A spherical concave mirror has a radius R=360 mm forms an image of a h(object)=12 cm high object located d0=p=0.9 m from the vertex of the mirror. Find the position di=q(cm) and the height cm of the image.
1/p + 1/q = 2/R
p = 0.9 m
R = 0.36 m
Solve for q
Image height = h*(q/p)
To find the position \(d_i\) and height \(h_i\) of the image formed by the spherical concave mirror, we can use the mirror equation and magnification formula.
1. Start by understanding the given information:
- Radius of the mirror, \(R = -360\) mm (negative because it is a concave mirror)
- Object height, \(h_{object} = 12\) cm
- Object distance, \(d_0 = p = 0.9\) m (converted to cm, \(p = 90\) cm)
2. Apply the mirror equation:
\(\frac{1}{f} = \frac{1}{p} + \frac{1}{q}\)
Here, \(f\) is the focal length and \(q\) is the image distance.
3. Find the focal length:
Since the mirror is concave, the focal length is half the radius of the mirror.
\(f = \frac{R}{2}\)
\(f = \frac{-360}{2} = -180\) mm (negative because it is a concave mirror)
4. Substitute the values into the mirror equation and solve for \(q\):
\(\frac{1}{-180} = \frac{1}{90} + \frac{1}{q}\)
To simplify, let's find the common denominator:
\(-\frac{1}{180} = \frac{1}{90} + \frac{1}{q}\)
Multiplying both sides by \(90q\) to eliminate the fractions:
\(-q = q + 90\)
Bringing the \(q\) terms to one side:
\(-q - q = 90\)
Simplifying:
\(-2q = 90\)
Solving for \(q\):
\(q = \frac{90}{-2} = -45\) cm (negative because it is to the left of the mirror)
Therefore, the position of the image is \(d_i = -45\) cm.
5. Find the height of the image using the magnification formula:
\(m = \frac{h_i}{h_{object}} = -\frac{q}{p}\)
Substituting the given values:
\(m = -\frac{-45}{90} = -\frac{1}{2}\)
Solving for \(h_i\):
\(\frac{h_i}{12} = -\frac{1}{2}\)
Cross-multiplying:
\(h_i = -\frac{1}{2} \times 12 = -6\) cm
Therefore, the height of the image is \(h_i = -6\) cm.
Note: The negative sign indicates that the image formed is inverted relative to the object.