(II) Four masses are arranged as shown in Fig. 6–28. Determine...

Question

(II) Four masses are arranged as shown in Fig. 6–28. Determine the x and y components of the gravitational force on the mass at the origin (m). Write the force in vector notation (î, ĵ) .

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. Four masses which are called M one equals 5 kg. M two equals 4 kg. M three equals 3 kg and M four equals 6 kg are positioned at the corners of a rectangle as shown below, determine what the X and Y components of the gravitational force exerted on the mass. M one due to the other three masses will be express the force in vector notation using I hat and J hat. So that's our end goal. Our goal is ultimately we're trying to figure out what the X and Y components of the gravitational force exerted on mass M one due to the other three masses will be. And we're also asked to a like write our final answer or express the force in vector notation using I and J hat. So that's what we're ultimately trying to do. Awesome. So looking at our figure that's provided to us, we have our rectangle denoted with a blue dotted line starting from M four or mass four going to mass three. So from left to right and then mass three down from top to bottom to M two. And then on the X axis we can have on the far left, we have M one going to M two on the right. Awesome. And note that M two is 2 m away from M one. I referring to our, our X axis here which indicates in meters in light gray, which we have our numbers in light gray corresponding to their meter values. So that will mean that M one is 2 m away from M two and M one is 3 m away from M four. So from bottom to top. So now that we have a visualization of what's going on in this particular problem, let's read off our multiple choice answers to see what our final answer might be. A is 2.86 multiplied by 10 to the power negative 10 newtons I hat plus 3.76 multiplied by 10 to the power negative 10 newtons. J hat B is 3.76 multiplied by 10 to the power of negative 10 newtons I hat plus 2.86 multiplied by 10 to the power of negative 10 newtons. J hat C is 5.73 multiplied by 10 to the power of negative 10 newtons I hat plus 7.52 multiplied by 10 to the power negative 10 newtons J hat and D is 7.52 multiplied by 10 to the power negative 10 Newton's I hat plus 5.73 multiplied by 10 to the power negative 10 Newton's J hat. Ok. So now, first off, let us note that the force on M one due to M two will point in the I hat direction. Let us also note that the force on M one due to M four will point in the J hat direction. Also the force on M one due to M three will point in the direction given by theta. Now we need to add the vectors together in order to find the net force. Thus, we need to recall and use the following net force equation which states that vector F is equal to the gravitational constant capital G multiplied by 4 kg multiplied by 5 kg all divided by 2.0 m squared I hat plus capital G multiplied by 6 kg multiplied by 5 kg, all divided by 3.0 m squared J hat plus capital G multiplied by 3 kg multiplied by 5 kg, all divided by 2.0 m squared plus 3.0 m squared plus oh, don't forget before. We have to add, we have to multiply oh sign of the I had. Now we need to add our last one because we're doing it for each of the masses. So M one M two M three M four. So now we need to add the gravitational constant capital G once again, multiplied by 3 kg, multiplied by 5 kg, divided by 2.0 m, 2.0 m squared plus 3.0 m 3.0 m squared. Awesome. And then this will be multiplied by sign of theta I had or J hat actually J hat. So moving right along here, so we quickly recap here. So we have our net force equation. So now we're considering M one, the M one mass M two mass M three mass M four mass. So that's four different equations that we're considering. So now what we're gonna be doing is we're gonna be simplifying this equation down to its most simple form. And then we're gonna plug in all of our known variables to solve for the numerical answer for our X and Y components. So when we start to simplify, we will get that vector F is equal to capital G multiplied by five and its units will be kilograms squared per meter squared. And this will be I had plus 13th. Well, 13th in its units will be kilograms squared per meter squared J hat plus 15 13th multiplied by two divided by the square root of 13. And its units will be kilograms squared per meter squared and this will be in the eye hat direction as well. Awesome and moving right along here. We now need to add 15 13th multiplied by three divided by the square root of 13. And don't forget that its units will also be kilograms squared per meters squared J hat and don't forget your bracket. So then simplifying one last time will give us vector F is equal to capital G, which is our gravitational constant multiplied by bracket bracket. Five plus 30 divided by 13 multiplied by the square root. 13 racket. Don't forget its units will still be kilograms squared per meter squared. And don't forget that this is, well, I hat or I hat value plus 13th plus 45 divided by 13 multiplied by the square root of 13. Don't forget that it's unit will be kilograms squared per meter squared and this will be in the J ha direction and don't forget your final bracket here. Awesome. So now at this point, we need to plug in all of our known variables to solve for vector F. So let us recall that the gravitational constant is 6.67 and it's gonna be multiplied by 10 to the power of negative 11 and its units will be newtons multiplied by meters squared per kilogram squared. Awesome. And this will be multiplied by when we multiply out our five plus 30 divided by 13 multiplied by the square of 13. When we multiply those values out or each of our bracket values out, we'll get 5.64 kilograms squared per meters squared. I hat plus 4.29 kilograms squared per meter squared. And don't forget that this is RJ hat value. Awesome. So when we plug that into our calculator, we will find that vector F is equal to 3.7 six multiplied by 10 to the power of negative 10 newtons I hat plus two point 86 multiplied by 10 to the power of negative 10 newtons J hat. Awesome. And that's it we've solved for this problem. Hooray, we did it. So this problem is relatively easy to solve. As long as you have a strong understanding of vector notation and like how to use the vector math. And if you have a strong understanding of that solving a problem, like this is pretty easy. So looking at our multiple choice answers, the correct answer has to be the letter B 3.76 multiplied by 10 to the power negative 10 Newton's I hat plus 2.86 multiplied by 10 to the power negative 10 Newton's J hat. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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(II) Four masses are arranged as shown in Fig. 6–28. Determine the x and y components of the gravitational force on the mass at the origin (m). Write the force in vector notation (î, ĵ) . <IMAGE>

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