Binary star: Two equal-mass stars maintain a constant distance...

Question

Binary star: Two equal-mass stars maintain a constant distance D apart (Fig. 6–34) of 8.0 x 10¹¹ m and revolve about a point midway between them at a rate of one revolution every 14.2 yr.

(b) What must be the mass m of each star?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem, consider a celestial system where two stars each of identical mass orbit around their common center of mass, which is located precisely at the midpoint of the line joining them, these stars are separated by 10, multiplied by 10 to the power of 11 m. They complete one full orbit around their common center of mass every 12.6 years, determine the mass of each star in this dual star system. So that's our goal. Our angles were ultimately trying to determine what the mass of each star is in this particular dual star system. Awesome. So let's look at our figure that's provided to us by the problem itself. On the far right, we have our mass M one, which is the mass of our first star. And then we have the mass of our second star M two. And let's note that M one is this orange spear and M two is this red spear which M two is the mass of our second star. And then in between our two stars, we have a black line joining them and then we have a red dot on our black line that's denoted as CM, which is our center of mass. Awesome. So from our center of mass to M one, we have some radius R one and then from our center of mass to our star two or M two, the mass of our star two is some distance R two, a radius distance of R two away. And, and they're separated by 10, multiplied by 10 to the power of 11 m between our second star and our first star, which is denoted by our purple arrow here with our purple lines. And then we can see that we have a blue dotted circle going around for M one mean M one has a larger orbit radius compared to M two, which is the green circle, the inner circle, a dotted line circle on the inside because it has a shorter radius. So M one has to travel a greater distance around whereas M two has to travel in a circle at a shorter distance which has a shorter radius. Awesome. So now that we have a visualization of what's going on in this particular problem, let's look at our multiple choice sensors to see what our final answer might be and let us know that they're all in units of kilograms. So A is 2.2 multiplied by 10 to the power of 29 B is 1.9 multiplied by 10 to the power of 30 C is 7.1 multiplied by 10 to the power of 29. And D is S is 8.7 multiplied by 10 to the power of 30. Awesome. So moving right along here. So first off, let us draw a free body diagram to help us better visualize all the forces that work in this particular problem. So looking at our free body diagram, which I put to the right of our figure that's provided to us, we have the distance between our two different stars and we have our star on the left and then we have our star on the right and there's some distance d apart which we know the distance is 10 multiplied by 10 to the power of 11 m. And as you can see, we have the gravitational force of the star on the left pointing towards the center. And we also have our star which is the force, the gravitational force, which is denoted as F subscript G pointing towards the center mass. So both stars on the left and the right are both have a gravitational force. They both have their gravitational force pointing towards the center, so they're pointing towards each other. Awesome. So now that we have an idea of what's going on force wise, let us now note that we are given our first piece of information, which states that the two stars have an equal mass. And secondly, that the distance between the two stars is equal two, 10 multiplied by 10 to the power of 11 m. And thirdly, the period is 12.6 years and we need to convert this to seconds. So let's take a moment here to pause and convert our period, which is 12.6 years to seconds in order to get it into the proper units to help us solve for this particular problem. So let us recall that a year has 365 days. So if we recall and use dimensional analysis, we could write that there are 365 days in one year. And ultimately, what we're trying to do is we're trying to take and we're trying to convert to seconds and then we're gonna multiply 12.6 years by how many other seconds there are in one year. So that's what we're trying to solve. We're right at this second. So note that there's 24 hours in one day. So we're trying to get all of our units to cancel out, but leaving us with just seconds. So as we should be able to recall, there's 3600 seconds in one hour. So when we plug that into our calculator, we will get 31 1000 536. So 31,536,000 seconds per year. Awesome. So it's gonna be 31 comma 536 comma 000 seconds per year. So now we need to convert seconds per year to just seconds. So we take 12.6 years and we multiply it by 31 comma 536 comma 000 seconds per year. As you could see our year units cancel out leaving us with just seconds. So that will mean that our final answer is approximately equal to 3.974 multiplied by 10 to the power of eight seconds when we round to three decimal places. So now we must set the gravitational force on one of the stars equal to the centripetal force using the following relationship that we should recall as V equals two multiplied by pi multiplied by R divided by capital T, which is equal to pi multiplied by D divided by capital T or capital T is our period. And R of course is our radius and D is the distance. So now we must solve for the mass. So now we need to set, so we need to note that when we set the gravitational force FG equal to the centripetal force FC, we can write that FG is equal to FC. So which we can also recall and write as so let's draw a little arrow. So we can write this equation also as capital G, which capital G is our gravitational constant. Multiplied by capital M squared divided by D squared, which is equal to capital M multiplied by V squared divided by D divided by two. Awesome. And let's call this equation one. So now we need to plug in the value of B into equation one. And then we could write, and let's call this equation two, we can write that capital G multiplied by capital M which I haven't said yet. But M is our mass. So capital M squared divided by D squared is equal to capital M multiplied by pi multiplied by D divided by capital T to the power of two divided by D divided by two. Awesome. So now we need to rearrange equation two to isolate and solve for capital M which is our final answer. And we will get that M is equal to two multiplied by pi squared multiplied by D to the power of three divided by capital G multiplied by T squared. So now at this point, we need to plug in all of our known variables in sulfur M which is our final answer. So let's do that. So two multiplied by pi squared multiplied by our distance D which is 10 multiplied by 10 to the power of 11 m. And don't forget that our distance is cubed or another way of saying it is to the power of three. And we need to divide this by our gravitational constant, which we should recall the gravitational constant is 6.67 multiplied by 10 to the power of negative 11. And its units are newtons multiplied by meters squared, divided by kilograms squared. And then we need to multiply this by our period which we found our period t in seconds to be 3.974 multiplied by 10 to the power of eight seconds. And don't forget that our period is squared. Therefore, when we plug this into our calculator, we will find that the mass M, when we round to one decimal place is equal to or approximately equal to 1.9 multiplied by 10 to the power of 30 kg. And that's it. We've solved for this problem. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter B 1.9 multiplied by 10 to the power of 30 kg. And that's it we've solved for this problem. Thank you so much for watching. Hopefully that helped and I can't wait to see in the next video. Bye.

Struggling with Physics?

Binary star: Two equal-mass stars maintain a constant distance D apart (Fig. 6–34) of 8.0 x 10¹¹ m and revolve about a point midway between them at a rate of one revolution every 14.2 yr.

(b) What must be the mass m of each star?
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Struggling with Physics?

Binary star: Two equal-mass stars maintain a constant distance D apart (Fig. 6–34) of 8.0 x 10¹¹ m and revolve about a point midway between them at a rate of one revolution every 14.2 yr.(b) What must be the mass m of each star?<IMAGE>

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Newton's Law of Gravity practice set

Binary star: Two equal-mass stars maintain a constant distance D apart (Fig. 6–34) of 8.0 x 10¹¹ m and revolve about a point midway between them at a rate of one revolution every 14.2 yr.

(b) What must be the mass m of each star?
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