A rotary snow plow mounted on a large truck eats its way...

A rotary snow plow mounted on a large truck eats its way through a snow drift on a level road at a constant speed of $20 \mathrm{km} / \mathrm{h}$. The plow discharges $60 \mathrm{Mg}$ of snow per minute from its $45^{\circ}$ chute with a velocity of $12 \mathrm{m} / \mathrm{s}$ relative to the plow. Calculate the tractive force $P$ on the tires in the direction of motion necessary to move the plow and find the corresponding lateral force $R$ between the tires and the road.

Key Concepts

Vector Decomposition of Forces

Since forces and velocities have both magnitude and direction, they must be broken down into their components (for example, forward and lateral) to analyze the effects in different directions. In the scenario, the discharge angle of the snow creates both a force component in the direction of motion (tractive force) and a lateral component (causing side force), which must be calculated by decomposing the overall momentum change vector.

Relative Velocity

Relative velocity is the velocity of one object as observed from another moving reference frame. When analyzing forces due to discharged snow, it is important to account for the snow’s ejection speed relative to the plow as well as the plow’s own velocity relative to the ground, in order to correctly determine the actual momentum change imparted to the snow.

Newton's Second Law (Rate of Change of Momentum)

Newton's second law relates the net force acting on an object to the rate of change of its momentum. In the context of continuous mass ejection, such as a snow plow discharging snow at a certain velocity, the net external force required to maintain constant speed is equal to the momentum flux of the ejected material, which is calculated from the mass flow rate and the velocity change.

Conservation of Momentum

This principle states that the momentum of an isolated system remains constant unless acted upon by an external force. In problems involving the discharge of mass (like snow being thrown from a plow), the change in momentum of the departing mass must be balanced by an equal and opposite change in the momentum of the system, leading to a reaction force on the moving object.

Mass Flow Rate

Mass flow rate is the amount of mass that passes a given point per unit time. In dynamics problems involving continuous mass ejection or accumulation, like a plow discharging snow, the force associated with a flow of mass is determined by multiplying the mass flow rate by the velocity change of the mass, thereby quantifying the momentum flux.

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