The engine that powers a crane burns fuel at a flame temperature of 2000℃. It is cooled by 20℃ air. The crane lifts a 2000 kg steel girder 30 m upward. How much heat energy is transferred to the engine by burning fuel if the engine is 40% as efficient as a Carnot engine?

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Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A rocket engine combusts fuel at a temperature of degrees Celsius. It is cooled by liquid hydrogen at a temperature of degrees Celsius. The rocket lifts a payload of 5000 kg to an altitude of 100 kg. How much heat energy is transferred to the engine by burning fuel? If the engine is 50% as efficient as a Carno engine, assume the specific heat capacity of a rocket exhaust gasses is 1.0 kilojoules per kilograms multiplied by Calvin. And that the gravitational acceleration is 9.8 m per second squared. Ok. So we're given some multiple choice answers. They're all in the same units of jewels. So let's read them off to see what our final answer might be. A is 4.97 multiplied by 10 to the power. Nine B is 9.87 multiplied by 10 to the power nine C is 7.89 multiplied by 10 to the power nine and D is 7.94 multiplied to the 10, multiplied by 10 to the power of nine. OK. So first off note that the work done by the rocket engine is equal to the gravitational potential energy. Thus, we can write that the work out is equal to delta U gravitational. So the potential gravitational potential energy is equal to mass multiplied by gravity multiplied by height. So now we could plug in our known variables. And so for the work out, so workout is equal to 5000 kilograms, which is the mass multiplied by gravity which is 9.81 meters per second squared multiplied by the height which is 100,000 meters is we need to convert the altitude which is 100 kilometers to meters. OK? So when we plug that into a calculator, we get the workout is equal to 4.9 multiplied by 10 to the power of nine jewels. So the efficiency of the rocket engine is 50% of the Carno engine. So we need to recall the efficiency of a Carno E engine equation. So A is equal to 0.50 multiplied by a ac. So a, a subscript Carnot or Carne, I should say Carne C for short. And this is equal to or the same thing as saying, 0.50 multiplied by. And this is the efficiency of a Carno Engine equation. One minus the temperature of the cold reservoir divided by the temperature of the hot reservoir. So now we need to plug in our known variables into this equation. So let's do that. So 0.50 multiplied by one minus the temperature of the cold reservoir, which we need to convert, you know, down below, I'm actually up here. So let's quickly convert our temperature for the cold reservoir and the temperature of the hot reservoir from degrees Celsius to degrees Kelvin. So for the cold reservoir, it was minus degrees C. So we need to take this and add 273. And when we do, we'll get 20 three Kelvin, same with the hot reservoir. So it was 3000 degrees Celsius and then we need to add 273. And when we do, we get Kelvin. OK. So let's plug that in. So 23 Kelvin is the temperature for the pulled reservoir divided by the hot reservoir which was 273 Calvin. Awesome. OK. So let me plug that into a calculator. We should get 0.4965. So we need to recall that Ada is equal to the work out divided by the heat transfer in the hot reservoir, which is represented by Qh. So we need to rearrange this equation to solve for qh the amount of heat transferred. So QH is equal to the work out divided by A. So when we plug in our known variables, the workout was 4.9 multiplied by 10 to the power of nine jewels divided by 0. 965. So that means that our QH value should be equal to when we put it into a calculator, 9.869 multiplied by 10 to the power of nine Jews, which is approximately equal to when we round 9.87 multiplied by 10 to the power of nine jewels. And that is our final answer and we did it. So looking at our multiple choice answers, the correct answer is the letter B 9. multiplied by 10 to the power of nine Jules. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

The engine that powers a crane burns fuel at a flame temperature of 2000℃. It is cooled by 20℃ air. The crane lifts a 2000 kg steel girder 30 m upward. How much heat energy is transferred to the engine by burning fuel if the engine is 40% as efficient as a Carnot engine?

Verified Solution

Key Concepts

Carnot Efficiency

Heat Transfer

Work Done by the Engine