A ramp is constructed with a parabolic shape such that the height $y$ at any point on its surface is given in terms of its horizontal distance $x$ from the bottom of the ramp $\left(x=y=0\right)$ by $y=\frac{x^2}{20}$. A small block is to be set on the ramp. The maximum height from the bottom level at which the block can be kept on the ramp without sliding is (Given that ${\mu }_s=0.5$).

Two blocks $A$ and $B$ of equal masses are sliding down along straight parallel lines on an inclined plane of $45°$. Their coefficients of kinetic friction are ${\mu }_{\mathrm{A}}=0.2$ and ${\dot{\mu }}_{\mathrm{B}}=0.3$, respectively. At $t=0$, both the blocks are at rest and block $A$ is $\sqrt{2} \mathrm{m}$ behind block $B$. The time and distance from the initial position where the front faces of the blocks come in line on the inclined plane as shown in figure. (Use, $g=10{ \mathrm{m} \mathrm{s}}^{-2}$.)

Two blocks $m_1=4 \mathrm{kg}$ and $m_2=2 \mathrm{kg}$ are connected by a weightless rod on a plane having inclination of $37°$. The coefficient of dynamic friction of $m_1$ and $m_2$ with the inclined plane are, $\mu =0.25$. Then, the common acceleration of the two blocks and the tension in the rod are,

A force $F=t$ is applied to block $A$ as shown in the figure. The force is applied at $t=0 \mathrm{seconds}$ when the system was at rest and string is just straight without tension. Which of the following graphs gives the force between $B$ and horizontal surface as a function of time $t.$

If the coefficient of friction between $A$ and $B$ is ${\mu }_1$ the maximum horizontal acceleration of the wedge $A$ for which $B$ will remain at rest with respect to the wedge is:

What is the minimum stopping distance for a vehicle of mass $m$ moving with speed $v$ along a level road if the coefficient of friction between the tyres and the road is $\mu$?

A block of mass $m_1=1 \mathrm{kg}$ another mass $m_2=2 \mathrm{kg}$ are placed together (see figure) on an inclined plane with the angle of inclination $\mathrm{\theta }.$ Various values $\mathrm{\theta }$ are given in List $\mathrm{I}.$ The coefficient of friction between the block $m_1$ and the plane is always zero. The coefficient of static and dynamic friction between the block $m_2$ and the plane is equal to$\mu =0.3$. In List$\mathrm{ll}$, an expression for the friction on block $m_2$ given. Match the correct expression of the friction in List $\mathrm{II}$ with the angles given in List $1$ and choose the correct option. The acceleration due to gravity is denoted by $g$ [useful information: $\tan \left(5.5°\right)\approx 0.1; \tan \left(11.5°\right)\approx 0.2; \tan \left(16.5°\approx 0.3\right)]$

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Assertion: While drawing a line on a paper, friction force acts on paper in the same direction along which line is drawn on the paper.

Reason: Friction always opposes motion.