A body of mass 478 grams has an
acceleration of negative four 𝐢 plus three 𝐣 meters per square second, where 𝐢
and 𝐣 are perpendicular unit vectors. What is the magnitude of the force
acting on the body?
Remember, to link mass,
acceleration, and net force, we can use Newton’s second law of motion. This says that force is equal to
mass times acceleration. Now, mass will always be a scalar
quantity. It will just have a magnitude. But force and acceleration can be
vector quantities. They will have a magnitude and a
direction. And so we can write this as the
vector sum of 𝐹 is equal to mass times the vector acceleration.
Now, since we are working in meters
per square second, we can’t use this formula until we convert from grams to
kilograms. We know that there are 1,000 grams
in a kilogram, and so we divide 478 by 1,000. And that tells us that it’s
equivalent to 0.478 kilograms. The vector sum of the forces acting
on the body then is equal to this mass times the vector acceleration. It’s 0.478 multiplied by negative
four 𝐢 plus three 𝐣.
We can of course distribute this
scalar quantity across our vector. 0.478 multiplied by negative four
is negative 1.912. And the 𝐣-component is 1.434. So we have the vector force, but we
need to find the magnitude. The magnitude of the vector is
found by finding the square root of the sum of the squares of each of its
components. So that’s the square root of
negative 1.912 squared plus 1.434 squared. That’s equal to 2.39. And so the magnitude of the force
that acts upon this body is 2.39 newtons.
Now, it is also worth noting that
we could have simply found the magnitude of the acceleration first and then
multiplied that by the mass. We would have achieved the same