An object is placed at a distance of $60\ cm$ from a concave lens of focal length $30\ cm$. Use lens formula to find the position of the image formed in this case.

To find the position of the image formed by the concave lens, we can use the lens formula: $$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$ where $v$ is the image distance, $u$ is the object distance, and $f$ is the focal length of the lens. Given that the object distance $u = -60\ cm$ (negative sign indicates that the object is placed in front of the lens) and the focal length $f = -30\ cm$ (negative sign indicates that the lens is concave), we can substitute these values into the lens formula: $$\frac{1}{v} - \frac{1}{-60} = \frac{1}{-30}$$ Simplifying the equation, we get: $$\frac{1}{v} + \frac{1}{60} = -\frac{1}{30}$$ $$\frac{1}{v} = -\frac{1}{30} - \frac{1}{60}$$ $$\frac{1}{v} = -\frac{2}{60} - \frac{1}{60}$$ $$\frac{1}{v} = -\frac{3}{60}$$ $$\frac{1}{v} = -\frac{1}{20}$$ $$v = -20\ cm$$ Therefore, the image is formed at a distance of $20\ cm$ from the lens, and the negative sign indicates that the image is formed on the same side of the lens as the object.