Step 1: Understanding the Concept:
We are given a convex lens forming a real and inverted image.
We need to determine the object distance using the given magnification and focal length.
We must apply standard Cartesian sign conventions.
Step 2: Key Formula or Approach:
The magnification formula for a lens is $m = \frac{v}{u}$.
For a real, inverted image, magnification $m$ is negative.
The thin lens formula is $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$.
Step 3: Detailed Explanation:
Given focal length $f = +\frac{1}{3} \text{ m}$ (convex lens).
The image is real, inverted, and twice the size of the object, so $m = -2$.
Using the magnification formula:
\[ m = \frac{v}{u} = -2 \implies v = -2u \]
Now, substitute $v$ and $f$ into the lens formula:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
\[ \frac{1}{-2u} - \frac{1}{u} = \frac{1}{1/3} \]
\[ \frac{-1 - 2}{2u} = 3 \]
\[ \frac{-3}{2u} = 3 \]
Solving for $u$:
\[ 2u = -1 \]
\[ u = -0.5 \text{ m} \]
The negative sign indicates the object is placed in front of the lens. The distance is the magnitude $|u| = 0.5 \text{ m}$.
Step 4: Final Answer:
The distance of the object from the lens is $0.5 \text{ m}$.