A locomotive accelerates a 25 -car train along a level...

A locomotive accelerates a 25 -car train along a level track. Every car has a mass of $5.0 \times 10^{4} \mathrm{~kg}$ and is subject to a frictional force $f=250 v$, where the speed $v$ is in meters per second and the force $f$ is in newtons. At the instant when the speed of the train is $30 \mathrm{~km} / \mathrm{h}$, the magnitude of its acceleration is $0.20 \mathrm{~m} / \mathrm{s}^{2}$. (a) What is the tension in the coupling between the first car and the locomotive? (b) If this tension is equal to the maximum force the locomotive can exert on the train, what is the steepest grade up which the locomotive can pull the train at $30 \mathrm{~km} / \mathrm{h}$ ?

Key Concepts

Newton's Second Law

This law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. It is central in analyzing the motion of any system, as it allows us to relate the applied forces and resulting acceleration. In multi-body systems, Newton's Second Law helps determine how forces are distributed among the different components of the system.

Frictional Forces

Friction is a force that opposes the motion of objects and can depend on factors like the speed of the moving object. In problems where friction varies with velocity, understanding its role is essential for calculating the net force. Frictional forces can reduce the effective force available for accelerating a system, thus influencing the motion.

Free-Body Analysis in Multi-Body Systems

A free-body diagram is an essential tool for visualizing and analyzing all the forces acting on each part of a multi-component system. In systems composed of several connected objects, such as train cars, analyzing the forces individually allows one to determine inter-component forces like tension in couplings as well as the net forces acting on each mass.

Inclined Plane Dynamics

When studying the motion of objects on an incline, the gravitational force is resolved into components parallel and perpendicular to the surface. The component along the incline directly affects the motion and must be overcome by the applied force. This concept is vital when determining the maximum grade a system can ascend, taking into account both frictional forces and the weight of the system.

Unit Conversion

Converting units appropriately is crucial in solving physics problems to ensure consistency across calculations. For instance, converting speeds from kilometers per hour to meters per second ensures that all quantities are expressed in standard SI units, reducing the risk of errors and facilitating the correct application of physical laws.

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