91Ó°ÊÓ

The mirror formula is a fundamental equation in optics, especially when dealing with concave mirrors. It connects the object distance \( u \, \text{m} \,\), the image distance \( v \, \text{m} \,\), and the focal length \( f \, \text{m} \,\) of a mirror through the relationship: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v}. \] This equation helps us calculate one of these distances when the other two are known. In this exercise, with the object distance \( u = 4.00 \, \text{m} \,\) and image distance \( v = -9.00 \, \text{m} \,\), substituting into the formula resolves to the focal length: \[ \frac{1}{f} = \frac{1}{4.00} + \frac{1}{-9.00}. \] Solving gives \( f = 7.20 \, \text{m} \,\). Knowing this helps in finding critical properties of the mirror.

Magnification is a measure of how much larger or smaller an image is compared to the object itself. For mirrors, the magnification formula is \( m = \frac{-v}{u} \,\), where \( m \,\) is the magnification, \( v \,\) is the image distance, and \( u \,\) is the object distance. In our exercise, the magnification was given as \( 2.25 \,\), meaning the image is \( 2.25 \,\) times the size of the object. To find \( v \,\), we rearrange the formula: \( 2.25 = \frac{-v}{4.00} \, \Rightarrow \, v = -9.00 \, \text{m} \,\). The negative sign here indicates that the image is real and inverted. This relationship helps determine the correct alignment and placement of the mirror.

The radius of curvature \( R \,\) of a mirror is an essential concept in optics. It is the radius of the sphere from which the mirror segment is derived. A concave mirror's radius is important because it determines the mirror's focal length. The relationship between the radius of curvature and the focal length is \( R = 2f \,\). In this exercise, we calculated the focal length \( f = 7.20 \, \text{m} \,\), then found the radius of curvature to be \( R = 2 \times 7.20 = 14.4 \, \text{m} \,\). Knowing the radius of curvature is crucial for understanding how a mirror focuses light and forms images. It tells us about the mirror's shape and its optical power.