A uniform board of length L is sliding along a smooth,...

A uniform board of length L is sliding along a smooth, frictionless, horizontal plane as shown in Figure P8.75a. The board then slides across the boundary with a rough horizontal surface. The coefficient of kinetic friction between the board and the second surface is $\mu_{k^{*}}$. (a) Find the acceleration of the board at the moment its front end has traveled a distance x beyond the boundary. (b) The board stops at the moment its back end reaches the boundary as shown in Figure P8.75b. Find the initial speed v of the board.

Key Concepts

Work-Energy Principle

The work-energy principle states that the work done by the forces acting on a body results in a change in its kinetic energy. This principle is particularly useful in friction problems, where the work done by friction can be equated to the loss of kinetic energy to predict stopping distances or determine the initial speed required for a given displacement.

Variable Friction Force Distribution

When an object partly transitions from a frictionless to a frictional surface, the friction force acting upon it is not uniform over its entire length but instead depends on the portion of the object that is in contact with the rough surface. This concept is essential for analyzing systems where the effective friction force changes continuously with the object's position.

Kinematic Equations

Kinematic equations form the backbone of analyzing motion, relating displacement, velocity, acceleration, and time. They are applied here not only to determine the board's motion under constant or variable acceleration but also to relate the distance traveled on the frictional surface with the board's initial and final speeds.

Kinetic Friction

Kinetic friction is the resistive force that opposes the relative motion of surfaces in contact when they slide against each other. Its magnitude is determined by the coefficient of kinetic friction and the normal force, and it plays a key role in calculating the deceleration of objects moving over rough surfaces.

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its mass and acceleration. In problems involving friction, this law is used to set up the equation of motion by equating the net friction force to the mass times the acceleration of the object, which is crucial for determining how the board slows down.

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