Two separate forces, 𝐹1 ⃗⃗⃗ = [200, 500, 450] and 𝐹2 ⃗⃗⃗ = [200, -400, 300], both measured in newtons, act on an object through a displacement defined by 𝑠 = [4, 2, 13], in metres

Question

a)Calculate the work done by force 𝐹1⃗⃗⃗ in the direction of travel.

b) Calculate the work done against gravity by force 𝐹2 .

Answer 1

For this problem we can use the equation W=F•d, where F and d are vectors for force and displacement respectively (sorry I can't easily use standard vector notation with the arrows above them while writing this out so instead I will make all vectors be in bold)

a) For this part we simply need to plug in the vectors and compute the dot product:

W = F₁ • s

W = <200,500,450> • <4,2,13>

W = (200)(4) + (500)(2) + (450)(13)

W = 800 + 1000 + 5850

W = 7650 J

b) To calculate the work done against gravity we just need to break apart force 2 along the direction gravity acts. It is unclear whether this problem gives the orientation of gravity (i.e., whether it acts along the y-axis or the z-axis) I will assume that it uses the standard convention of gravity acting along the z-axis. Since gravity acts in the negative z-axis, the component of force 2 that goes against gravity is <0,0,300>. Let's call this component F_2,z to denote the breakdown of the force F₂ along the z-axis. Now we can plug into our formula again:

W = F_2,z • s

W = <0,0,300> • <4,2,13>

W = (0)(4) + (0)(2) + (300)(13)

W = 0 + 0 + 3900

W = 3900 J