An object with a mass of five kilograms accelerates left at three meters per second squared, while a force of 20 newtons is applied to it, acting to the right, and an unknown force is applied to it, acting to the left. What is the magnitude of the unknown force?
Okay, so there’s a fair amount of information in this question. So let’s break it down step by step. First of all, we’re told that an object which has a mass of five kilograms is accelerating. And it’s accelerating left at three meters per second squared. This is happening, while a force of 20 newtons is applied to it, acting to the right, and another unknown force is applied to it, acting to the left.
What we’ve been asked to do is to find the magnitude or size of the unknown force. So let’s say that this is our object. Now, the first thing we know about this object is that it has a mass of five kilograms. Secondly, we know it’s accelerating to the left and it’s accelerating at a rate of three meters per second squared.
Now, we also know something about the forces acting on the object. Firstly, a 20-newton force is acting on the object to the right and secondly an unknown force is acting to the left. We’ll call this force 𝑥. Now, it’s the value of 𝑥 that we’re actually trying to work out here. And to do this, we first need to recall Newton’s second law of motion.
Newton’s second law of motion tells us that the net force on an object 𝐹 is equal to the mass of that object multiplied by the acceleration experienced by the object. Now, it’s important to know that we are actually talking about the net or resultant force on the object. And we can use Newton’s second law to work out what the resultant force on the object should be because we know the mass of the object already and we know its acceleration.
Now, since the object is accelerating to the left, to make life easier for ourselves, let’s say that left is positive and to the right is negative. This way when we have a final answer for the value of 𝑥, we’ll have it as a positive value because 𝑥 is acting towards the left and left is positive.
Anyway, let’s work out the resultant or net force on the object. So we say that the net force 𝐹 is equal to the mass of the object which is five kilograms multiplied by the acceleration of the object, which is three meters per second squared to the left. But remember the left is positive. So it’s just three meters per second squared. We don’t need to worry about any negative signs.
At which point, when we evaluate the right-hand side of this equation, we find that the net force on the object is 15 newtons. So what does that tell us? Well, it tells us that the two competing forces — 𝑥 newtons to the left and 20 newtons to the right — have magnitudes such that the overall force is 15 newtons to the left.
Now, to find what the resultant force out of these two forces should be, we just add them together, whilst accounting for the directions in which they act. In other words, that overall or resultant force on the object, which is 𝐹, once again calculated using the two forces that we already know is equal to 𝑥 newtons which is positive because it’s acting to the left minus 20 newtons because the 20-newton force is acting to the right.
But as we’ve already worked out earlier, the resultant force 𝐹 is 15 newtons. So we can replace the left-hand side of the equation with 15 newtons, at which point we can add 20 to both sides of the equation. This way, the negative 20 cancels out with the positive 20. And so what we’re left with is 15 plus 20 is equal to 𝑥.
Hence, 𝑥 is equal to 35. And so the magnitude of the unknown force which is 𝑥, which is what we’re trying to work out, is 35 newtons. And that is our final answer.