Newton's Second Law — Question 8

Question 8

A desired acceleration determines the required control force

Use $\sum\vec F=m\vec a$ for constant mass in an inertial frame. A 5 kg body must have acceleration $(2,-1)$ m/s$^2$. An unavoidable environmental force is $(-3,4)$ N. A controllable force is the only other interaction.

1. Find the required total force.

2. Find the control force and its magnitude.

3. Can a controller limited to 15 N achieve the target?

Question 8 - Solution

A desired acceleration determines the required control force

(a) Translate the target into a resultant.

$$\vec F_{\mathrm{req}}=m\vec a=5(2,-1)=(10,-5)\,\mathrm N$$ $\vec F_{\mathrm{required}}=5(2,-1)=(10,-5)$ N.

(b) Cancel the environment and supply the remainder.

$$\boxed{\vec F_C=\vec F_{\mathrm{req}}-\vec F_E=(13,-9)\,\mathrm N},\quad F_C=\sqrt{13^2+9^2}=15.811\,\mathrm N$$ $\vec F_C=(10,-5)-(-3,4)=(13,-9)$ N, magnitude $\sqrt{250}=5\sqrt{10}\approx15.811$ N.

(c) Compare the required magnitude with the bound.

$$F_C^2=250\,\mathrm{N^2}>15^2=225\,\mathrm{N^2}$$ No. The unique required vector exceeds the 15 N limit. Checking only the 13 N x component would miss the simultaneous vertical demand.