Two stars, one twice as massive as the other, are 1.0 light year (ly) apart. One light year is the distance light travels in one year at the speed of light, 3.00 ✕ 10⁸ m/s . The gravitational potential energy of this double-star system is ─8.0 ✕ 10³⁴ J. What is the mass of the lighter star?

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Hi, everyone in this practice problem, we're being asked to find the mass of the heavier star in the binary star system where one of the stars is four times heavier than the other. It is known that the gravitational potential energy within the two stars is negative 1.2 times 10 to the power of 34 Jules. And we're being asked to find the mass of the heavier star. If the distance between the two stars is 1.5 light years. The options given are a 3.2 times 10 to the power of kg b eight times 10 to the power of 30 kg C 6.3 times 10 to the power of 59 kg and D 8.0 times 10 to the power of 59 kg. So we will consider the stars as spherical masses. And I'm gonna start us with listing the given information in the problem statement. The first one is going to be the gravitational potential energy which is represented by this U which is given to be 1.2 times 10 to the power of Jules. Next, we have the distance which is represented by R which is 1.5 light years, which is represented by this L Y right here. And we need to convert that into meter so that we can use it in our further calculation. So to do so we want to multiply it by three times 10 to the power of eight, which is the speed of light meters per second multiply that again by 3600 seconds for every hour. And then we wanna multiply that again with 24 hours in a day. And then we wanna multiply that as well with 365 days per year or why? So then that will result in an R value of 1.42 times 10 to the power of 16 m just like so awesome. Next, we want to recall the equation that we will use, which is the equation for the gravitational potential energy. So you can be calculated by multiplying G multiplied by M one multiplied by M two, divided by R and a negative sign right in front just like. So in this case, it is given that M one equals four M two. So we are interested in finding the heavier star or the mass of the heavier star which is M one. So I'm just gonna put for uh I'm gonna substitute M one with four M two and then recalculate M one in the end. Awesome. So let's substitute uh four M one and equals to four M two into this equation. So that will give negative G multiplied by four M two squared divided by R. In this case, we want to find M two. So I'm gonna rearrange this equation that we have to then find an equation for M two. So M two will then equals two. Um the square root of negative U multiplied by R divided by four G Awesome. So this is going to be the final equation that we need. In order for us to get M two, we also already know all the information needed because the G here is essentially just a constant. So we know all the information needed to calculate M two. So let's substitute everything into this equation right here. So M two will then equals to the square root of negative U which is going to be negative 1. times 10 to the power of 34 Jule multiply that by R which is 1.42 times 10 to the power of 16 m multi uh defied that by four G which is four multiplied by G is going to be the constant which is 6.67 times 10 to the power of negative 11 Newton multiplied by a meter squared divided by kilograms squared, putting that into the calculator or calculating everything, we will then get M two to be the value of 7.99 times 10 to the power of kg. Awesome. I'm just gonna round it up right now for simplicity purposes. So M two will be 8 10.0 times 10 to the power of 29 kg. So then we are, what we're interested to find is M one which is going to equals to four M two. So in this case, M one will then be equals to four multiplied by 8.0 times 10 to the power of kg or M one essentially will equals to 3.2 times 10 to the power of kg. So the mass of the heavier star is going to be 3.2 times 10 to the power of 30 kg, which will correspond to option A in our answer choices. So answer A will be the answer to this particular practice problem and that'll be it for this video. If you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topics and that'll be it for this one. Thank you.

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Key Concepts

Gravitational Potential Energy

Mass and Distance Relationship

Units of Measurement in Astronomy