A convex mirror forms an image of height of 2.4 cm, If the radius of curvature of thr mirror is 40 cm, find the heihgt and distance of the object from the mirror.
Noting
To solve this problem, we can use the mirror formula for a convex mirror, which is given by:
1/f = 1/v - 1/u
Where:
- f is the focal length of the mirror,
- v is the image distance from the mirror (positive for virtual image),
- u is the object distance from the mirror (positive in front of the mirror).
First, we need to find the focal length (f) of the convex mirror. The focal length for a convex mirror is equal to half the radius of curvature (R). Therefore, f = R/2.
Given that the radius of curvature (R) is 40 cm, we can calculate the focal length (f) as follows:
f = R/2 = 40 cm / 2 = 20 cm
Now, let's find the image distance (v). The height of the image formed (h') is given as 2.4 cm. However, for a convex mirror, the image formed is always virtual and diminished.
The magnification formula for a convex mirror is given by:
m = -v/u
Since the image is diminished, the magnification (m) is less than 1. Therefore, when we substitute the given height value (h' = 2.4 cm) in the magnification formula, we get:
m = h'/h = 2.4 cm / h
Where h is the height of the object.
From the given information, we can see that the height of the image is positive (2.4 cm), indicating that it is an upright image. Hence, the magnification is negative, which means it's a virtual image.
By substituting the magnification formula into the mirror formula, we get:
1/f = 1/v - 1/u
Since the image is virtual (v is positive), we can rewrite the formula as:
1/f = 1/(h'/h) - 1/u
Substituting the values we know:
1/20 cm = 1/(2.4 cm / h) - 1/u
Next, we can simplify the formula:
1/20 cm = h/(2.4 cm) - 1/u
We need to find two unknowns: the height of the object (h) and the object distance (u). To eliminate one of the unknowns, we can use a correlation between the magnification (m) and the object/image distances:
m = -v/u
Since m = h'/h and v is positive, we get:
h'/h = -v/u
Substituting the given values:
2.4 cm / h = -2.4 cm / u
Now we have two equations:
1/20 cm = h/(2.4 cm) - 1/u (Equation 1)
2.4 cm / h = -2.4 cm / u (Equation 2)
We can solve this system of equations simultaneously to find the values of h and u.