question_answer

A car of mass \[m\] moves in a horizontal circular path of radius \[r\] meter. At an instant its speed os \[v\,\,m/s\] and is increasing at a rate a \[m/{{s}^{2}}\], then the acceleration of the car is:

A)  \[\sqrt{a\left( \frac{{{v}^{2}}}{r} \right)}\]                          

B)  \[\sqrt{{{a}^{2}}+{{\left( \frac{{{v}^{2}}}{r} \right)}^{2}}}\]

C)  \[\frac{{{v}^{2}}}{r}\]                                   

D)  \[a\]

Correct Answer: B

Solution :

Key Idea: In the non-uniform circular motion, car will have radial and tangential acceleration. In the non-uniform circular motion of the car, the car will have radial \[({{a}_{R}})\] and tangential acceleration \[({{a}_{T}})\] and both these accelerations will be perpendicular to each other. Radial or centripetal acceleration, \[a=\frac{{{v}^{2}}}{r}\] and tangential acceleration, \[{{a}_{T}}=a.\] \[\therefore \] Magnitude of resultant acceleration is \[a'=\sqrt{a_{R}^{2}+a_{T}^{2}}\] \[a'=\sqrt{{{\left( \frac{{{v}^{2}}}{r} \right)}^{2}}+{{a}^{2}}}\]