The radius of curvature of a convex mirror is 1.60 x 102 cm. An object that is 11.0 cm high is placed 40.0 cm in front of this mirror. Determine (a) the location and (b) the height of the image.
To determine the location and height of the image formed by a convex mirror, we can use the mirror equation:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
where:
f = focal length of the mirror (radius of curvature)
d_o = object distance from the mirror
d_i = image distance from the mirror
In this case, we are given:
f = 1.60 x 10^2 cm
d_o = -40.0 cm (negative because the object is in front of the mirror)
To find the image distance (d_i), we need to rearrange the mirror equation:
\[ \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \]
\[ \frac{1}{d_i} = \frac{1}{1.60\times 10^2} - \frac{1}{-40.0} \]
Calculating this expression will give us the image distance.
Once we know the image distance, we can use the magnification formula to find the height of the image:
\[ \text{magnification (m)} = - \frac{d_i}{d_o} \]
Substituting the known values into the formula, we can find the height of the image:
\[ \text{height of image} = \text{magnification} \times \text{height of object} \]
Let's calculate the values step by step:
First, let's find the image distance (d_i):
\[ \frac{1}{d_i} = \frac{1}{1.60\times 10^2} - \frac{1}{-40.0} \]
To evaluate this expression, we calculate the reciprocal of each value on the right-hand side of the equation, then add them together, and finally take the reciprocal of that sum.
\[ \frac{1}{d_i} = \frac{1}{1.60\times 10^2} + \frac{1}{40.0} \]
Now, we calculate the sum:
\[ \frac{1}{d_i} = 0.00625 + 0.025 \]
\[ \frac{1}{d_i} = 0.03125 \]
Taking the reciprocal of both sides gives:
\[ d_i = \frac{1}{0.03125} \]
This evaluates to:
\[ d_i = 32 \text{ cm} \]
So the image distance is 32.0 cm.
Now, let's find the magnification:
\[ \text{m} = - \frac{d_i}{d_o} = - \frac{32}{-40.0} \]
This gives us:
\[ \text{m} = 0.8 \]
Finally, we can find the height of the image:
\[ \text{height of image} = \text{magnification} \times \text{height of object} = 0.8 \times 11.0 \]
Thus, the height of the image is 8.8 cm.
To summarize:
(a) The location of the image is 32.0 cm from the convex mirror.
(b) The height of the image is 8.8 cm.