A box of mass 8 kg is placed on a rough inclined plane of inclination θ. Its downward motion ca

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A box of mass 8 kg is placed on a rough inclined plane of...

A box of mass $8 \mathrm{~kg}$ is placed on a rough inclined plane of inclination $\theta$. Its downward motion can be prevented by applying an upward pull $F$ and it can be made to slide upwards by applying a force $2 F .$ The co-efficient of friction between the box and the inclined plane is (A) $\frac{1}{3} \tan \theta$ (B) $3 \tan \theta$ (C) $\frac{1}{2} \tan \theta$ (D) $2 \tan \theta$

A box of mass $8 \mathrm{~kg}$ is placed on a rough inclined plane of inclination $\theta$. Its downward motion can be prevented by applying an upward pull $F$ and it can be made to slide upwards by applying a force $2 F .$ The co-efficient of friction between the box and the inclined plane is
(A) $\frac{1}{3} \tan \theta$
(B) $3 \tan \theta$
(C) $\frac{1}{2} \tan \theta$
(D) $2 \tan \theta$

Key Concepts

Effect of Applied Forces

Applying additional forces, such as pulling or pushing, affects the net force acting on the object along the plane. Analyzing these effects involves incorporating the applied force into the force balance equations. Adjusting the magnitude and direction of the applied force can shift the equilibrium state, which is critical for determining parameters like the coefficient of friction in various scenarios.

Force Decomposition on an Inclined Plane

This concept involves breaking down the gravitational force acting on an object into two components: one parallel to the plane's surface that causes the object to slide down, and one perpendicular to the surface that determines the normal reaction force. Understanding these components is essential for analyzing the forces acting on an object on an incline.

Frictional Force

Friction is the resistive force that acts opposite to the direction of motion (or impending motion) along a surface. It is commonly modeled as the product of the coefficient of friction and the normal force. Recognizing how friction interacts with other forces is crucial when determining the conditions for equilibrium or motion on an inclined plane.

Equilibrium Conditions in Mechanics

In problems involving forces on an inclined plane, equilibrium conditions are used to set up equations where the sum of the forces in each direction is zero. This holds true whether the object is at rest or moving with constant velocity. The balance of forces, including applied forces, frictional forces, and gravitational components, allows one to solve for unknown quantities such as the coefficient of friction.

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