Circular motion on banked road

Question

Independent of the specific problem in question, I do not understand why $Rsin\theta=mv^2/r$. Surely the centripetal force and $Rsin\theta$ are acting in the same direction(left on diagram), since centripetal is acting towards the centre of the bend and the road itself is pointing towards the centre of the circle.

Answer 1

Left to its own devices the car wants to travel in a straight line. Newton's First Law: every object continues in its state of rest or uniform motion unless acted on by a force.

We are told that the car is travelling in a circle. It must therefore be acted on by a force. The only force shown in your diagram that has a component towards the centre of the circle is $R$, and its component in that direction is $R \sin \theta$. So the force making the car travel in a circle is $R \sin \theta$.

This force can be shown to be $\frac{mv^2}r$, so we have $R \sin \theta =\frac{mv^2}r$