a concave lens of focal length 15cm forms an image 1/3 the size of the object. the object distance is

To find the object distance when using a concave lens, we can use the lens formula:

\(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\)

Where:
- \(f\) is the focal length of the lens
- \(v\) is the image distance
- \(u\) is the object distance

We are given that the focal length (\(f\)) is 15 cm and the image size is one-third the size of the object.

Now, let's break down the problem step by step:

1. The image is \(1/3\) the size of the object. This means that the magnification (\(m\)) is \(-1/3\), as the image is formed on the same side as the object in a concave lens, making it negative.

The magnification formula is given by:
\(m = -\frac{v}{u}\)

Substituting the given value for magnification, we get:
\(-\frac{1}{3} = -\frac{v}{u}\)

2. Now, we can substitute the focal length (\(f\)) and the magnification (\(m\)) into the lens formula:
\(\frac{1}{15} = -\frac{v}{u} - \frac{1}{u}\)

3. Rearranging the equation and simplifying, we have:
\(\frac{1}{15} = \frac{-v - u}{u}\)

4. Cross-multiplying and simplifying further, we get:
\(1 = \frac{-v - u}{15}\)

5. Multiplying both sides by 15 to eliminate the fraction, we have:
\(15 = -v - u\)

6. Rearranging the equation to isolate the object distance (\(u\)), we get:
\(u = -v - 15\)

Therefore, the object distance (\(u\)) is \(-v - 15\) cm.