Question Video: Finding the Magnitude of a Force Acting on a Body given the Acceleration Vector and Mass of the Body

A body of mass 478 g has an acceleration of (−4𝐢 + 3𝐣) m/s², where 𝐢 and 𝐣 are perpendicular unit vectors. What is the magnitude of the force acting on the body?

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Video Transcript

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 result.

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