Torque

If force $\overrightarrow{F}=3\hat{\mathrm{i}}+4\hat{\mathrm{j}}-2\hat{\mathrm{k}}$ acts on a particle having position vector $2\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ then, the torque about the origin will be:

A cylinder of mass $m$ is kept on the edge of a plank of mass $2\mathrm{m}$ and length$12 \mathrm{meter}$ , which in turn is kept on smooth ground. Coefficient of friction between the plank and the cylinder is $0.1$. The cylinder is given an impulse, which imparts it a velocity $7 \mathrm{m}/\mathrm{s}$ but no angular velocity. Find the time after which the cylinder falls off the plank. $\left(g=10\mathrm{m}/{\mathrm{s}}^2\right)$.

A cylinder of mass $M$ is placed on top of a block of mass $2M$. Neglecting the friction between the block and the horizontal plane. But friction coefficient between the cylinder and the top surface of block is $\mu$. A small body of mass $M$ travelling with a velocity $v_0$ hits the block horizontally. Collision is completely elastic. If the maximum value of $v_{\mathit{0}}$ for which the cylinder starts pure rolling before it topples from the top of the block is ${\left(v_0\right)}_{\max }=n\sqrt{m\mu gd}$. The value of $m\times n$ is ____

A uniform rod of length $4 \mathrm{m}$ and mass $2\sqrt{2} \mathrm{kg}$ revolves with constant angular velocity $\omega$ about a vertical axis through a smooth joint A at one extremity of the rod so that it describes a cone of semi vertical angle $53°$ as shown in the figure. choose the INCORRECT statement(s). $\left(g=10 \mathrm{m} {\mathrm{s}}^{-1}\right)$

A light beam suspended with the help of two ropes supports two loads as shown in the figure on the next page. The load at the left end is of $75 \mathrm{kg}$ and the load$P$ at the right end is unknown. The maximum tension that either of the rope can bear is $\mathrm{N}$. In an experiment, mass of the load $\mathrm{P}$ is gradually increased from zero. If the rod remains horizontal in equilibrium, which of the following conditions must be satisfied?

Two vertical strings having different lengths joined to a uniform rod at points A and B. The rod is at rest. Now string connected to point B is burnt. Which of the following is/are correct just after this?

A half section of pipe of mass $m$ and radius $r$ rests on a rough horizontal surface. Now a vertical force $F$ is applied as shown. Assuming that the section rolls without sliding. Then [Centre of mass of half ring is $\frac{2r}{\pi }$ below the centre]

Three massless straight rigid rods, each of length $L$ are rigidly joined to give shape as shown. Two particles each of mass $M$ are fixed at ends as shown. Whole system is free to rotate about a fixed axes passing through the cross section of middle rod as shown in fig. in a gravity free space. What is the net torque acting by the fixed axes on the whole system

A uniform rod rests against a smooth vertical wall and a rough horizontal surface A horizontal force $F$ is applied at the center of mass of the rod such that the rod is in static equilibrium and the resultant of $F$ and the weight of the rod intersect at the point $P$ shown in the figure. The angle of friction between the rod and the horizontal surface is :

A uniform slender rod of length $l$ and mass $m$ is rigidly to a circular hoop of radius $l$ as shown. The mass of the hoop is negligible. If the rod and hoop are released from rest on a fixed horizontal surface in the position illustrated, determine the initial value of the friction force $F$, if friction is sufficient to prevent slipping.

A uniform cylinder of height $10 \mathrm{cm}$, diameter $5 \mathrm{cm}$ and mass $\mathrm{M}$ is placed on a horizontal smooth table. One end of a string passing over an ideal pulley is connected to top point of the cylinder and the other end is connected to a block of mass $m$ as shown in the figure. The system is released from rest from the situation shown. Then, mark the CORRECT option(s) (${\overrightarrow{a}}_M$ represents acceleration of centre of mass of $M$)

A cord is wrapped around the rim of a solid cylinder of radius $0.25\mathrm{m}$ and a constant force of $40 \mathrm{N}$ is exerted on the cord. The cylinder is mounted on frictionless bearings and its moment of inertia is $4 \mathrm{kg}-{\mathrm{m}}^2$ The angular velocity of the cylinder after $5\mathrm{m}$ of cord has been unwound, is ${\omega }_1$. If the $40\mathrm{N}$ force is replaced by a $40\mathrm{N}$ weight and released from rest at the top, the angular velocity of the cylinder after $5\mathrm{m}$ of cord has been unwounded is ${\omega }_2$ (Given, $\mathrm{g}=10{ \mathrm{ms}}^{-2}$ ). Then

Three identical cylinders of rough surface are mounted on frictionless horizontal axles passing through their centres. They are all given angular velocity ${\omega }_0$ as shown. They are brought closer in contact and their axles are held stationary till slipping between them ceases. The final angular velocities of the three cylinders in order are

Consider a gyroscope, in which a uniform disc of radius $R=5.0\mathrm{cm}$ at the end of a light rod of length $\ell =10\mathrm{cm}$ is mounted on the floor of an elevator going up with a constant acceleration $a=2{\mathrm{ms}}^{-2}$. The other end of the rod is hinged at $O$. Plane of the disc is perpendicular to rod

Let mass of the disc is $200$ grams and mass of rod and friction are to be neglected. The disc has angular velocity $\omega$ about the rod, and the rod-disc system (gyroscope) is precessing about vertical axis passing through $O$ at the rate $n=0.5\mathrm{rps}$. Then

[Take $g=9.8 {\mathrm{ms}}^{-2}$]

A particle of mass M tied to a string of length L is rotating along a circular path with constant speed $\mathrm{\vartheta }$. The torque on the particle is

Let $\overrightarrow{F}$ be the force acting on a particle having position vector $\overrightarrow{r}$ and $\overrightarrow{\tau }$ be the torque of this force about the origin. Then

A force $\overrightarrow{F}=4\hat{\mathrm{i}}-10\hat{\mathrm{j}}$ acts on a body at a point having position vector $\overrightarrow{r}=-5\hat{\mathrm{i}}-3\hat{\mathrm{j}}$ relative to origin of coordinates on the axis of rotation. If the torque acting on the body about origin is $a\hat{\mathrm{k}}$, then find the value of $a$ (in SI units). [Given, $\overrightarrow{F}$ and $\overrightarrow{r}$ are in SI units]

What are the rotational equivalents for the physical quantity force?

Relation between Torque and angular momentum is similar to the relation between

A turbine fan in a jet engine has moment of inertia $2.5 \mathrm{kg} {\mathrm{m}}^2$ about its axis of rotation. Its angular velocity is $40t^2$. The net torque is