### Video Transcript

The body shown being acted on by three forces has a mass of 10.0 kilograms. Find the acceleration of the body.

Looking at the diagram given, we see that the mass ๐ is being acted on by ๐น one, ๐น two, and ๐น three. To solve for the acceleration of mass ๐, weโll want to know what are the ๐ฅ- and ๐ฆ-components of these three forces acting on the mass. We can do that and organize our information more easily using a table. In this table, weโll have our three forces: ๐น one, ๐น two, and ๐น three as well as the components of those forces: ๐น sub ๐ฅ and ๐น sub ๐ฆ. And what weโre trying to do is after figuring out the ๐ฅ- and ๐ฆ-components of each of the three forces, we want to add them together to find the total ๐ฅ- and the total ๐ฆ-component of the force acting on the mass ๐.

So here is our table that weโll fill in and again we want to look at the ๐ฅ- and ๐ฆ-components of the three forces acting on ๐. So letโs start with ๐น one. As we look at ๐น one in the diagram, we see that it has no horizontal component; itโs all vertical. So the ๐ฅ-component of ๐น one is zero. Moving on to its ๐ฆ-component, we see ๐น one points in the negative ๐ฆ-direction so that ๐ฆ-component of the force is negative 10.0 newtons.

Filling that into the table, we move on to ๐น two. Now, ๐น two, unlike ๐น one, has both ๐ฅ- and ๐ฆ-components. And to figure those out, we want to use the fact that we can draw a horizontal dotted line between the dotted vertical line we see and the end of ๐น two. So letโs draw that in now. What we see with that line drawn in is that our two dotted lines are at right angles to one another and that these dotted lines together with ๐น two form a right triangle.

So we can solve for the ๐ฅ- and ๐ฆ-components of ๐น two using that fact. The ๐ฅ-component of ๐น two is equal to the magnitude of ๐น two, 20.0 newtons, multiplied by the sine of 37 degrees. You maybe more used to seeing the cosine associated with horizontal or ๐ฅ-direction motion or forces, but in this case because of the particular way weโve drawn our triangle, indeed weโre working with the sine function to find that horizontal component of ๐น two.

With that ๐ฅ-component of ๐น two added to our table, now letโs work on the ๐ฆ-component. That will be equal to again the magnitude of ๐น two, 20.0 newtons, multiplied this time by the cosine of 37 degrees because the side of our triangle we want to solve for is the side adjacent to our 37-degree angle. Letโs fill that value in in our table.

Now, we move on to ๐น three, the last force of the three acting on mass ๐. We see that just like with ๐น two, we can draw in a dotted horizontal line that joins the end or the arrowhead of ๐น three with the dotted vertical line weโve drawn in. That line shown here forms the opposite side of our triangle.

So the ๐ฅ-component of ๐น three is equal to the magnitude of ๐น three multiplied again by the sine of 37 degrees. But notice that in this case that value is negative because the ๐ฅ-component of ๐น three points in the negative ๐ฅ-direction. So overall that will be negative 10.0 newtons multiplied by the sine of 37 degrees. Then finally, the ๐ฆ-component of ๐น three is equal to positive 10.0 newtons multiplied by the cosine of 37 degrees.

Great! Now our table is all filled out and weโre ready to add up the ๐ฅ- and ๐ฆ-columns to solve for the ๐ฅ- and ๐ฆ-components of the net force โ the total force โ acting on mass ๐. Adding up the three values in our ๐ฅ-column, we get an answer of 6.02 newtons. And in the ๐ฆ-direction, negative 10 newtons plus 20 newtons times the cosine of 37 degrees plus 10 newtons times the cosine of 37 degrees equals 14.0 newtons.

Now again, these are the components of the net force that acts on ๐. So letโs write it out and make it official; weโll call that net force ๐น sub net. And we now know that that force is equal to 6.02 ๐ plus 14.0 ๐ newtons. And remember that this is all in service to figuring out what is the acceleration that the mass ๐ takes on under the influence of all three of these forces.

So weโve solved for that net force, resulting from ๐น one, ๐น two, and ๐น three. And weโve been given the mass of our object, 10.0 kilograms, and wanna solve for its acceleration. One of Newtonโs laws of motion, the second law connects all three of these values. The second law says that the net force on a mass ๐ is equal to that mass multiplied by its acceleration. We can use this law to solve for acceleration. Letโs start by dividing our net force by the mass of our object ๐.

Now when we do this, what weโve done has created a fraction that is equal to the acceleration of our mass. Thatโs true by Newtonโs second law. So now we can plug in for our mass ๐, the mass of 10.0 kilograms. Having done that, we now divide these numbers using our calculator to solve for the net acceleration of the mass ๐.

And we find that that net acceleration that mass ๐ experiences under the influence of these three forces is 0.602 ๐ plus 1.40 ๐ meters per second squared. That is the net acceleration of mass ๐.