A concave shaving mirror has a radius of curvature of 22.0 cm. For each of the following cases, find the magnification and determine whether the image formed is real or virtual and upright or inverted.
(a) an upright pencil placed 41.4 cm from the mirror.
Have you considered a simple sketch of a ray diagram? f=11cm, do=41cm.
I did it in about five seconds.
real, inverted. Do that sketch.
To find the magnification and determine the nature of the image formed by a concave shaving mirror, we can use the mirror formula and magnification formula.
First, let's define the given data:
- Radius of curvature (R) = 22.0 cm (since it's a concave mirror, the radius of curvature is positive)
- Object distance (u) = 41.4 cm (distance of the pencil from the mirror)
Using the mirror formula, we have:
1/u + 1/v = 1/f,
Where:
- v is the image distance (distance of the image from the mirror),
- f is the focal length of the mirror.
For a concave mirror, the focal length (f) is half of the radius of curvature (R):
f = R/2 = 22.0 cm / 2 = 11.0 cm.
Now, we can substitute the given values into the mirror formula to solve for v:
1/41.4 + 1/v = 1/11.0.
Simplifying the equation:
1/v = 1/11.0 - 1/41.4,
1/v = (41.4 - 11.0)/(11.0 * 41.4),
1/v = 30.4/454.8,
v = 454.8/30.4,
v ≈ 14.95 cm (approximated to two decimal places).
Now, let's determine the magnification (m) using the magnification formula:
m = -v/u,
Substituting the given values:
m = -14.95 cm / 41.4 cm,
m ≈ -0.36 (approximated to two decimal places).
The negative sign indicates that the image is inverted.
Now, let's determine whether the image formed is real or virtual and upright or inverted:
Since the magnification (m) is negative, the image is inverted.
To determine whether the image is real or virtual, we can look at the image distance (v). If v is positive, the image is real, and if v is negative, the image is virtual.
In this case, the image distance (v) is positive (14.95 cm), so the image formed is real.
Summary:
The magnification (m) of the image is approximately -0.36, indicating an inverted image. The image formed is real and inverted.