A light bulb is placed 21.7 cm in front of a concave mirror. When the concave mirror is replaced by a plane mirror in the same location, the bulb's image moves 4.60 cm closer to the mirror. Calculate the focal length of the concave mirror.
97.4
To solve this problem, we can use the mirror formula, which states:
1/f = 1/v - 1/u
where:
f is the focal length of the mirror,
v is the image distance,
u is the object distance.
Let's start by finding the object distance, u, for the concave mirror.
Since the light bulb is placed 21.7 cm in front of the concave mirror, u is equal to -21.7 cm (as per the convention where distances in front of a mirror are positive, and distances behind the mirror are negative).
Now, let's find the image distance, v, for the concave mirror.
When the concave mirror is replaced by a plane mirror, the image moves 4.60 cm closer to the mirror. This means that v is equal to -4.60 cm.
Substituting these values into the mirror formula, we have:
1/f = 1/v - 1/u
1/f = 1/(-4.60 cm) - 1/(-21.7 cm)
Now, let's simplify this equation further.
Note that when we add or subtract fractions with different signs, we take the sign of the larger fraction.
1/f = -21.7 cm/(-4.60 cm * -21.7 cm) - (-4.60 cm)/(-4.60 cm * -21.7 cm)
1/f = -21.7 cm - (-4.60 cm)/(4.60 cm * 21.7 cm)
1/f = 21.7 cm + 4.60 cm/(4.60 cm * 21.7 cm)
1/f = 26.3 cm/(4.60 cm * 21.7 cm)
Now, let's calculate the value of f.
Dividing both sides of the equation by 26.3 cm, we get:
1/f = 1/(4.60 cm * 21.7 cm)
To find the value of f, we can take the reciprocal of both sides:
f = (4.60 cm * 21.7 cm)/1
f = 99.62 cm
Therefore, the focal length of the concave mirror is approximately 99.62 cm.