A circular racetrack of radius $300\; m$ is banked at an angle of $15^o$. If the coefficient of friction between the wheels of a race-car and the road is $0.2$, what is the
$(a)$ optimum speed of the racecar to avoid wear and tear on its tyres, and
$(b)$ maximum permissible speed to avoid slipping ?
On a banked road, the horizontal component of the normal force and the frictional force contribute to provide centripetal force to keep the car moving on a circular turn without slipping. At the optimum speed, the normal reaction’s component is enough to provide the needed centripetal force, and the frictional force is not needed. The optimum speed $v_o$ is given by Eq.
$v_{o}=(R g \tan \theta)^{1 / 2}$
Here $R=300 \,m , \theta=15^{\circ}, g=9.8 \,m s ^{-2} ;$ we
have $v_{o}=28.1 \,m s ^{-1}$
The maximum permissible speed $v_{\max }$ is given by Eq.
$v_{\max }=\left(R g \frac{\mu_{s}+\tan \theta}{1-\mu_{s} \tan \theta}\right)^{1 / 2}=38.1 \,m s ^{-1} $
A body of mass $2 \,kg$ is kept by pressing to a vertical wall by a force of $100\, N$. The coefficient of friction between wall and body is $0.3.$ Then the frictional force is equal to ........ $N$
A block of mass $m$ (initially at rest) is sliding up (in vertical direction) against a rough vertical wall with the help of a force $F$ whose magnitude is constant but direction is changing. $\theta = {\theta _0}t$ where $t$ is time in sec. At $t$ = $0$ , the force is in vertical upward direction and then as time passes its direction is getting along normal, i.e., $\theta = \frac{\pi }{2}$ .The value of $F$ so that the block comes to rest when $\theta = \frac{\pi }{2}$ , is
Someone is using a scissors to cut a wire of circular cross section and negligible weight. The wire slides in the direction away from the hinge until the angle between the scissors blades becomes $2 \alpha$. The friction coefficient between the blades and the wire, is :-
In figure, the coefficient of friction between the floor and the block $B$ is $0.2$ and between blocks $A$ and $B$ is $0.3$. ........ $N$ is the maximum horizontal force $F$ can be applied to the block $B$ so that both blocks move together .
A $20\, kg$ block is initially at rest on a rough horizontal surface. A horizontal force of $75 \,N$ is required to set the block in motion. After it is in motion, a horizontal force of $60\, N$ is required to keep the block moving with constant speed. The coefficient of static friction is