Newton's Second Law — Question 3

Question 3

Recover an unknown force from a measured trajectory

Use $\sum\vec F=m\vec a$ for constant mass in an inertial frame. A 4 kg particle follows $\vec r=(t^2,3t-t^2/2)$ m for numerical t in seconds. One force is $(3,5)$ N; one other force acts.

1. Find the required net force.

2. Find the other force.

3. Check whether the particle’s velocity must be parallel to that net force.

Question 3 - Solution

Recover an unknown force from a measured trajectory

(a) Differentiate twice.

$$\vec v=(2t,3-t),\quad\vec a=(2,-1),\quad\boxed{\vec F_{\mathrm{net}}=4(2,-1)=(8,-4)\,\mathrm N}$$ $\vec v=(2t,3-t)$ m/s and $\vec a=(2,-1)$ m/s$^2$. Thus $\vec F_{\mathrm{net}}=(8,-4)$ N.

(b) Subtract the known interaction.

$$\vec F_2=\vec F_{\mathrm{net}}-\vec F_1=(8,-4)-(3,5)=(5,-9)\,\mathrm N$$ $\vec F_2=(8,-4)-(3,5)=(5,-9)$ N.

(c) Test a concrete instant.

$$v_xF_y-v_yF_x=(2t)(-4)-(3-t)(8)=-24\ne0$$ At t=0, velocity is $(0,3)$ m/s, not parallel to $(8,-4)$ N. Newton’s second law aligns acceleration with the resultant, not velocity.