A cup of mass 125 grams is at rest on a desk when it is accidentally knocked by a person’s arm. The cup accelerates in the direction that it is pushed at a rate of 0.25 meters per second squared. How much force is applied by the arm that knocks the cup? Give your answer to two significant figures.
Okay, so in this question we know that we’ve got a cup which has a mass of 125 grams and initially it’s at rest on a desk. We know that it’s then accidentally knocked by a person’s arm. When the person knocks the cup, the cup accelerates in the direction that it is pushed at a rate of 0.25 meters per second squared. What we need to do is to work out the amount of force applied by the arm that knocks the cup and we need to give our answer to two significant figures.
Okay, so here is our cup and here is the elbow of the person who’s about to accidentally knock the cup over. Oh, oops! They’ve just exerted a force on the cup. We’ll call this force 𝐹 because that’s what we’re trying to work out. We do know however that the cup accelerates in the direction that it’s pushed. And we’ll call this acceleration 𝑎, which is 0.25 meters per second squared. We also know the mass of the cup which, we’ll call 𝑚 and it’s 125 grams.
So what we need to do is to find a relationship between the acceleration of the cup, the force exerted on the cup, and the mass of the cup. This relationship is Newton’s second law of motion and it tells us that the force exerted on an object is equal to the mass of that object multiplied by the acceleration of the object. So we can use this equation to work out what the forces. However, we need to consider units first.
If we want to find the force in newtons, which is the standard unit of force, then we need to have the mass and the acceleration in their standard units as well. The standard unit of mass is the kilogram and the standard unit of acceleration is meters per second squared. Now, we do want to find the force in newtons because that’s the standard unit and we’ve already got the acceleration in meters per second squared.
However, the mass that we have is not in kilograms, but rather it’s in grams. We know that the mass of the cup is 125 grams. So the first thing we need to do is to convert from grams to kilograms. We can recall that the conversion factor is that 1000 grams is equal to one kilogram. If we divide both sides of the equation by 1000, then we find that one gram is equal to one thousandth of a kilogram. So how many kilograms is 125 grams equal to? Well, 125 grams is equal to 125 times a thousandth of a kilogram or one hundred and twenty-five thousandth of a kilogram.
And when we evaluate the fraction, we find that this is 0.125 kilograms. So at this point, we can replace the mass in our diagram with 0.125 kilograms down here. And now, we can use 𝐹 is equal to 𝑚𝑎 to work out the force in newtons. So we say that the force 𝐹 is equal to the mass in kilograms multiplied by the acceleration in meters per second squared. So we have 𝐹 is equal to 0.125 times 0.25 and this evaluates to 0.03125 newtons.
However, this is not our final answer. Remember we need to give our answer to two significant figures. So here’s the first significant figure and here’s the second. It’s the one after the second is this two that will tell us what happens to the second significant figure. Now this third significant figure — that two — is less than five. So the second significant figure — the one — stays the same. It doesn’t round up. And therefore, to two significant figures, we have that 𝐹 is equal to 0.031 newtons. And that is our final answer.