Question

9, 11, 13 Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance$\mathit{p}\mathit{s}$(centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $\mathit{r}$(including sign), (b) the image distance $\mathit{i}$, and (c) the lateral magnification m. Also, determine whether the image is (d) real $\left(R\right)$or virtual $\left(V\right)$, (e) inverted $\left(I\right)$from object O or non-inverted$\left(NI\right)$, and (f) on the same side of the mirror as O or on the opposite side.

Short answer

(a) The radius of curvature is $r=+72 \text{cm}$.

(b) The image distance is $i=-72 \text{cm}$.

(c) Lateral magnification is $m=+3.0$.

(d) The image is virtual$\left(V\right).$

(e) The image is non-inverted$\left(NI\right).$

(f) The image is on the opposite side of the object

Step-by-step solution

Step 1: Identification of the given data

$f=36$

The mirror is concave.

Determining the concept

The object distance, type of mirror, and focal length is given in the problem. First, find the radius of curvature from the focal length. Then by using the mirror formula, find the image distance. Use the formula for magnification to find the lateral magnification. Using these quantities, determine whether the image is real or virtual and inverted or non-inverted. Aslo, find the position of the image.

The formulas as follows:

$\mathit{r}\mathbf{=}\mathbf{2}\mathit{f}$

$\frac{\mathbf{1}}{\mathbf{f}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{p}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{i}}$

$\mathit{m}\mathbf{=}\frac{\mathbf{-}\mathbf{i}}{\mathbf{p}}$

Where, $\mathit{m}$is the magnification, $\mathit{p}$is the pole,$\mathit{f}$is the focal length.

(a) Determining the radius of curvature r

The radius of curvature:

Use the following formula to find the radius of curvature;

$r=2\times f$

Since the mirror is concave, the focal length must be positive, i.e, role="math" localid="1663069537437" $f=+36$

$$\begin{gathered} r=2\times \left(36 \text{cm}\right) \\ =+72 \text{cm} \end{gathered}$$

Therefore, the radius of curvature is,$r=+72 \text{cm}$.

(b) Determining the image distance i

Image distance$\mathit{i}$:

$\frac{1}{f}=\frac{1}{p}+\frac{1}{i}$

Rearranging for$i,$

$i=\frac{pf}{\left(p-f\right)}$

Plugging the values,

$$\begin{gathered} i=\frac{24 \text{cm}\times \left(36 \text{cm}\right)}{24 \text{cm}-\left(36 \text{cm}\right)} \\ i=-72 \text{cm} \end{gathered}$$

Therefore, the image distance is $i=-72 \text{cm}$.

(c) Determining the lateral magnification

Lateral magnification:

$m=\frac{-i}{p}$

Therefore,

$$\begin{gathered} m=-\frac{\left(-72 \text{cm}\right)}{24 \text{cm}} \\ m=+3.0 \end{gathered}$$

Therefore, the lateral magnification is, $m=+3.0$.

(d) Determining whether the image is real or virtual

Whether the image is real or virtual:

Since the image distance is negative, the image is virtual$\left(\text{V}\right)$.

Therefore, the image is virtual$\left(\text{V}\right)\text{.}$

(e) Determining whether the image is inverted or non-inverted

Whether the image is inverted or non-inverted:

As the magnification is positive, the image is non-inverted$\left(\text{NI}\right)$.

Therefore, the image is non-inverted$\left(\text{NI}\right)\text{.}$

(f) Determining the position of the image

Position of image:

For spherical mirrors, real images form on the side of the mirror where the object is located and virtual images form on the opposite side. Since the image is virtual, it is formed on the opposite side as the object.

Therefore, the image is on the opposite side as the object.

The mirror formula and equation for magnification to find the required quantities can be used.