A block of mass m lying on a rough horizontal plane is acted...

A block of mass $m$ lying on a rough horizontal plane is acted upon by a horizontal force $P$ and another force $Q$ inclined at an angle $\theta$ to the vertical. The block will remain in equilibrium if the coefficient of friction between it and the surface is (a) $\frac{P+Q \sin \theta}{m g+O \cos \theta}$ (b) $\frac{P \cos \theta+Q}{m g-0 \sin \theta}$ (c) $\frac{P+Q \cos \theta}{m g+Q \sin \theta}$ (d) $\frac{P \sin \theta-Q}{m g-O \cos \theta}$

A block of mass $m$ lying on a rough horizontal plane is acted upon by a horizontal force $P$ and another force $Q$ inclined at an angle $\theta$ to the vertical. The block will remain in equilibrium if the coefficient of friction between it and the surface is
(a) $\frac{P+Q \sin \theta}{m g+O \cos \theta}$
(b) $\frac{P \cos \theta+Q}{m g-0 \sin \theta}$
(c) $\frac{P+Q \cos \theta}{m g+Q \sin \theta}$
(d) $\frac{P \sin \theta-Q}{m g-O \cos \theta}$

Key Concepts

Static Equilibrium

Static equilibrium refers to the condition in which an object remains at rest, meaning that the vector sum of all forces and torques acting on it is zero. This concept is essential for analyzing problems where objects do not accelerate, ensuring that all horizontal and vertical force components cancel each other out.

Force Decomposition

Force decomposition involves breaking a force into its orthogonal components, typically along the horizontal and vertical axes. This is crucial in problems where forces act at angles, as it allows one to analyze and balance the components separately based on trigonometric functions such as sine and cosine.

Normal Reaction and Friction

The normal reaction force is the perpendicular contact force exerted by a surface on an object, and it plays a key role in friction calculations. The maximum static friction force is proportional to this normal reaction, with the coefficient of friction as the proportionality constant. Understanding this relationship is vital for determining the conditions required to prevent sliding.

Trigonometric Relationships in Force Analysis

Trigonometric relationships are used in resolving forces into their horizontal and vertical components when forces are applied at angles. By using sine and cosine functions based on the angle of application, one can effectively incorporate the effects of inclined forces on both the equilibrium in the vertical direction (affecting the normal force) and the horizontal motion.

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