A 3.5-cm tall object is placed 10.0 cm in front of a concave mirror having a focal length of 14.5 cm. Find the location position and height by drawing a ray diagram to scale. Verify your answer using the mirror and magnification equations.
cm (location)
cm (height)
To find the position and height of the image formed by a concave mirror using a ray diagram, follow these steps:
Step 1: Draw the Principal Axis
Draw a straight line in the center of your paper, representing the principal axis of the mirror. This line should be perpendicular to the mirror's surface.
Step 2: Draw the Mirror
Draw the concave mirror, making sure the center of the mirror coincides with the principal axis.
Step 3: Locate the Object
Measure 10.0 cm from the center of the mirror along the principal axis, and mark it as the location of the object. Draw an arrowhead pointing toward the mirror to represent the object.
Step 4: Draw Incident Ray
Draw a straight line from the top of the object (the arrowhead) to the mirror. The incident ray should be drawn in a way that it passes through the center of the mirror.
Step 5: Draw Normal Line
At the point where the incident ray intersects the mirror, draw a perpendicular line (known as the normal line) that touches the mirror's surface.
Step 6: Draw Reflected Ray
From the point where the incident ray intersects the mirror, draw a reflected ray. According to the laws of reflection, this ray should bounce off the mirror at the same angle as the incident ray but in the opposite direction.
Step 7: Locate the Intersection Point
Extend the reflected ray behind the mirror, and locate the point where it intersects with the principal axis. This point represents the location of the image formed by the mirror.
Step 8: Measure the Image Height
Measure the distance from the principal axis to the top of the image. This distance represents the height of the image.
Now, to verify the answer using the mirror and magnification equations, take the following steps:
1. Use the mirror equation:
1/f = 1/di + 1/do
Where:
- f is the focal length (14.5 cm),
- di is the image distance (unknown),
- do is the object distance (-10.0 cm, since the object is located in front of the mirror)
Rearrange the equation to solve for di:
1/di = 1/f - 1/do
1/di = 1/14.5 - 1/-10.0
1/di = 0.069 - (-0.1)
Simplifying further, we get:
1/di = 0.169
Taking the reciprocal of both sides, we obtain:
di = 5.92 cm (rounded to two decimal places)
2. Use the magnification equation:
hi/ho = -di/do
Where:
- hi is the image height (unknown),
- ho is the object height (3.5 cm),
- di is the image distance (5.92 cm),
- do is the object distance (-10.0 cm)
Substituting the known values into the equation, we have:
hi/3.5 = -5.92/-10.0
Simplifying further, we get:
hi/3.5 = 0.592
Cross-multiplying, we obtain:
hi = 2.072 cm (rounded to three decimal places)
Therefore, the position of the image is approximately 5.92 cm from the mirror, and the height of the image is approximately 2.072 cm.