Problem 3 - University Physics with Modern Physics

Question

A pencil that is 9.0 cm long is held perpendicular to the surface of a plane mirror with the tip of the pencil lead 12.0 cm from the mirror surface and the end of the eraser 21.0 cm from the mirror surface. What is the length of the image of the pencil that is formed by the mirror? Which end of the image is closer to the mirror surface: the tip of the lead or the end of the eraser?

Step-by-Step Solution

Answer

The image of the pencil is 9.0 cm long, and the tip of the lead is closer to the mirror.

Step 1: Understand the scenario

We have a pencil that is 9.0 cm long, positioned perpendicularly to a plane mirror. The tip of the pencil is 12.0 cm from the mirror, and the eraser is 21.0 cm from it. We need to find how the mirror's reflection affects the image of the pencil.

Step 2: Calculate Distance from Mirror for Each End

The tip of the pencil lead is 12.0 cm from the mirror, and the end of the eraser is 21.0 cm from the mirror. In a plane mirror, each point of the object reflects a similar distance onto the opposite side of the mirror. Hence, the distances for both the tip and eraser in the image will be the same as in reality.

Step 3: Determine Image Length

In a plane mirror, the image formed is of the same size as the object. So, the length of the image of the pencil will be equal to the pencil itself, which is 9.0 cm.

Step 4: Determine which end of the image is closer

Since the image replicates the real object, the point on the image corresponding to the pencil lead will be at 12.0 cm away on the other side. Likewise, the point corresponding to the eraser will be 21.0 cm away. Therefore, the tip of the lead, when reflected, is closer to the mirror on the image's side too.

Key Concepts

Plane Mirror

A plane mirror is a flat, reflective surface. It is commonly used in household mirrors and other places where we need to see a clear and undistorted image of an object.
In a plane mirror, light rays strike the mirror and then bounce back, creating a reflection. This reflection is called the virtual image. Unlike curved mirrors, which can distort reflections, a plane mirror always produces an image that is the same size as the object.

• Flat reflective surface
• Produces a virtual image
• Images are not distorted
• Image size equals object size

Image Formation

When discussing image formation with plane mirrors, it's important to know how reflections behave. In a plane mirror, the image formed is a virtual image. This means the image can't be projected on a screen, as light doesn't actually pass through the virtual image's location.
The image is laterally inverted, meaning left and right are swapped in the image. Yet, the size of the image remains the same as the actual object. This characteristic of plane mirrors allows you to see a clear representation of the object without size distortion.

• Virtual and laterally inverted
• Can't be projected onto a screen
• Size of image equals size of object

Reflection

The concept of reflection is crucial in understanding how plane mirrors work. Reflection occurs when light rays hit a surface and bounce back. In plane mirrors, this is referred to as specular reflection.
Here, the incident angle (angle at which the light hits the mirror) equals the reflection angle (angle at which light leaves the mirror). Because of this equal angle principle, the image formed by a plane mirror appears directly opposite the object, maintaining its orientation but laterally inverted.

• Light bounces back from surfaces
• Specular reflection in plane mirrors
• Incident angle equals reflection angle
• Images appear directly opposite

Distance Calculation

When calculating distances in mirror reflections, an important principle is that the image appears as far behind the mirror as the object is in front.
For instance, if an object is 12 cm from the mirror surface, the corresponding image will appear 12 cm on the opposite side. This symmetry helps determine the position of any point of the image relative to the mirror.
In the exercise example, the tip of the pencil is 12.0 cm and the eraser is 21.0 cm from the mirror. Consequently, the image of the pencil tip will appear 12.0 cm behind the mirror and the image of the eraser 21.0 cm behind the mirror. This understanding explains why both ends maintain their respective distances in their mirrored image.

• Image distance equals object distance
• Symmetrical reflection principles
• Calculations apply equally to all points