A balance supported by a thin string is shown below. A distributed force is applied on the right side of the balance, with linear variation from $0$ $N/m$ (at 2 metres from the string) to $10$ $N/m$ (3 metres from the string). What mass $M$ must be hung from the left side to keep the balance horizontal?

A. $0.38$ $kg$
B. $0.42$ $kg$
C. $0.45$ $kg$
D. $0.67$ $kg$
E. $1.27$ $kg$