Consider a concave mirror with radius R. An

upright object is placed in between the interval R/2 and R.
R

To understand the behavior of a concave mirror when an upright object is placed between the intervals R/2 and R, let's break down the steps:

Step 1: Understanding the concave mirror
A concave mirror is a mirror with a reflecting surface that curves inward, resembling a portion of the inner surface of a sphere. The center of the sphere, which the mirror is a part of, is known as the center of curvature (C).

Step 2: Radius of curvature (R)
The radius of curvature (R) is the distance between the center of curvature (C) and the reflecting surface of the mirror. It determines the strength and the focal length of the mirror.

Step 3: Object position
The upright object is placed between the intervals R/2 and R. This means that the distance of the object from the mirror's vertex is greater than R/2 but less than R.

Step 4: Real or virtual image
To determine whether the image formed by the concave mirror is real or virtual, we need to consider the object's position in relation to the focal point (F) of the mirror. The focal point is located at a distance of R/2 from the vertex of the mirror.

- If the object is placed between the mirror and the focal point (F), i.e., at a distance less than R/2, the image formed will be virtual, erect, and magnified.
- If the object is placed between the focal point (F) and the center of curvature (C), i.e., at a distance greater than R/2 but less than R, the image formed will be real, inverted, and magnified.
- If the object is placed beyond the center of curvature (C), the image will be real, inverted, and diminished.

Therefore, based on the object's position between the intervals R/2 and R, the image formed would be real, inverted, and magnified.

To understand the behavior of the concave mirror with a radius of R, we need to consider the different cases based on the placement of the object.

Case 1: Object placed at R/2
When an upright object is placed at a distance of R/2 from the concave mirror, we can use the mirror formula to find the image distance and the magnification of the image.
The mirror formula is given by:

1/f = 1/v - 1/u

Where:
- f is the focal length of the mirror
- v is the image distance
- u is the object distance

In this case, the object is placed at a distance of R/2, so u = R/2.

For a concave mirror, the focal length is positive. The focal length for a mirror with a radius R is given by:

f = R/2

Substituting the values into the mirror formula, we get:

1/(R/2) = 1/v - 1/(R/2)

Simplifying the equation, we find:

2/R = 1/v - 2/R

Combining like terms:

2/R + 2/R = 1/v

4/R = 1/v

v = R/4

The image distance is R/4.

Using the magnification formula:

m = -v/u

Substituting the values, we get:

m = -(R/4) / (R/2)

Simplifying:

m = -1/2

So, the magnification of the image is -1/2. The negative sign indicates that the image is inverted with respect to the object's upright position. The magnitude of 1/2 tells us that the image is half the size of the object.

Case 2: Object placed at R
When the object is placed at the distance of R, it is at the center of curvature of the mirror. In this case, the image is formed at the same position as the object, but it is inverted and of the same size.

So, the image distance, v, is equal to the object distance, u, which is R.
The magnification, m, is equal to -1, indicating that the image is inverted.

In both cases, the object is upright, but the image formed is inverted due to the nature of a concave mirror.