An automobile having a mass of 2 Mg travels up a 7^∘ slope at a constant speed of v=100 km / h.

step 1 In this problem, you have a car that's traveling up the slope, and you want to find the power ne

An automobile having a mass of 2 Mg travels up a 7^∘ slope...

An automobile having a mass of $2 \mathrm{Mg}$ travels up a $7^{\circ}$ slope at a constant speed of $v=100 \mathrm{km} / \mathrm{h}$. If mechanical friction and wind resistance are neglected, determine the power developed by the engine if the automobile has an efficiency $\varepsilon=0.65.$

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Key Concepts

Inclined Plane Mechanics

This concept involves analyzing the forces acting on an object moving on a sloped surface, where gravity is decomposed into components parallel and perpendicular to the incline. It is crucial for understanding how much force is needed to counteract gravity when moving upward on an incline.

Power in Mechanical Systems

Power is defined as the rate at which work is done or energy is transferred. In the context of a vehicle moving at constant speed, it can be computed by multiplying the force opposing the motion (such as the gravitational component along the slope) by the vehicle’s velocity.

Mechanical Efficiency

Mechanical efficiency measures how effectively a system converts input energy into useful work output. It accounts for energy losses in the system, requiring that the engine produces more power than the minimal work necessary to overcome resistive forces due to these inefficiencies.

Dynamic Equilibrium

When an object moves at constant speed, it is in dynamic equilibrium, meaning that the net force acting on it is zero. This principle allows for the simplification of calculations by equating the engine's power output, adjusted for efficiency, to the work done against forces such as gravity.

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