A given object takes n times more time to slide down 45^∘ rough inclined plane as it takes to sl

Step 1: We are given two scenarios. In the first scenario, an object is sliding down a smooth in

A given object takes n times more time to slide down 45^∘...

A given object takes n times more time to slide down $45^{\circ}$ rough inclined plane as it takes to slide down a perfectly smooth $45^{\circ}$ incline. The coefficient of kinetic friction between the object and the incline is (A) $\left[1 /\left(2-\mathrm{n}^{2}\right)\right]$ (B) $\left[1-\left(1 / \mathrm{n}^{2}\right)\right]$

A given object takes n times more time to slide down $45^{\circ}$ rough inclined plane as it takes to slide down a perfectly smooth $45^{\circ}$ incline. The coefficient of kinetic friction between the object and the incline is
(A) $\left[1 /\left(2-\mathrm{n}^{2}\right)\right]$
(B) $\left[1-\left(1 / \mathrm{n}^{2}\right)\right]$

Key Concepts

Decomposition of Gravitational Force

In problems involving motion on an inclined plane, gravity is resolved into two components: one perpendicular and one parallel to the surface. The parallel component (g sin?) drives the motion down the incline, while the perpendicular component (g cos?) determines the normal reaction force, which is essential in calculating frictional forces.

Coefficient of Kinetic Friction

The coefficient of kinetic friction is a dimensionless value that characterizes the frictional resistance between moving surfaces. It is defined as the ratio of the frictional force to the normal force and plays a crucial role in determining the net acceleration of an object sliding on an inclined plane where friction opposes the motion.

Uniform Acceleration Kinematics

When an object slides down an incline, it experiences a constant acceleration if the net force remains constant. The kinematic equations for uniformly accelerated motion relate displacement, time, and acceleration, allowing us to compare different scenarios (such as motion with and without friction) by analyzing their time of travel and acceleration profiles.

Newton's Second Law in the Context of Inclined Planes

Newton's second law, F = ma, is fundamental in analyzing the forces acting on an object on an inclined plane. By applying it along the slope, one can relate the net force – which is a combination of gravitational components and frictional resistance – to the resulting acceleration. This analysis is used to understand how friction alters the motion compared to a frictionless scenario.

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