Force — Question 7

Question 7

A resultant can be reconstructed from an incomplete inventory

The total force on a particle is $(2,-3,6)$ N. Two identified forces are $(4,1,2)$ N and $(-1,2,1)$ N. One additional force acts.

1. Find the unidentified force.

2. Find its magnitude.

3. Verify all three components by resumming.

Question 7 - Solution

A resultant can be reconstructed from an incomplete inventory

(a) Subtract the known sum.

$$\vec F_U=\vec R-\vec F_A-\vec F_B=(2,-3,6)-(3,3,3)=(-1,-6,3)\,\mathrm N$$ The known sum is $(3,3,3)$ N. The missing force is $(2,-3,6)-(3,3,3)=(-1,-6,3)$ N.

(b) Use the spatial norm.

$$\boxed{F_U=\sqrt{(-1)^2+(-6)^2+3^2}=\sqrt{46}=6.782\,\mathrm N}$$ $|\vec F_{\mathrm{missing}}|=\sqrt{1+36+9}=\sqrt{46}\approx6.782$ N.

(c) Close the component audit.

$$(4-1-1,\ 1+2-6,\ 2+1+3)=(2,-3,6)\,\mathrm N$$ Adding gives $(4-1-1,1+2-6,2+1+3)=(2,-3,6)$ N. Checking only magnitudes would not establish the required resultant direction.