A freight train consists of two 8.00 ×10^4 -kg engines and 45 cars with average masses of 5.50 ×

A freight train consists of two 8.00 ×10^4 -kg engines and...

A freight train consists of two $8.00 \times 10^{4}$ -kg engines and 45 cars with average masses of $5.50 \times 10^{4}$ kg . (a) What force must each engine exert backward on the track to accelerate the train at a rate of $5.00 \times 10^{-2} \mathrm{m} / \mathrm{s}^{2}$ if the force of friction is $7.50 \times 10^{5} \mathrm{N}$ , assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

Key Concepts

Internal (Coupling) Forces

These are the forces transmitted between connected cars (or between an engine and a car) within a train. Analysis of internal forces involves considering the train as a series of connected bodies and often requires using free-body diagrams for sections of the train to determine the tension in the couplings, which is essential for understanding how force is propagated through the entire system.

Force Distribution Among Engines

When multiple engines are used to pull a massive system like a train, the total required force (including both accelerating the mass and overcoming friction) must be distributed among them. Assuming identical engines, each engine contributes an equal share of the total force needed to accelerate the train.

Total Mass Calculation

For systems composed of multiple objects, such as the two engines and numerous freight cars in a train, calculating the total mass is crucial. The total mass is the sum of the masses of all individual parts. This calculation is necessary to determine the net force required for a given acceleration using Newton's Second Law.

Newton's Second Law

This principle states that the net force acting on an object (or a system of objects) is equal to the product of its mass and acceleration (F = ma). It is fundamental for determining how much force is needed to accelerate a system, and it forms the basis for analyzing both external forces, like the propulsion and friction forces, and internal forces, such as the tension in the couplings.

Friction Force

Friction, particularly rolling friction in this context, opposes the motion of the train and must be overcome by the engines. Even though rolling friction for trains is relatively small compared to the mass, it still requires careful consideration when computing the net force needed to achieve a specific acceleration.

Transcript

00:01 First, we can say that the net force is going to be equalling the capital m.

00:06 This capital m essentially is going to be the combined mass of the two engines and all of the cars that are attached to one another to make up the train.

00:19 And then this would be multiplied by the acceleration.

00:21 We also know that this is equaling two times the force.

00:25 This would be the force, two times the force exerted by one engine, so two engines.

00:30 Minus the force of friction.

00:32 And so we can say that then the force exerted by each engine is going to be equaling the total mass times the acceleration plus the force of friction divided by two.

00:47 And so we can say that the force exerted by each engine.

00:51 The total mass is going to be two times 80 ,000 kilograms plus 45 cars times 55 ,000.

01:04 Kilograms multiplied by the acceleration of 0 .05 meters per second squared.

01:15 And then plus the force of friction, we know to be 7 .50 times 10 to the 5th newton's.

01:27 And then this is going to be divided all by two.

01:31 And so the force exerted by each engine is going to be 4 .41 times.

01:37 Times 10 to the 5th newton's.

01:40 This would be our answer for the force exerted by each engine.

01:44 And then this would be your answer for part a.

01:46 For part b now, essentially, we have the 37th car has eight cars.

02:04 The 37th car has eight cars behind it.

02:09 And we can then say that the force net prime, force net prime simply means the net force in this case.

02:17 This would be eight times the mass times the acceleration...

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