A jeweler inspects a watch with a diameter of 3.6 cm by placing it 8.0 cm in front of a concave mirror of 12.6 cm focal length.
(a) Where will the image of the watch appear?
(b) What will be the diameter of the image?
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To answer these questions, 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 distance from the mirror (positive for a real image, negative for a virtual image),
- u is the object distance from the mirror (positive for an object in front of the mirror, negative for an object behind the mirror).
Let's calculate the answers step by step:
(a) To find the position of the image, we need to calculate the image distance (v). Given the object distance (u = -8.0 cm) and the focal length (f = -12.6 cm for a concave mirror):
1/f = 1/v - 1/u
1/-12.6 = 1/v - 1/-8.0
Solving this equation will give us the value of v, which represents the image distance. To make the calculation easier, we can take the reciprocal of both sides:
-12.6 = v/8.0 - 1
Rearranging the equation:
v/8.0 = -12.6 + 1
v/8.0 = -11.6
Now, multiply both sides by 8.0 to obtain v:
v = -11.6 * 8.0
v = -92.8 cm
Therefore, the image of the watch will appear at a distance of -92.8 cm from the mirror. The negative sign indicates that the image is formed on the same side as the object (in front of the mirror), which means it is a real image.
(b) To find the diameter of the image, we can use the magnification formula:
magnification (m) = -v/u
The negative sign indicates that the image is inverted compared to the object.
Calculating the magnification:
m = -v/u
m = -(-92.8 cm)/(-8.0 cm)
m = 11.6 cm/8.0 cm
m = 1.45
Now, we can use the magnification to find the diameter of the image. Since the diameter of the object is 3.6 cm, the diameter of the image (d') is:
d' = m * d
d' = 1.45 * 3.6 cm
d' = 5.22 cm (rounded to two decimal places)
Therefore, the diameter of the image of the watch will be approximately 5.22 cm.