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Chapter 15: Problem 4

A model steam engine of \(1.00-\mathrm{kg}\) mass pulls eight cars of \(1.00-\mathrm{kg}\) mass each. The cars start at rest and reach a velocity of \(3.00 \mathrm{m} / \mathrm{s}\) in a time of \(3.00 \mathrm{s}\) while moving a distance of \(4.50 \mathrm{m} .\) During that time, the engine takes in $135 \mathrm{J}$ of heat. What is the change in the internal energy of the engine?

Short Answer

Expert verified
Answer: The change in internal energy of the engine is 99 J.

Step by step solution

01

Calculate the net force acting on the cars from the given information

First, let's find the net force acting on the cars. We know their initial velocity (u), final velocity (v), time taken (t), and mass (m), so we can use the equation of motion v = u + at to find the acceleration (a). Then we can use Newton's second law (F = ma) to find the force. v = 3.00 m/s u = 0 m/s t = 3.00 s Rearrange the equation of motion to solve for acceleration: a = (v - u) / t Plug the values into the equation: a = (3.00 - 0) / 3.00 a = 1.00 m/s² Now, find the net force on the cars using F = ma: F = (1.00 kg per car × 8 cars) × 1.00 m/s² F = 8.00 kg × 1.00 m/s² F = 8.00 N
02

Calculate the work done by the steam engine on the cars

Now that we have the net force acting on the cars, let's find the work done by the steam engine. We know the distance traveled by the cars (d) and the net force (F), so we can use the formula for work: W = F × d. Plug the values into the equation: W = 8.00 N × 4.50 m W = 36.00 J
03

Apply the First Law of Thermodynamics to find the change in internal energy

Now that we have the work done by the system (W) and the heat absorbed (Q), we can use the First Law of Thermodynamics to find the change in internal energy (ΔU). ΔU = Q - W Plug the values into the equation: ΔU = 135 J - 36.00 J ΔU = 99 J The change in the internal energy of the engine is 99 J.

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Most popular questions from this chapter

Suppose 1.00 mol of oxygen is heated at constant pressure of 1.00 atm from \(10.0^{\circ} \mathrm{C}\) to \(25.0^{\circ} \mathrm{C} .\) (a) How much heat is absorbed by the gas? (b) Using the ideal gas law, calculate the change of volume of the gas in this process. (c) What is the work done by the gas during this expansion? (d) From the first law, calculate the change of internal energy of the gas in this process.
In a heat engine, 3.00 mol of a monatomic ideal gas, initially at 4.00 atm of pressure, undergoes an isothermal expansion, increasing its volume by a factor of 9.50 at a constant temperature of \(650.0 \mathrm{K} .\) The gas is then compressed at a constant pressure to its original volume. Finally, the pressure is increased at constant volume back to the original pressure. (a) Draw a \(P V\) diagram of this three-step heat engine. (b) For each step of this process, calculate the work done on the gas, the change in internal energy, and the heat transferred into the gas. (c) What is the efficiency of this engine?
A town is considering using its lake as a source of power. The average temperature difference from the top to the bottom is \(15^{\circ} \mathrm{C},\) and the average surface temperature is \(22^{\circ} \mathrm{C} .\) (a) Assuming that the town can set up a reversible engine using the surface and bottom of the lake as heat reservoirs, what would be its efficiency? (b) If the town needs about \(1.0 \times 10^{8} \mathrm{W}\) of power to be supplied by the lake, how many \(\mathrm{m}^{3}\) of water does the heat engine use per second? (c) The surface area of the lake is $8.0 \times 10^{7} \mathrm{m}^{2}$ and the average incident intensity (over \(24 \mathrm{h})\) of the sunlight is \(200 \mathrm{W} / \mathrm{m}^{2} .\) Can the lake supply enough heat to meet the town's energy needs with this method?
Estimate the entropy change of \(850 \mathrm{g}\) of water when it is heated from \(20.0^{\circ} \mathrm{C}\) to \(50.0^{\circ} \mathrm{C} .\) [Hint: Assume that the heat flows into the water at an average temperature. \(]\)
A system takes in \(550 \mathrm{J}\) of heat while performing \(840 \mathrm{J}\) of work. What is the change in internal energy of the system?
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