Newton's First Law — Question 10

Question 10

The first law constrains the sum, not each interaction

A body moves uniformly in an inertial frame. Known forces are $(3,4)$ N and $(-1,2)$ N. Exactly one additional force acts.

1. Find the known resultant.

2. Find the missing force and its magnitude.

3. Could all individual forces be zero in this situation?

Question 10 - Solution

The first law constrains the sum, not each interaction

(a) Add components.

$$\vec F_K=(3,4)+(-1,2)=(2,6)\,\mathrm N$$ The known resultant is $(2,6)$ N.

(b) Require zero total resultant.

$$\boxed{\vec F_3=-\vec F_K=(-2,-6)\,\mathrm N},\quad |\vec F_3|=\sqrt{40}=6.325\,\mathrm N$$ The remaining force must be $(-2,-6)$ N, with magnitude $\sqrt{40}=2\sqrt{10}\approx6.325$ N. This cancels both components.

(c) Distinguish resultant from terms.

$$\vec F_1+\vec F_2+\vec F_3=\vec0,\quad\vec F_1\ne\vec0,\quad\vec F_2\ne\vec0$$ The known forces are explicitly nonzero. Uniform motion requires their vector sum with all other forces to vanish; it does not erase individual interactions. Omitting one known interaction would give an incomplete inventory.