Question

(1)
A car travels around a bend of radius 400m on a road which is banked at an angle to the horizontal, if the car has no tendency to slip when travelling at 35m/s. Find the angle ..

Answer 1

According to Newton’s second law of motion:

$F=m*a;$

Centripetal acceleration a can be expressed: 

$a=\upsilon^2/r;$

So the centripetal force acting on the car :

$F=(m*\upsilon^2)/r;$

Force due to friction:

$f=\mu*N;$

If car is situated in the origin, Force equations at this maximum speed for the car are:

$(m*\upsilon^2)/r=N*sin\theta+\mu*N*cos\theta$

$m*g+\mu*N*sin\theta=N*cos\theta;$

Due to condition, $\mu=0;$

After equations reduce:

$(m*\upsilon^2)/r=N*sin\theta;$

$m*g=N*cos\theta;$

Dive first with second:

$(M*\upsilon^2/r)/mg=(N*sin\theta)/(N*cos\theta);$

As a result we get:

$tan\theta=\upsilon^2/(m*g);$

The final formula is:

$\theta=arctan(\upsilon^2/(m*g));$

$\theta=arctan(35^2/(400*9.8))$ $=arctan(0.3125)=17.35;$

Answer: $\theta=17.35.$