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Chapter 25: Problem 35

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?

Short Answer

Expert verified
The object distance is approximately 29.30 cm.

Step by step solution

01

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 \(d_o\).
02

Rearranging the Mirror Equation

The mirror equation can be rearranged to solve for the object distance \(d_o\). The formula becomes: \[ \frac{1}{d_o} = \frac{1}{f} - \frac{1}{d_i} \]
03

Substituting Known Values

Now, substitute the known values into the rearranged equation: \[ \frac{1}{d_o} = \frac{1}{42} - \frac{1}{-97} \]
04

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: \[ \frac{1}{d_o} = 0.02381 + 0.01031 = 0.03412 \]
05

Finding the Object Distance

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

(a) For an inverted image that is in front of a mirror, is the image distance positive or negative and is the image height positive or negative? Explain. (b) Given the image distance, what additional information is needed to determine the focal length? Explain. (c) Given the object and image heights and a statement as to whether the image is upright or inverted, what additional information is needed to determine the object distance? Problem A small statue has a height of \(3.5 \mathrm{~cm}\) and is placed in front of a concave mirror. The image of the statue is inverted, \(1.5 \mathrm{~cm}\) tall, and is located \(13 \mathrm{~cm}\) in front of the mirror. Find the focal length of the mirror.

A concave mirror \((f=45 \mathrm{~cm})\) produces an image whose distance from the mirror is onethird the object distance. Determine (a) the object distance and (b) the (positive) image distance.

A clown is using a concave makeup mirror to get ready for a show and is \(27 \mathrm{~cm}\) in front of the mirror. The image is \(65 \mathrm{~cm}\) behind the mirror. Find (a) the focal length of the mirror and (b) the magnification.

The drawing shows a laser beam shining on a plane mirror that is perpendicular to the floor. The beam's angle of incidence is \(33.0^{\circ} .\) The beam emerges from the laser at a point that is \(1.10 \mathrm{~m}\) from the mirror and \(1.80 \mathrm{~m}\) above the floor. After reflection, how far from the base of the mirror does the beam strike the floor?

A small mirror is attached to a vertical wall, and it hangs a distance \(y\) above the floor. A ray of sunlight strikes the mirror, and the reflected ray forms a spot on the floor. (a) From a knowledge of \(y\) and the horizontal distance \(x\) from the base of the wall to the spot, describe how one can determine the angle of incidence of the ray striking the mirror. If it is morning and the mirror is facing due east, would (b) the angle of incidence and (c) the distance \(x\) increase or decrease in time? Why? Problem Suppose the mirror is \(1.80 \mathrm{~m}\) above the floor. The reflected ray of sunlight strikes the floor at a distance of \(3.86 \mathrm{~m}\) from the base of the wall. Later in the morning, the ray is observed to strike the floor at a distance of \(1.26 \mathrm{~m}\) from the wall. The earth rotates at a rate of \(15.0^{\circ}\) per hour. How much time (in hours) has elapsed between the two observations?
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