Question

A rod of mass M and length L is kept on horizontal plane as shown. A force F is applied as shown what should be x so that the hinge offers no reaction ?

• A. Zero
• B. $\frac{2L}{3}$
• C. $\frac{L}{3}$
• D. L

Answer 1

The correct option is B $\frac{2L}{3}$

$\begin{array}{c}\text{From Translatory motion equation -}\\ \begin{array}{cc}F& -F_H=Ma\\ \Rightarrow F_H=F-Ma& -11\end{array}\end{array}$

$\begin{array}{c}\text{From torque equation-}\\ \begin{array}{cc}F\cdot x=I\alpha & I\to \text{moment of Inertia}\\ \alpha \to \text{angular acleleration of Rod.}& \\ (\frac{L}{2}\alpha =a)& \end{array}\\ \Rightarrow Fx=\frac{M{L}^2}{3}\cdot \alpha \end{array}$

$\begin{array}{c}\Rightarrow Fx=\frac{2MLa}{3}-\frac{2}{2L}\cdot x\\ \text{From ea}D\text{and}2\text{we get -}\\ F_H=F-\frac{3F}{2L}\\ \text{For}{F}_H=0\text{we get}x=\frac{2L}{3}\end{array}$