Question

Two forces F1, and F2 (and no other forces) act on a particle P causing it to move from rest through a distance of 2 m. Find the work done by the resultant force if F = i- j (N) and F2 is of magnitude 10 N and acts in the direction of 4i + 3j.​

Answer 1

It looks as though the first force is

`\vec F_1 = (\vec\imath-\vec\jmath)\,\mathrm N`

The second force acts in the direction of
`4\vec\imath+3\,\vec\jmath`. Normalize this direction vector to make it have length 1:

`\left\|4\,\vec\imath+3\,\vec\jmath\right\| = √(4^2+3^2) = √(25) = 5 \\\\ \implies \left\|\frac{4\,\vec\imath+3\,\vec\jmath}5\right\| = 1`

Then the second force is

`\vec F_2 = (10\,\mathrm N)\frac{4\,\vec\imath+3\,\vec\jmath}5 = (8\,\vec\imath+6\,\vec\jmath)\,\mathrm N`

The resultant force is

`\vec F_1 + \vec F_2 = (9\,\vec\imath+5\,\vec\jmath)\,\mathrm N`

Assuming the particle moves 2 m in the same direction as the resultant force, the work perfomed on the particle by it is

`\|\vec F_1 + \vec F_2\| (2\,\mathrm m) = 2√(9^2+5^2)\,\mathrm{Nm} \approx \boxed{20.6\,\mathrm J}`