Video Transcript
A force of 200 plus or minus two
newtons is applied to an object with a mass of 10 plus or minus 0.1 kilograms. What is the uncertainty in the
acceleration of the object?
Let’s say that this is our object,
which has a mass we’ll call 𝑚 and a force being applied to it that we’ll call
𝐹. Newton’s second law of motion tells
us that the net force acting on an object is equal to the mass of that object
multiplied by its acceleration. Dividing both sides of this
equation by an object’s mass 𝑚 so that that mass cancels on the right, we see that
acceleration equals force divided by mass. In our case then, 𝑎 is equal to 𝐹
divided by 𝑚, or 200 plus or minus two newtons divided by 10 plus or minus 0.1
kilograms. We can see that here we’re dividing
two numbers each of which has an uncertainty associated with it. Whenever this happens, the
resultant number, in this case the acceleration of our object 𝑎, has an uncertainty
that we find out according to this rule.
First, we find the percent
uncertainty of our value in the numerator. Then, we do the same thing for the
value in the denominator. To differentiate these values,
let’s call the numerator percent uncertainty percent 𝑢 sub of 𝑛 and the
denominator percent uncertainty percent 𝑢 sub 𝑑. In general, given some value, we’ll
call that value 𝑎, with an associated uncertainty we’ll call 𝜎 sub 𝑎, the percent
uncertainty in the value 𝑎 equals the uncertainty 𝜎 sub 𝑎 divided by 𝑎 all
multiplied by 100 percent. Let’s now apply this rule to the
numerator and denominator of our expression for the acceleration 𝑎.
The percent uncertainty in our
numerator is equal to the uncertainty in the force, that’s two, divided by the force
itself, that’s 200, all multiplied by 100 percent. Note that when we calculate a
percent uncertainty, we often leave out the units involved. We know that two divided by 200 is
equal to one divided by 100. And that is the same thing as
0.01. 0.01 times 100 percent equals one
percent. So that is the percent uncertainty
in our numerator. When it comes to our denominator,
the uncertainty there is 0.1, while the value itself is 10. Interestingly, 0.1 divided by 10 is
equal exactly to 0.01, which when multiplied by 100 percent equals one percent. So the percent uncertainties in our
numerator and denominator are the same.
Once we know these values, the next
step is to add them together. One percent plus one percent is two
percent, and this will be the percent uncertainty in our final value, that is, the
percent uncertainty in the acceleration 𝑎. Note though that our question
doesn’t ask about the percent uncertainty in the acceleration, but just the
uncertainty itself. So let’s actually find out what our
acceleration will be with this force and this mass.
200 newtons of force applied to a
mass of 10 kilograms is equal to 20 meters per second squared plus or minus our
overall percent uncertainty of two percent. The last question then is, what is
two percent of 20 meters per second squared? That will be the uncertainty in the
acceleration of our object. If we write two percent in its
decimal form, that’s equal to 0.02. If we multiply this decimal
representation of two percent by our acceleration of 20 meters per second squared,
then we get a result of 0.4 meters per second squared. This then is the uncertainty,
rather than the percent uncertainty, in the acceleration of our object. The uncertainty in the acceleration
of the object is 0.4 meters per second squared.
