Embibe Experts Solutions for Chapter: Laws of Motion, Exercise 1: JEE Advanced Paper 2 - 2018

A student skates up a ramp that makes an angle $30°$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity)

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m=0.4 \mathrm{kg}$ is at rest on this surface. An impulse of $1.0 \mathrm{N}\mathrm{ }\mathrm{s}$ is applied to the block at time $t=0$ so that it starts moving along the x-axis with a velocity $v\left(t\right)=v_0{e}^{-\frac{t}{\tau }}$, where $v_0$ is a constant and $\tau =4 \mathrm{s}$ . The displacement of the block, in meter, at $t=\tau$ is __________. Take $e^{-1}=0.37$.

Put a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \mathrm{cm}$ and the right one at $90.00 \mathrm{cm}$. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_R$ from the center $(50.00 \mathrm{cm})$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are $0.40$ and $0.32$, respectively, the value of $x_R$ (in cm) is___________.

A block of weight $100\mathrm{ }\mathrm{N}$ is suspended by copper and steel wires of same cross-sectional area $0.5\mathrm{ }\mathrm{c}{\mathrm{m}}^2$ and, length $\sqrt{3}\mathrm{ }\mathrm{m}$ and $1\mathrm{ }\mathrm{m},$ respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}},$ respectively. If elongation in copper wire is $\left(\mathrm{\Delta }{l}_{\mathrm{C}}\right)$ and elongation in steel wire is $\left(\mathrm{\Delta }{l}_{\mathrm{S}}\right),$ then the ratio $\frac{\mathrm{\Delta }{l}_{\mathrm{C}}}{\mathrm{\Delta }{l}_{\mathrm{S}}}$ is $\frac{x}{1}$.Find the value of $x$.
[Young's modulus for copper and steel are $1\times {10}^{11}\mathrm{N} {\mathrm{m}}^{-2}$ and $2\times {10}^{11}{\mathrm{Nm}}^{-2}$ respectively]

The potential energy of a particle of mass m at a distance r from a fixed point O is given by $V\left(r\right)=k{r}^2/2$, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle $\mathrm{\theta }$ with the horizontal floor. The coefficient of friction between the wall and the ladder is ${\mathrm{\mu }}_1$ and that between the floor and the ladder is ${\mathrm{\mu }}_2$ . The normal reaction of the wall on the ladder is ${\mathrm{N}}_1$ and that of the floor is ${\mathrm{N}}_2$. If the ladder is about to slip, then

A particle of mass $m$ is projected from the ground with an initial speed $u_{\circ }$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_{\circ }$. The angle that the composite system makes with the horizontal immediately after the collision is