A lamp which is 9.0 cm tall is in front of a concave mirror which has a focal length of 53.0 cm.
(a) How tall is the lamp's image if the lamp is a distance of 26.5 cm from the mirror (include sign if negative)?
(b) How tall is the lamp's image if the lamp is a distance of 63.6 cm from the mirror (include sign if negative)?
(c) How tall is the lamp's image if the lamp is a distance of 111.3 cm from the mirror (include sign if negative)?
To solve these questions, we can use the mirror equation:
1/f = 1/d_i + 1/d_o
where:
f = focal length of the mirror (given as 53.0 cm)
d_i = distance of the image from the mirror (what we need to find)
d_o = distance of the object from the mirror
We can rearrange the equation to solve for d_i:
1/d_i = 1/f - 1/d_o
Now, let's calculate the height of the image in each case:
(a) d_o = 26.5 cm
Using the equation, we have:
1/d_i = 1/53.0 - 1/26.5
1/d_i = 0.0189 - 0.0377
1/d_i = -0.0188
To find d_i, we can take the reciprocal of both sides:
d_i = -1 / 0.0188
d_i ≈ -53.19 cm
Since the negative sign indicates that the image is real and inverted, the height of the lamp's image is 53.19 cm.
(b) d_o = 63.6 cm
Using the equation, we have:
1/d_i = 1/53.0 - 1/63.6
1/d_i = 0.0189 - 0.0157
1/d_i = 0.0032
To find d_i, we can take the reciprocal of both sides:
d_i = 1 / 0.0032
d_i ≈ 312.5 cm
The positive sign indicates that the image is virtual and upright, so the height of the lamp's image is 312.5 cm.
(c) d_o = 111.3 cm
Using the equation, we have:
1/d_i = 1/53.0 - 1/111.3
1/d_i = 0.0189 - 0.0090
1/d_i = 0.0099
To find d_i, we can take the reciprocal of both sides:
d_i = 1 / 0.0099
d_i ≈ 101.0 cm
The positive sign indicates that the image is virtual and upright, so the height of the lamp's image is 101.0 cm.