A concave mirror has a radius of curvature of 3.4 cm.
(a) Determine the object position for which the resulting image is upright and seven times the size of the object.
cm?
(b) Draw a ray diagram to determine the position of the image. (Use a negative number if the image is behind the mirror.)
cm from the mirror?
To determine the object position for which the resulting image is upright and seven times the size of the object, we can use the mirror formula:
1/f = 1/v - 1/u
where f is the focal length of the mirror, v is the image position, and u is the object position.
Given that the radius of curvature (R) is twice the focal length, we have:
R = 2 * f => f = R/2
Substituting this value into the mirror formula, we get:
1/(R/2) = 1/v - 1/u
Simplifying the equation:
2/R = 1/v - 1/u
Now, we can substitute the given values:
R = 3.4 cm
Plugging this into the equation:
2/3.4 = 1/v - 1/u
Simplifying further:
0.588 = 1/v - 1/u
Given that the resulting image is upright and seven times the size of the object, we have:
v/u = -7
Substituting this value into the equation:
0.588 = 1/(-7u) - 1/u
Now, we replace 1/u with x to simplify the equation:
0.588 = x - 1/7x
To solve this equation, we can multiply through by 7x to eliminate the fractions:
4.116x = 7x - 1
Rearranging the equation:
7x - 4.116x = 1
2.884x = 1
x = 1 / 2.884
x ≈ 0.346 cm
Therefore, the object position for which the resulting image is upright and seven times the size of the object is approximately 0.346 cm.
Moving on to part (b), to draw a ray diagram to determine the position of the image, we can use the following steps:
1. Draw a vertical line representing the concave mirror.
2. Mark the center of the mirror on the line.
3. Draw an arrow perpendicular to the line to represent the object.
4. From the tip of the object arrow, draw a straight line towards the mirror parallel to the principal axis.
5. At the point where the line intersects the mirror, draw a line perpendicular to the mirror (normal).
6. From the same point, draw a line towards the center of curvature (twice the radius of curvature).
7. The point where the line intersects the principal axis is the position of the image.
In this case, since the object is upright and the image is upright, the image will be on the same side of the mirror as the object. Therefore, the image position will be a positive value.
Using the value we obtained for the object position (0.346 cm), we can draw the ray diagram and determine the position of the image.