At time $t=0$, a disk of radius $1 m$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha=\frac{2}{3} rad s ^{-2}$. A small stone is stuck to the disk. At $t=0$, it is at the contact point of the disk and the plane. Later, at time $t=\sqrt{\pi} s$, the stone detaches itself and flies off tangentially from the disk. The maximum height (in $m$ ) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$. The value of $x$ is. . . . . . .[Take $g=10 m s ^{-2}$.]

The coefficient of friction between the tyres and the road is $0.25$. The maximum speed with which a car can be driven round a curve of radius $40 \,m$ without skidding is ........ $ms^{-1}$ (assume $g = 10 \,ms^{-2}$)

A car of mass $1000\,kg$ negotiates a banked curve of radius $90\,m$ on a frictionless road. If banking angle is $45^o$ , the maximum speed of car is ............ $m/s$     $[g = 10\,m/s^2]$

A gramophone record is revolving with an angular velocity $\omega$. A coin is placed at a distance $r$ from the centre of the record. The static coefficient of friction is $\mu  .$ The coin will revolve with the record if

A cyclist is travelling with velocity $v$ on a curved road of radius $R$. The angle $\theta$  through which the cyclist leans inwards is given by

A $100 \,kg$ car is moving with a maximum velocity of $9 \,m/s$ across a circular track of radius $30\,m$. The maximum force of friction between the road and the car is ........ $N$