Question

What is the minimum value of $F$ needed so that block begins to move upward on frictionless incline plane as shown

• A. $M g \tan \left(\frac{\theta}{2}\right)$
• B. $M g \cot \left(\frac{\theta}{2}\right)$
• C. $\frac{M g \sin \theta}{(1+\sin \theta)}$
• D. $M g \sin \left(\frac{\theta}{2}\right)$

Answer 1

Answer: (A)

$F+F \cos \theta=m g \sin \theta$
$F=\frac{m g \sin \theta}{1+\cos \theta}$
$F=\frac{m g 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}}$
$=m g \tan \frac{\theta}{2}$

$\left(\because \sin \theta=2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}\right.$ and $\left.1+\cos \theta=2 \cos ^{2} \frac{\theta}{2}\right)$