Pulling force making an angle θ to the horizontal is applied on a block of weight W placed on a horizontal table. If the angle of friction is $\alpha$ , then the magnitude of force required to move the body is equal to
(a) $\frac{W \sin \alpha}{g \tan (\theta - \alpha)}$ (b) $\frac{W \cos \alpha}{\cos (\theta - \alpha)}$ (c) $\frac{W \sin \alpha}{\cos (\theta - \alpha)}$ (d) $\frac{W \tan \alpha}{\sin (\theta - \alpha)}$
Text Solution
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$\uparrow : N + F \sin \theta = W \Rightarrow N = W - F \sin \theta$
The block will move if
$F \cos \theta \ge f_{\max}$ $F \cos \theta \ge \mu (W - F \sin \theta)$ $\mu = \tan \alpha = \frac{\sin \alpha}{\cos \alpha}$ $F \cos \theta \ge \frac{\sin \alpha}{\cos \alpha} (W - F \sin \theta)$ $F (\cos \theta \cos \alpha + \sin \theta \sin \alpha) \ge W \sin \alpha$ $F \ge \frac{W \sin \alpha}{\cos (\theta - \alpha)}$ $F_{\min} = \frac{W \sin \alpha}{\cos (\theta - \alpha)}$
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