An image formed by a convex mirror with a radius of curvature of 48 cm has a magnification of 0.150. Which way and by how much should the object be moved to double the size of the image?
To determine how the object should be moved to double the size of the image formed by a convex mirror, we can use the mirror formula and the magnification formula.
Here are the steps to find the answer:
1. Let's start by using the mirror formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the convex mirror
- v is the image distance (distance of the image from the mirror)
- u is the object distance (distance of the object from the mirror)
In this case, the radius of curvature (R) is twice the focal length, so f = R/2 = 48 cm / 2 = 24 cm.
2. The magnification formula is given by:
magnification (m) = -v/u
In this case, the magnification (m) = 0.150.
3. We can rearrange the magnification formula to find the image distance (v):
v = -m * u
4. Substitute the given values into the magnification formula:
-0.150 = -v/u
5. Solve for v:
v = (0.150) * u
6. Now, we want to double the size of the image. This means the new magnification (m') will be twice the initial magnification (m):
m' = 2 * m
m' = 2 * 0.150
m' = 0.300
7. Substitute the new magnification into the magnification formula:
0.300 = -v'/u
Where v' is the new image distance.
8. Since we want to double the size, the new image distance (v') should be twice the initial image distance (v):
v' = 2 * v
v' = 2 * (0.150 * u)
v' = 0.300 * u
9. Substitute this value of v' into the magnification formula with the new magnification:
0.300 = -(0.300 * u) / u
10. Simplify the equation:
0.300 = -0.300
The equation simplifies to 0.300 = -0.300, which means that the equation is true for any value of u. Therefore, the object can be moved in any direction and by any distance, and the size of the image will still be doubled.
Thus, to double the size of the image formed by the convex mirror with a magnification of 0.150, the object can be moved in any direction and by any distance.