Banked Circular Motion without friction

[hqjb]

Homework Statement

A roadway is designed for traffic moving at a speed of 28 m s . A curved section of the
roadway is a circular arc of 190 m radius. The roadway is banked so that a vehicle can go
around the curve with the lateral friction forces equal to zero

Homework Equations

$F_C = \frac{mv^2}{r}$

The Attempt at a Solution

$N\sin\beta = \frac{mv^2}{r}$
$mg\cos\beta\sin\beta = \frac{mv^2}{r}$
$2\sin\beta\cos\beta = \frac{2v^2}{rg}$
$\sin(2\beta) = \frac{2(28)(28)}{(190)(9.8)}$

I got the right answer if I didnt assume $N = mg\cos\beta$

Edit : Nevermind, careless mistake I was using different coordinate systems.

 

[PeterO]

Homework Statement

A roadway is designed for traffic moving at a speed of 28 m s . A curved section of the
roadway is a circular arc of 190 m radius. The roadway is banked so that a vehicle can go
around the curve with the lateral friction forces equal to zero

Homework Equations

$F_C = \frac{mv^2}{r}$

The Attempt at a Solution

$N\sin\beta = \frac{mv^2}{r}$
$mg\cos\beta\sin\beta = \frac{mv^2}{r}$
$2\sin\beta\cos\beta = \frac{2v^2}{rg}$
$\sin(2\beta) = \frac{2(28)(28)}{(190)(9.8)}$

I got the right answer if I didnt assume $N = mg\cos\beta$

Edit : Nevermind, careless mistake I was using different coordinate systems.

I assume you found that using $mg\tan\beta = \frac{mv^2}{r}$ was more fruitfull?