An object 6.70 cm high is placed 15.1 cm in front of a convex mirror with radius of curvature of 8.55 cm. Where is the image formed?
1/q + 1/p = 1/f f=r/2 p=15.1
f= 8.55/2= 4.275
1/q= 1/f+1/p= 1/4.275 + 1/15.1 = 19.375/64.5525
take inverse to get q
q= 3.33
do=15.1 cm, R=8.55 cm, H= 6.7 cm
di= ? h=?
1/do – 1/di = -2/R
1/di = 1/do+2/R = 1/15.1 +2/8.55.
di= 3.33 cm
The virtual image is smaller and closer to the mirror than the object
h/di =H/do =>
h=H•di/do = 6.7•3.33/15.1 =1.48 cm
To determine the position of the image formed by a convex mirror, you can use the mirror equation:
1/f = 1/d₀ + 1/dᵢ
where:
f is the focal length of the convex mirror,
d₀ is the object distance (distance of the object from the mirror), and
dᵢ is the image distance (distance of the image from the mirror).
In this case, the radius of curvature (R) is given, and for a convex mirror, the focal length (f) is half the radius of curvature:
f = R/2
Substituting the values:
R = 8.55 cm
f = 8.55 cm / 2 = 4.275 cm
Given:
h₀ (object height) = 6.70 cm
d₀ (object distance) = -15.1 cm (negative value since the object is in front of the mirror)
To find the image distance (dᵢ), we can rearrange the mirror equation:
1/dᵢ = 1/f - 1/d₀
Substituting the given values:
1/dᵢ = 1/4.275 - 1/(-15.1)
Simplifying the equation:
1/dᵢ = 0.2337 + 0.06622
1/dᵢ = 0.29992
Taking the reciprocal of both sides:
dᵢ = 1/0.29992
dᵢ = 3.334 cm
Since the image distance (dᵢ) is positive, the image is formed on the same side as the object. Therefore, the image is formed 3.334 cm from the convex mirror.
To determine where the image is formed by a convex mirror, we can use the mirror equation:
1/f = 1/d₀ + 1/dᵢ
Where:
f = focal length of the mirror
d₀ = object distance (distance of the object from the mirror)
dᵢ = image distance (distance of the image from the mirror)
In this case, we are given:
d₀ = -15.1 cm (negative because the object is placed in front of the mirror)
f = R/2 = 8.55 cm / 2 = 4.275 cm (where R is the radius of curvature)
To find the image distance, we need to rearrange the mirror equation:
1/dᵢ = 1/f - 1/d₀
Let's substitute the values into the equation:
1/dᵢ = 1/4.275 - 1/-15.1
Now, we can calculate the image distance by taking the reciprocal of the sum on the right side of the equation:
1/dᵢ = (1/4.275 + 1/15.1)^-1
Calculating this expression will give us:
1/dᵢ ≈ 0.415
Taking the reciprocal once again, we find:
dᵢ ≈ 2.41 cm
Therefore, the image is formed approximately 2.41 cm behind the convex mirror.