Final answer:
Using the mirror equation and magnification formula, the object distance is calculated to be 60 cm from the mirror when the image is twice the size of the object using a converging mirror with a focal length of 30 cm.
Step-by-step explanation:
To calculate the distance of the object (pencil) from the mirror when an image is formed that is twice the size of the object using a converging mirror (concave mirror), we use the mirror equation and magnification formula. The mirror equation is 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance. The magnification formula is m = -di/do, where m is the magnification.
Given that the image is twice the size of the object, the magnification m is -2 (it is negative because the image by a converging mirror is inverted). Hence, with the focal length f being -30 cm (negative because the mirror is concave), we must solve the following equations simultaneously:
- m = -2 = -di/do
- 1/f = 1/do + 1/di
Using the second equation with the known f, we get 1/-30 = 1/do + 1/di. To find do, we can substitute di using di = -2do from the magnification equation into the mirror equation. The two equations become:
- 1/-30 = 1/do - 1/(2do)
- 1/-30 = (2do - do)/(2do^2)
- 1/-30 = 1/(2do)
- do = 2(-30)
- do = -60 cm
The negative sign indicates that the object is placed in front of the mirror. Therefore, the object distance do is 60 cm from the mirror.