Question

A rail-road car is moving towards right with acceleration a. A man A accelerating toward left with an acceleration of magnitude a/3 w.r.t. to car. A dog of mass m is following man A with an acceleration a/3 relative to the car. Observing the dog and man A. Find the
a. net force experienced by the dog as seen by observer B standing on ground.
b. rate of change of linear momentum of the dog relative to the man A moving on trolley.
c. pseudo-force on the dog as seen from man A.

Answer 1

a. The net force acting on dog, as seen from observer'B' on ground is

$\overrightarrow{F}=m\overrightarrow{a_{dog,ground}}=m\overrightarrow{a_d}$..........(i)

Acceleration of dog= Acceleration of dog w.r.t. car + Acceleration of car

$\Rightarrow$ $\overrightarrow{a_d}=\overrightarrow{a_{d,c}}+\overrightarrow{a_c}=-\frac{a}{3}\hat{i}+a\hat{i}=\frac{2a}{3}\hat{i}$

Hence, from (i), $\overrightarrow{F}=\frac{2ma}{3}\hat{i}$

b. The rate of change of momentum of the dog relative to the man $A\frac{d\overrightarrow{p}}{dt}=m\widehat{a_{d,A}}$.......(ii)

Where $\overrightarrow{a_{d,A}}=\overrightarrow{a_{d,c}}-\overrightarrow{a_{A,C}}=-\frac{a}{3}\hat{i}\left(\frac{-a}{3}\hat{i}\right)=0$

Hence, from (ii)$\frac{d\overrightarrow{p}}{dt}=0$

c. Pseudo-force acting on dog viewed by the man A is

$\overrightarrow{F_{pseudo}}=-m\overrightarrow{a_{A,ground}}$

Where $\overrightarrow{a_{A,ground}}=\overrightarrow{a_{A,C}}+\overrightarrow{a_C}=-\frac{a}{3}\hat{i}+a\hat{i}=\frac{2a}{3}\hat{i}$

This gives $\overrightarrow{F_{pseudo}}=\frac{2ma}{3}\hat{i}$