Video Transcript
A body is resting on a rough horizontal plane. The coefficient of friction between the body and the plane is 0.2 and the limiting friction force that is acting on the body is 80 newtons. Given that 𝑅 is the resultant of the force of friction and the normal reaction force, find the magnitude of 𝑅.
With questions like this, it can be really sensible to begin by sketching what you know out. We have a body resting on a horizontal plane. We don’t know anything about the mass of this body. So we could define the mass to be equal to 𝑚, meaning the downwards force of the body on the plane is 𝑚𝑔, mass times gravity. We then know that there is a normal reaction force of the plane on the body. This acts in the opposite direction. Now, normally, we might define that to be equal to 𝑅. But 𝑅 is defined as a resultant force, so we’ll define this to be equal to 𝑁.
Since the plane is rough, we know that there will be some sort of frictional force on the body. For there to be a frictional force though, we also need to assume that there is a force acting in the opposite direction to this. Let’s define that force to be equal to 𝐹 and the frictional force to be equal to 𝐹 sub 𝑟.
Now, we actually have a formula that links the friction and the normal reaction force. We usually define it as friction is equal to 𝜇𝑅, where 𝜇 is the coefficient of friction. In this case, we’re going to redefine it as 𝜇𝑁 since 𝑁 is our normal reaction force. We’re told that the coefficient of friction is 0.2, so we can rewrite this as friction equals 0.2𝑁.
We’re also told that the limiting friction force acting on the body is 80 newtons. This tells us two things. We can rewrite this equation as 80 equals 0.2𝑁. And then, we can solve this by dividing through by 0.2 to get 𝑁 is equal to 400 or 400 newtons. What it also tells us is that the body is on the point of moving, so the forwards force, that’s 𝐹, must also be equal to 80 newtons.
So, we now know the normal reaction force and the force of friction. To find the resultant force 𝑅, we’re going to use a force triangle. This force triangle can be represented using a right-angled triangle since we know the normal reaction force and the frictional force meet at 90 degrees. We’re looking to find the resultant. That’s this length of the triangle here. And so, since we’re working with a right-angled triangle, we can use the Pythagorean theorem.
The Pythagorean theorem tells us that the sum of the squares of the two shorter sides is equal to the square of the longer side. Remember, the longer side is the hypotenuse. It’s the side opposite the right angle. So, here, that’s the side that is 𝑅 or 𝑅 newtons. And so, our equation is 80 squared plus 400 squared equals 𝑅 squared. 80 squared plus 400 squared is 166,400. We can, therefore, find the value of 𝑅 by taking the square root of this number. The square root of 166,400 is 80 root 26. And so, the magnitude of 𝑅 is 80 root 26.
Now, you might be wondering what the point was of including the word “magnitude”. Well, remember, when we take the square root on an equation, we need to consider both the positive and negative square roots. The magnitude just tells us the size of this force. So, the size is 80 root 26 newtons, whether it’s positive or negative.
