What is the total gravitational potential energy of the three ...

Question

What is the total gravitational potential energy of the three masses in FIGURE P13.36?

Accepted Answer

Hello, fellow physicists today we solved 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. Three masses are arranged as shown in the figure below, determine what the total gravitational potential energy is. So that's our goal, our angles, we trying to figure out what the total gravitational potential energy is. So now that we have an idea of what we're solving for. Let's read o let's read off our multiple choice answers to see what our final answer might be. And let us note that they're all in the same units of jewels. So A is negative 0.40 multiplied by 10 to the power negative seven B is negative 0.47 multiplied by 10 to the power of negative seven C is negative 0.47 multiplied by 10 to the power of negative nine and D is negative 0.40 multiplied by 10 to the power of negative nine. OK. So now really quick, let's look at the diagram that's provided to us by the prom itself. That shows how our three masses are arranged. So at the very top of our three masses, which it appears when you connect them with, with black lines, they form a triangular shape. So at the very top at 4.0 kg is our green circle. And then to the left is our 7.0 kg blue mass. And then to the right is our 9.0 kg red mass. So we have our 4.0 kg green mass at the top. So from our green mass to our blue mass is 25.0 centimeters. And then from our blue mass to our red mass is 27.0 centimeters apart. And from our green mass to our red masts is 14.0 centimeters. OK. So with that in mind, now that we have a visualization of what's going on in this particular problem, let's write down all of our known variables. So let us denote our three masses like so ma is equal to 4.0 kg and let us call MB is equal to 7.0 kg. And finally, let's call MC is equal to nine 0.0 kilograms. OK. So now at this point, we need to write down all the distance values between these specific masses which we will call rab. So rab is equal to 25.0 centimeters, which is also equal to 0.25 m. So you can use dimension analysis to perform this conversion or you can quickly look it up. I just gave it to you to save some time. We also know that R BC is equal to 27.0 centimeters, which is also equal to 0.27 m. And we know that our eight seed is equal to 14.0 centimeters, which is equal to 0.14 m. OK. So now at this point, we need to recall and use the equation for the total gravitational potential energy U which we can write as U is equal to UAB plus U BC plus UC A which we can also recall and expand our equation to write that U is equal to negative capital G multiplied by ma multiplied by MB divided by rab minus capital G multiplied by MB multiplied by MC divided by R BC minus capital G multiplied by ma multiplied by MC all divided by R ac. Awesome where capital G is our gravitational constant. And let's make a quick little note here off to the side in blue that capital G, the numerical value for G, the gravitational constant is equal to 6.674 multiplied by 10 to the power of negative 11. And its units are newtons multiplied by meters squared divided by or per kilograms swear fantastic. So now we can simplify our equation to write that you is equal to capital G multiplied by negative ma multiplied by MB divided by R A B minus MB multiplied by MC, divided by R BC minus ma multiplied by MC divided by R ac Fantastic. So now at this point, we can plug in all of our known variables to solver the numerical value of U which is our final answer that we're ultimately trying to solve for. So when we plug in our known variables, we will find that U is equal to G which is 6.674 multiplied by 10 to the power of negative 11 newtons multiplied by meters squared per kilogram square multiplied by negative 4.0 kilograms, multiplied by 7.0 kg, divided by 0.25 meters minus 7.0 kg, multiplied by nine 0.0 kg, divided by 0.27 meters minus 4.0 kg multiplied by 9.0 kg divided by 0.14 m. Fantastic. So let me plug that into our calculator and we round to two decimal places. We will get negative 0.40 multiplied by 10 to the power of negative seven jewels. And that's it. That's our final answer. Hooray. We did it, we found our final answer. So looking at our multiple choice answers, the correct answer has to be the letter A negative 0.40 multiplied by 10 to the power of negative seven jewels. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

What is the total gravitational potential energy of the three masses in FIGURE P13.36?

Key Concepts

Gravitational Potential Energy

Mass and Weight

Reference Point in Potential Energy Calculations

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