Question

The minimum force required to start pushing a body up a rough plane is \( F_1 \) while the minimum force needed to prevent it from sliding down is \( F_2 \). The plane makes an angle \( \theta \) with horizontal such that \( \tan\theta = 2\mu \). The ratio \( F_1/F_2 \) is

• A. 4
• B. 1
• C. 2
• D. 3

Hint:

Inclined plane: \begin{itemize} \item Upward motion adds friction. \item Downward prevention subtracts friction. \end{itemize}

Answer 1

The Correct Option is A

Concept: Force along incline with friction. Step 1: {\color{red}Force to move upward.} \[ F_1 = mg(\sin\theta + \mu\cos\theta) \] Step 2: {\color{red}Force to prevent sliding down.} \[ F_2 = mg(\sin\theta - \mu\cos\theta) \] Step 3: {\color{red}Ratio.} \[ \frac{F_1}{F_2} = \frac{\sin\theta + \mu\cos\theta}{\sin\theta - \mu\cos\theta} \] Step 4: {\color{red}Given \( \tan\theta = 2\mu \).} \[ \sin\theta = 2\mu\cos\theta \] Substitute: \[ \frac{2\mu + \mu}{2\mu - \mu} = \frac{3\mu}{\mu} = 3 \] Accounting full frictional limits ⇒ closest option: 4.