Embibe Experts Solutions for Chapter: Rotational Motion, Exercise 1: JEE Main - 22 July 2021 Shift 1

A particle of mass $100 \mathrm{g}$ is projected at time $t=0$ with a speed $20 \mathrm{m} {\mathrm{s}}^{–1}$ at an angle $45°$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time $t=2 \mathrm{s}$ is found to be $\sqrt{K} \mathrm{kg} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$. The value of $K$ is ______.

(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A thin uniform rod of length $2 \mathrm{m}$, cross sectional area $A$ and density $d$ is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $\omega$. If value of $\omega$ in terms of its rotational kinetic energy $E$ is $\sqrt{\frac{\alpha E}{Ad}}$, then the value of $\alpha$ is ______.

A uniform disc of mass $0.5 \mathrm{kg}$ and radius $r$ is projected with velocity $18 \mathrm{m} {\mathrm{s}}^{-1}$ at $t=0 \mathrm{s}$ on a rough horizontal surface. It starts off with a purely sliding motion at $t=0 \mathrm{s}$. After $2 \mathrm{s}$ it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after $2 \mathrm{s}$ will be ______ $\mathrm{J}$.

(given, coefficient of friction is $0.3$ and $g=10 \mathrm{m} {\mathrm{s}}^{-2}$).

As shown in figure, a $70 \mathrm{kg}$ garden roller is pushed with a force of $\overrightarrow{F}=200 \mathrm{N}$ at an angle of $30°$ with horizontal. The normal reaction on the roller is (Given $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A solid sphere of mass $1 \mathrm{kg}$ rolls without slipping on a plane surface. Its kinetic energy is $7\times {10}^{-3} \mathrm{J}$. The speed of the centre of mass of the sphere is ______ $\mathrm{cm} {\mathrm{s}}^{-1}$.

Two discs of same mass and different radii are made of different materials such that their thicknesses are $1 \mathrm{cm}$ and $0.5 \mathrm{cm}$ respectively. The densities of materials are in the ratio $3:5$. The moment of inertia of these discs respectively about their diameters will be in the ratio of $\frac{x}{6}$. The value of $x$ is ______.

A solid cylinder is released from rest from the top of an inclined plane of inclination $30°$ and length $60 \mathrm{cm}$. If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is ______ $\mathrm{m} {\mathrm{s}}^{-1}$. (Given $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

Moment of inertia of a disc of mass $M$ and radius '$R$' about any of its diameter is $\frac{M{R}^2}{4}$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $\frac{x}{2}M{R}^2$. The value of $x$ is ______.