Newton's Second Law for a System of Particles — Question 9

Question 9

A moving observer needs an inertial correction

A fixed-mass system has mass $10\,\mathrm{kg}$. Relative to a cart accelerating right at $2.0\,\mathrm{m/s^2}$, its center accelerates left at $0.50\,\mathrm{m/s^2}$. The cart axes remain parallel to an inertial laboratory frame.

1. Find the actual total external horizontal force.

2. What fictitious force makes the cart-frame second-law equation valid?

Question 9 - Solution

A moving observer needs an inertial correction

(a) Derivation and calculation

For translating axes with no rotation, $$a_C=a_{\mathrm{rel}}+A=-0.50+2.0=1.50\ \mathrm{m/s^2}.$$ $$\boxed{F_{\mathrm{ext}}=Ma_C=10(1.50)=15\ \mathrm N}.$$

(b) Result and interpretation

The accelerated frame requires the inertial-force term $$\boxed{F_{\mathrm{fict}}=-MA=-20\ \mathrm N}.$$ $$F_{\mathrm{ext}}+F_{\mathrm{fict}}=15-20=-5\ \mathrm N=10(-0.50)=Ma_{\mathrm{rel}}.$$ Using relative acceleration alone would mistake the effective frame force for the actual external force.

Physical check

Adding the cart acceleration restores the inertial acceleration and the physical force.