equation can be rearranged to solve for the object distance \(d_o\). The formula becomes: \[ \frac{1}{d_o} = \frac{1}{f} - \frac{1}{d_i} \]

Step 3: Substituting Known Values

Now, substitute the known values into the rearranged equation: \[ \frac{1}{d_o} = \frac{1}{42} - \frac{1}{-97} \]

Step 4: Calculating the Fractions

First, calculate the right-hand side of the equation: \[ \frac{1}{42} = 0.02381 \text{ and } \frac{1}{-97} = -0.01031 \] Substituting these values, we get: \[ \frac{1}{d_o} = 0.02381 + 0.01031 = 0.03412 \]

Step 5: Finding the Object Distance

Finally, take the reciprocal of \(0.03412\) to find the object distance: \[ d_o = \frac{1}{0.03412} \approx 29.30 \text{ cm} \]

Key Concepts

Mirror Equation

The mirror equation is essential in understanding how mirrors form images. It connects three vital aspects: the focal length ( f ), the object distance ( d_o ), and the image distance ( d_i ). It is expressed as:

• \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]

For a concave mirror, the sign conventions are crucial. The focal length is positive, while the image distance is often negative if the image is real and in front of the mirror. This equation helps determine unknown variables by plugging in the known ones.
For example, if you know the focal length and the image distance, you can find the object distance using this equation. This relationship allows us to solve real-world problems and understand how lenses and mirrors manipulate light.

Object Distance

Object distance, denoted as \(d_o\), is the distance between the object and the mirror. It plays a critical role in determining the characteristics of the image formed by the mirror. By using the mirror equation, we can solve for the object distance if the focal length and the image distance are known.
The calculation involves rearranging the mirror equation to find \(d_o\):

• \[ \frac{1}{d_o} = \frac{1}{f} - \frac{1}{d_i} \]

This step is vital as it allows us to focus on the unknown and solve it easily. Once the fractions are calculated and added, taking the reciprocal gives us the object distance. Knowing \(d_o\) helps us understand where the object should be placed in relation to the mirror to achieve a desired image size and type.
This concept is essential in practical applications like cameras and telescopes, where precise placement is crucial.

Focal Length

The focal length ( f ) of a mirror is the distance between its surface and the focal point, where parallel light rays converge. For a concave mirror, this value is positive.
Understanding and calculating the focal length is significant because it directly affects how the mirror magnifies or reduces the appearance of objects.
For instance, a small focal length indicates a strong converging power, meaning it can form images that are significantly different in size compared to the object. In our example, a focal length of 42 cm helps us pinpoint how the light converges, allowing us to apply the mirror equation effectively.
Identifying the focal length is necessary for designing systems that use mirrors, like headlights, satellite dishes, and solar cookers.