Question

A car initially travelling eastwards turns north by travelling in a quarter circular path of radius R metres at uniform speed as shown in figure. The car completes the turn in T second.
(a) What is the acceleration of the car when it is at B located at an angle of ${37}^o$. Express your answers in terms of unit vectors $\hat{i}$ and $\hat{j}$
(b) The magnitude of car's average acceleration during T second period.

Answer 1

$\left(a\right)$At B, acceleration of car is centripetal and is along the center O of the circle.

Hence, unit vector along BO is $\hat{a}=-sin{37}^o\hat{i}+cos{37}^o\hat{j}$

$⟹\hat{a}=\frac{-3i+4j}{5}$

And, magnitude of acceleration is $a=\frac{v^2}{R}=\frac{1}{R}{\left(\frac{\pi R}{2T}\right)}^2$

$⟹\overrightarrow{a}=\frac{{\pi }^{2}R}{4T^2}\left(\frac{-3\hat{i}+4\hat{j}}{5}\right)$

$\left(b\right)$ $\overrightarrow{a}=\frac{\overrightarrow{v}-\overrightarrow{u}}{T}$

$⟹\overrightarrow{a}=\frac{v\hat{j}-v\hat{i}}{T}$

$⟹\overrightarrow{a}=\frac{\pi R}{2T^2}(\hat{j}-\hat{i})$

$⟹a=\frac{\sqrt{2}\pi R}{2T^2}$