An object, 12 cm tall, is placed 8.0 cm in front of a convex mirror with a radius of curvature of 7.0 cm.
Use the mirror formula to determine the position and height of the image.
To determine the position and height of the image using the mirror formula, we can use the following steps:
Step 1: Identify the given values:
- Object height (h_o) = 12 cm (positive, since it is above the principal axis)
- Object distance (d_o) = -8.0 cm (negative, since it is in front of the convex mirror)
- Radius of curvature (R) = 7.0 cm (positive)
Step 2: Calculate the image distance (d_i) using the mirror formula:
1/f = 1/d_o + 1/d_i
Since the convex mirror has a negative focal length, we can use the formula for the image distance:
1/f = -1/d_i
-1/7.0 cm = -1/8.0 cm + 1/d_i
Simplifying the equation:
-1/7.0 cm + 1/8.0 cm = 1/d_i
(8.0 - 7.0)/56.0 cm = 1/d_i
1/56.0 cm = 1/d_i
So, the image distance (d_i) is 56.0 cm.
Step 3: Determine the image height (h_i) using the magnification formula:
magnification (m) = h_i/h_o = -d_i/d_o
Using the given values:
m = h_i/12 cm = -56.0 cm / (-8.0 cm)
Simplifying the equation:
h_i / 12 cm = 7
h_i = 7 * 12 cm
h_i = 84 cm
Therefore, the position of the image is 56.0 cm behind the mirror, and the height of the image is 84 cm.
To determine the position and height of the image using the mirror formula, we need to use the following formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the convex mirror (which can be calculated as f = R/2, where R is the radius of curvature),
- v is the position of the image,
- u is the position of the object.
Given values:
- Object height (h_o) = 12 cm
- Object position (u) = -8.0 cm (since the object is placed in front of the mirror, the distance is negative)
- Radius of curvature (R) = 7.0 cm
To determine the position and height of the image, we will follow these steps:
1. Calculate the focal length (f):
f = R/2 = 7.0 cm / 2 = 3.5 cm
2. Substitute the given values into the mirror formula:
1/f = 1/v - 1/u
Substituting the values, we get:
1/3.5 = 1/v - 1/-8.0
Simplifying, we get:
1/3.5 = 1/v + 1/8.0
3. Solve for v:
To solve the equation, we multiply both sides by 3.5 * 8.0 (which is the common denominator):
8.0(1) = 3.5(1/v + 1/8.0)
Simplifying further, we get:
8.0 = 3.5 + 0.4375/v
Rearranging the equation, we have:
0.4375/v = 8.0 - 3.5
Simplifying:
0.4375/v = 4.5
Multiplying both sides by v, we get:
0.4375 = 4.5v
Dividing both sides by 4.5, we get:
v = 0.4375 / 4.5 ≈ 0.097 cm
Therefore, the position of the image (v) is approximately 0.097 cm.
4. Calculate the height of the image (h_i):
To calculate the height of the image, we can use the magnification formula: h_i / h_o = -v / u
Substituting the given values:
h_i / 12 = -0.097 / -8.0
Simplifying, we get:
h_i / 12 = 0.012125
Multiplying both sides by 12, we get:
h_i = 0.012125 * 12
Simplifying further:
h_i ≈ 0.1455 cm
Therefore, the height of the image (h_i) is approximately 0.1455 cm.
So, the position of the image is approximately 0.097 cm and the height of the image is approximately 0.1455 cm.