A package of mass m is placed onto a horizontal conveyor belt moving at speed v (Fig. 7–32). The coefficient of kinetic friction between package and belt is μₖ. What is the package's displacement d during this time?

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Identify the forces acting on the package: The forces include the gravitational force (mg), the normal force (N), and the kinetic friction force (fₖ). Since the package is on a horizontal surface, the normal force is equal in magnitude to the gravitational force, so N = mg.

Determine the kinetic friction force: The kinetic friction force is given by the formula fₖ = μₖ * N. Substituting N = mg, we get fₖ = μₖ * m * g.

Apply Newton's second law in the horizontal direction: The net force acting on the package is the frictional force, which causes the package to accelerate. Using F = ma, we have fₖ = m * a. Substituting fₖ = μₖ * m * g, we get μₖ * m * g = m * a. Simplify to find the acceleration: a = μₖ * g.

Use kinematic equations to find the displacement: The package starts from rest relative to the conveyor belt and accelerates until it matches the belt's speed v. Using the kinematic equation v = u + at, where u = 0, we solve for the time t: t = v / a. Substituting a = μₖ * g, we get t = v / (μₖ * g).

Substitute t into the displacement formula: The displacement d is given by the kinematic equation d = u * t + (1/2) * a * t². Since u = 0, this simplifies to d = (1/2) * a * t². Substituting a = μₖ * g and t = v / (μₖ * g), we get d = (1/2) * (μₖ * g) * (v / (μₖ * g))². Simplify this expression to find d in terms of v, μₖ, and g.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It is proportional to the normal force and is calculated using the coefficient of kinetic friction (μₖ) multiplied by the normal force (N). In this scenario, it plays a crucial role in determining how the package interacts with the conveyor belt as it moves.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma). This principle is essential for analyzing the forces acting on the package, including the frictional force, to determine its acceleration and subsequent displacement over time.

Displacement and Time Relationship

Displacement refers to the change in position of an object and can be calculated using the equation d = vt + (1/2)at², where d is displacement, v is initial velocity, a is acceleration, and t is time. Understanding this relationship is vital for solving the question, as it allows us to determine how far the package travels on the conveyor belt during the time it is in motion.

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