A powerful laser portrayed in a movie provides a 3-mm-diameter...

Question

A powerful laser portrayed in a movie provides a 3-mm-diameter beam of green light with a power of 3 W. A good agent inside a spacecraft aims the laser beam at an enemy astronaut hovering outside. The mass of the enemy astronaut is 120 kg and the spacecraft 185,000 kg. (a) Determine the “radiation-pressure” force exerted on the enemy by the laser beam assuming her suit is perfectly reflecting. (b) If the enemy is 30 m from the spacecraft’s center of mass, estimate the gravitational force the spacecraft exerts on the enemy. (c) Which of the two forces is larger, and by what factor?

Accepted Answer

Welcome back. Everyone in this problem. A space station uses an advanced laser system that emits red light with a power output as high as seven watts to deflect space debris from its path. The diameter of this laser beam measures four millimeters. If we consider an instance where it aims at deflecting an object weighing around 80 kg, it happened, it happens to be located 20 m away from its center. First. What would be the radiation pressure force exerted on this object? If we assume it perfectly reflects all incident light. Second, can you calculate how much gravitational pull this space station which weighs nearly 300,000 kg exerts on this object? And third, which of these two forces exerts more influence over the other. For our answer choices. A says the radiation pressure force is 4.7 multiplied by 10 to the negative eight newtons. The gravitational pull is four multiplied by 10 to the negative sixth newtons. And that the gravitational force is greater than the radiation pressure force by a factor of 85 B says they are 4.9 multiplied by 10 to the negative eight and four multiplied by 10 to the negative sixth newtons respectively. And the gravitational force is greater than the radiation pressure force by a factor of 90 C says they are 5.2 multiplied by 10 to the negative F and the six multiplied by 10 to the negative sixth newtons. And the gravitational force is less than the radiation pressure force by a factor of 95. While B says they are 5.2 multiplied by 10 to the negative eight and six, multiplied by 10 to the negative sixth newtons respectively. And the gravitational force is less than the radiation pressure force by a factor of 98. No, we're given a lot of information in this problem. And we have three questions that essentially we want to answer. First, we want to find the radiation pressure force that's exerted on the object given those conditions. Next, we want to figure out how much gravitational pull the space station exerts given its width or its mass rather. And finally, we want to figure out which of the two forces exerts more influence over the other. So we're going to compare the radiation pressure to the gravitational pull force. Now, let's make a note of all the information we have before we continue. So what do we already know? Well, so far we know that the power of the laser, OK is equal to seven watts. Next, we also know that the diameter of the laser beam, let's call that D is equal to four millimeters. OK. We know that the mass of our object based on our problem says it weighs around 80 kg. So we can say MOBG is 80 kg. J subscript. OK. We also know that the distance between the object and the space station is 20 m. So let's call that capital D OK. Uh 20 m. The mass of the space station, let's call that MS is 300,000 kg. OK. And from this information, we want to figure out our radiation pressure force, let's call that FP our gravitational pull FG. And then we want to find out how FR P compares to FG. So let's start with the first problem, the radiation pressure force. Now how can we figure that force out? Well, recall, OK, recall that the force, the force due to the radiation pressure. OK. Let's call that FRP is going to be equal to the pressure multiplied by the area. Now, we can figure the area out. OK. Based on the information we have or, or we can get to that later, but we also need the radiation pressure. So how can we figure out what the pressure on the object is due to the laser? Well, we also know, OK, we also know, let me make a note here. OK. We also know that our power is the change in energy over time. So we can call that DUDT which is seven watts just to make sure we distinguish it from peeing this problem because they are not the same thing. This is not the pressure. OK. It's the power. So because we don't want to confuse them, I I rewrote it as DUDT. Now we also know that the power is related to the average magnitude of the pointing vector, the pointing vector S is equal to our power. DUDT. OK. Divided by a, in other words, it is the power per unit of area. It's our average uh energy flux. OK. But we also know that the pressure on the object due to the laser, OK. The pressure on the object P due to the laser is equal to two multiplied by the pointing vector divided by the, the, the speed of light in a vacuum. OK. And it's too multiplied by the pointing vector divided by the speed of light. Because in our problem, remember it told us that it perfectly reflects all incident light. OK. So because it's perfectly, the wave is perfectly reflected, that means the momentum is doubled. So now here, since we have a value for our pressure, OK, then we can try to substitute it into our formula including the pointing vector to see if we can simplify, to get the force due to the pressure. OK. Because using this idea, then that means the force due to the pressure, the radiation pressure force is going to be equal to the pressure that's two s divided by C OK. Multiplied by the area where like we said, s our pointing vector equals our power divided by our area. So in other words, that's going to be two multiplied by our power. OK? Divided by our area. OK? Multiplied well, multiplied by ac multiplied by our area. And now the area cancers, they leave us with two multiplied by our power divided by the speed of light. Now we can go ahead and solve because when we substitute those values, OK, then we get that radiation pressure force to be equal to two multiplied by seven watts, divided by three, multiplied by 10 to the eighth meters per second, which simplifies to 4.7 multiplied by 10 to the negative eight newtons. OK. So this is going to be the radiation pressure force. Next, let's try to figure out the b part the gravitational pull the space station exerts on this object. Now, what do we know about our gravitational pull? Well, recall Newton's law of universal gravitational force which tells us that the gravitational force is going to be equal to the gravitational constant G multiplied by the product of both masses. In this case, the mass of the station and the mass of the object divided by the distance between them squared. In this case, that distance between them earlier, we defined as D OK. So now that we have all of this information, we can go ahead and substitute, we know that the gravitational constant is 6.674 multiplied by 10 to the negative 11th Newton square meters per kilogram squared. OK. The mass of the space station is three multiplied by 10 to the 6 kg. OK. And the mass of the object is 80 kg. And then we know that the distance between the space station and the object is 20 m that's going to be squared. Now, we can go ahead and solve for our gravitational force. And when we do that, you should find that the gravitational force equals four multiplied by 10 to the negative sixth newtons. So this is going to be our gravitational force. Let me use a different color here. OK. So we have our radiation pressure force, we have our gravitational force. The last thing we need to do is to compare both of them because remember we want to figure out which of the two forces exerts more influence over the other. And by a factor of what? Well, when we compare our radiation pressure force to our gravitational force, notice that the gravitational force is much larger than the force due to the pressure of the laser. So here FG is greater than FRP. And now the factor that it's greater than it by is going to be equal to the gravitational force divided by the radiation pressure force. That's going to be four multiplied by 10 to the negative sixth newtons divided by 4.7 multiplied by 10 to the negative eighth newtons. And by dividing, you should find that a factor equals 85. Thus, to sum up all of what we've done so far, the radiation pressure force is 4.7 multiplied by 10 to the negative eight newtons. The gravitational force is four multiplied by 10 to the negative sixth newtons. And the gravitational force is greater than the radiation pressure force by a factor of 85. If we go back to our answer choices, that means a is the correct answer. Thanks for watching everyone. I hope this video helped.

Key Concepts

Radiation Pressure

Gravitational Force

Comparative Force Analysis

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