Question

A cylinder rests in a supporting carriage as shown. The side $AB$ of carriage makes an angle ${30}^{\circ }$ with the horizontal and side $BC$ is vertical. The carriage lies on a fixed horizontal surface and is being pulled towards left with an horizontal acceleration ${}^{\prime }{a}^{\prime }$. The magnitude of normal reactions exerted by sides $AB$ and $BC$ of carriage on the cylinder be $N_{AB}$ and $N_{BC}$ respectively. Neglect friction everywhere. Then as the magnitude of acceleration ${}^{\prime }{a}^{\prime }$ of the carriage is increased, pick up the correct statement.

• A. $N_{AB}$ increases and $N_{BC}$ decreases.
• B. Both $N_{AB}$ and $N_{BC}$ increase
• C. $N_{AB}$ remains constant and $N_{BC}$ increases
• D. $N_{AB}$ increases and $N_{BC}$ remains constant

Answer 1

The correct option is C $N_{AB}$ remains constant and $N_{BC}$ increases

FBD of the sphere (rest frame of the carriage)

For equilibrium,
Along $y-$ axis,
$N_{AB}\cos {30}^{\circ }=mg$
$N_{AB}=\frac{2mg}{\sqrt{3}}$ (constant)
Along $x-$ axis,
$N_{BC}=ma+N_{AB}\sin {30}^{\circ }$
$N_{BC}=ma+\frac{mg}{\sqrt{3}}$
$\Rightarrow$ as ${}^{\prime }{a}^{\prime }$ increases, $N_{BC}$ increases.