A convex mirror has a radius of curvature of 36.0 cm. When a 5.60 cm tall object is placed in front of the mirror, the magnification is 0.250. (a) What is the height of the image? (b) How far is the object from the mirror?
a) Di = (1/4)*5.6 = 1.4 cm
b) Use the equation
1/Do + 1/Di = 1/f = -2/R
Di = -0.25*Do
(A minus sign is needed for Di since the mirror is convex and the image is virtual)
1/Do -4/Do = -3/Do = -1/18
Do = 54 cm
Di = -13.5 cm
To find the height of the image and the distance of the object from the mirror, we can use the mirror equation and the magnification formula.
The mirror equation is given by:
1/f = 1/dO + 1/dI
Where:
- f is the focal length of the mirror
- dO is the distance of the object from the mirror
- dI is the distance of the image from the mirror
The magnification formula is given by:
m = hI / hO
Where:
- m is the magnification
- hI is the height of the image
- hO is the height of the object
Given:
- Radius of curvature (R) = 36.0 cm
- Height of the object (hO) = 5.60 cm
- Magnification (m) = 0.250
First, we need to find the focal length (f) of the mirror using the formula:
f = R / 2
f = 36.0 cm / 2 = 18.0 cm
(a) To find the height of the image (hI), we can rearrange the magnification formula:
m = hI / hO
hI = m * hO
Substituting the given values:
hI = 0.250 * 5.60 cm = 1.40 cm
Therefore, the height of the image is 1.40 cm.
(b) To find the distance of the object from the mirror (dO), we can rearrange the mirror equation:
1/f = 1/dO + 1/dI
Rearranging and substituting the given values:
1/dO = 1/f - 1/dI
1/dO = 1/18.0 cm - 1/dI
We know that at magnification (m = 0.250):
dI = -m * dO
Substituting this into the equation:
1/dO = 1/18.0 cm - 1/(-0.250 * dO)
Multiplying through by dO and simplifying:
dO = 18.0 cm * (-0.250 * dO - 1)
dO = -4.5 cm * dO - 18.0 cm
dO + 4.5 cm * dO = -18.0 cm
(1 + 4.5 cm) * dO = -18.0 cm
5.5 cm * dO = -18.0 cm
dO = -18.0 cm / 5.5 cm
dO ≈ -3.27 cm
Since distance cannot be negative, we take the absolute value:
dO ≈ 3.27 cm
Therefore, the object is approximately 3.27 cm away from the mirror.
To find the height of the image and the distance of the object from the mirror, we can use the mirror formula and the magnification formula.
The mirror formula is given by:
1/f = 1/do + 1/di
Where:
- f is the focal length of the mirror
- do is the distance of the object from the mirror (known)
- di is the distance of the image from the mirror (unknown)
And the magnification formula is given by:
magnification = -di/do
Where:
- magnification is the ratio of the height of the image to the height of the object (known)
- di is the distance of the image from the mirror (unknown)
- do is the distance of the object from the mirror (known)
Now, let's substitute the known values into these formulas:
Given:
Radius of curvature (R) = 36.0 cm (Note: for a convex mirror, R is positive)
Height of the object (ho) = 5.60 cm
Magnification (magnification) = 0.250
(a) To find the height of the image (hi):
We know that magnification = hi/ho
Substituting the given values, we have:
0.250 = hi/5.60
Rearranging the formula:
hi = magnification * ho
hi = (0.250) * (5.60)
hi = 1.40 cm
Therefore, the height of the image is 1.40 cm.
(b) To find the distance of the object from the mirror (do):
We will use the mirror formula:
1/f = 1/do + 1/di
Since it's a convex mirror, the focal length (f) will be negative, given by:
f = -R/2
f = -36.0 cm / 2
f = -18.0 cm
Substituting this value into the formula:
1/-18.0 cm = 1/do + 1/di
Now, rearranging the formula and substituting the known values:
1/do = 1/f - 1/di
1/do = 1/-18.0 cm - 1/di
1/do = -1/18.0 cm - 1/di
From the magnification formula, we know that:
magnification = -di/do
Substituting the given magnification value:
0.250 = -di/do
Solving for di, we get:
di = -do * (0.250)
Substituting this value back into the mirror formula:
1/do = -1/18.0 cm - 1/(-do * 0.250)
1/do = -1/18.0 cm + 4/do
Now, finding a common denominator:
1/do = (-1 + 72/do) / (18.0 cm * do)
Simplifying further:
1/do = (71/do) / (18.0 cm * do)
Cross-multiplying:
do = (18.0 cm * do) / (71/do)
Simplifying:
do^2 = 18.0 cm * 71
do^2 = 1278 cm^2
Taking the square root:
do = √1278 cm
do = 35.77 cm (rounded to two decimal places)
Therefore, the distance of the object from the mirror (do) is approximately 35.77 cm.
To summarize:
(a) The height of the image is 1.40 cm.
(b) The distance of the object from the mirror is approximately 35.77 cm.