Welcome to ElectroOnline. Here's our second example of how to calculate the forces acting on a block on an inclined plane. And in this case, contrary to the previous example, there is indeed a friction force.
Matter of fact, the parallel component to the surface of the applied force, F cosine theta, is going to be equal to the sum of the, what we call, parallel component of the weight, plus... the friction force between the block and the surface in other words f cosine theta equals mg sine theta plus the maximum friction force that can exist between the block and the inclined plane in other words this block is on the verge of moving if this force becomes any larger the block will begin to move up the incline so in this particular case notice that the reaction force is no longer perpendicular to the surface because we now have a friction force. And we're asked to find the applied force F and we're asked to find the reaction force R.
And just so we can compare things to one another, we're also going to calculate those three forces right there, or at least those three components. Also notice we still have a nice triangle where we have the applied force. the weight of the block and the reaction force. But in this case, notice that the angle between the reaction force and the vertical is equal to theta plus an additional angle phi.
So this angle becomes the sum of theta plus phi. Only theta is known, phi is not. If I now also know phi, so that I know the total angle here, and knowing the weight of the object, I can very easily calculate the applied force and the reaction force.
So, The path to success here would be to find this phi right there. And to find that, we're going to look at this right triangle. Notice that this is a right triangle here, and that this here is equal to the friction force.
So this here would be equal to... I have this arrow in the wrong direction. All right, let me change this here.
Okay, I need it in this direction. That's better. There we go.
And so I have this force right here and this force. When we add them together... There we go.
So we have the reaction force caused by the friction force and the normal force combined. So the reaction force is the vector force or the vector sum of the normal force pushing back and the friction force that's fighting against the normal force. this component of the force that's applied right there.
All right, now I'm ready to go. Now I can figure out this angle right here. If this is equal to the friction force, then this is equal to the normal force times the static coefficient of friction. And because this is a right triangle, I can say that the tangent of phi is equal to the ratio of the opposite side divided by the adjacent side. And the opposite side is going to be the normal force times mu sub s, and the adjacent side is going to be equal to the normal force, which means that the tangent of phi is equal to this fraction.
Notice that the normal forces cancel out. Makes it easy. That'd be it. that means that the tangent of phi is equal to the static coefficient of friction, or the angle phi is equal to the arc tangent of the static coefficient of friction. Notice how easy that makes things.
So we know that the static coefficient of friction is 0.35, which means that the angle phi is equal to the arc tangent of 0.35, which is equal to... 0.35 times the inverse tangent which is 19.29 degrees. Now that we know that, now we can see that since theta is 25 degrees, that theta plus phi is going to be 25 degrees plus another 19.29 degrees.
which is 44.29 degrees. So now we understand what this angle is. Now we can very easily find F and R. We can use the concept of the cosine of, of course, this is going to be theta plus phi, and the concept of the tangent of theta plus phi.
We know that the cosine of theta plus phi by definition is equal to the ratio of the adjacent side divided by the hypotenuse. So in this case, the cosine of 44.29 degrees is equal to the adjacent side to the angle is mg, that's 100 newtons, divided by the hypotenuse, which is equal to r. which means that r is equal to 100 newtons divided by the cosine of 44.29 degrees and with a calculator we take the cosine of that bring it to the denominator times 100 is r the reaction force is equal to 139.7 newtons wow that was rather easy wasn't it so now we'll use the tangent of the angle We can say that the tangent of theta plus phi by definition is equal to the ratio of the opposite side divided by the adjacent side.
In other words, the tangent of 44.29 degrees is equal to the opposite side of the angle is F and the adjacent side of the angle is mg, which is 100 newtons. is equal to 100 newtons times the tangent of 44.29 degrees. There's a dot right there.
All right. So we take the, let's see here, we take 44.29, take the tangent of that, and multiply it times 100, and we get the applied force is equal to 97.55 newtons. I'll just keep my next to decimal place. and so we found the reaction force and the applied force in a very easily straightforward manner now just so we can verify that we do things correctly let's calculate mg sine theta the maximum friction force and f cosine theta and see that f cosine theta is indeed equal to the sum of these two all right mg sine theta is equal to 100 newtons for mg times the sine of 25 degrees which is 25 take the sine times 100 equals 42.26 42.26 newtons all right next we're going to find the maximum friction force friction force max which is equal to the normal force times mu. So the normal force is 100 newtons times the cosine of 25 plus F, which we found to be 97.55, times the sine of 25. And now we have to take the whole thing and multiply it times the static coefficient of friction times 0.3.
equals and we get 46.15 Newtons. So now we found mg sine theta which is this component of the weight and we found the maximum friction force which is this force right here. And the sum of these two should add up to F cosine theta.
Let's find out if it does. So F cosine theta is equal to, we found F, 97.55 newtons. And we multiply it times the cosine of 25 degrees. All right, 25 take the cosine times 97.55 equals, and that gives us 88.41 newtons. Now, does 46.15 plus 42.26 add up to this?
Well, 46 plus 42 is 88, and 0.15 and 0.26 is 0.41. And sure enough, we've just shown that we got the right results by doing it both ways. Now you can see again the power of this technique right here. All we have to do is calculate this angle. We need to find phi, and phi can be found by simply taking the relationship of this right triangle right here.
Once we find phi, we get the combined angle, and then we can very easily find the reaction force and the applied force F. And that's how it's done.