Torque, Couple & Equilibrium of Rigid Bodies

A force of $3\hat{i}+2\hat{j}-\hat{k}$ N acts on a particle with position vector $\hat{i}+\hat{j}-\hat{k}$ m. The magnitude of torque of given force is

In the case of rotational dynamics, which of the following statements is correct? {$S$ = linear displacement, $\alpha$ = angular acceleration, $r$ = radius vector, $a$ = linear acceleration, $L$ = angular momentum, $p$ = linear momentum, $\theta$ = angular displacement}

1. $S = r \times \theta, a = r \times \alpha, L = p \times r$

2. $S = \theta \times r, a = \alpha \times r, L = r \times p$

3. $S = r \cdot \theta, a = r \cdot \alpha, L = p \cdot r$

4. $S = \omega \times r, a = r \times p, L = r \times f$

Which of the following sentence is true?

a. Moment of a couple is independent of the axis of rotation or the points of application of forces. b. Moment of a couple produces translational acceleration.

c. Moment of a couple can be balanced by single force.

d. Moment of a couple depends upon the axis of rotation or the points of application of forces.

If force $\vec{F} = -3\hat{i}+\hat{j}+5\hat{k}$ acts along $\vec{r} = 7\hat{i}+3\hat{j}+\hat{k}$ then the torque acting at that point is

If $\vec{F} = (5\hat{i}-10\hat{j})$ and $\vec{r} = (4\hat{i}-3\hat{j})$, then the torque acting on the object will be

The torque due to the force $(2\hat{i}+\hat{j}+2\hat{k})$ about the origin, acting on a particle whose position vector is $(\hat{i}+\hat{j}+\hat{k})$, would be

The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force of $\vec{F} = \hat{i} - \hat{j} + \hat{k}$ acts on the particle, then the magnitude of torque (with respect to origin) in z- direction is __________.

Which of the following are correct expression for torque acting on a body?

A. $\vec{\tau} = \vec{r} \times \vec{L}$

B. $\vec{\tau} = \frac{d}{dt}(\vec{r} \times \vec{p})$

C. $\vec{\tau} = \vec{r} \times \frac{d\vec{p}}{dt}$

D. $\vec{\tau} = I\vec{\alpha}$

E. $\vec{\tau} = \vec{r} \times \vec{F}$

( $\vec{r} =$ position vector; $\vec{p} =$ linear momentum; $\vec{L} =$ angular momentum; $\vec{\alpha} =$ angular acceleration; $I =$ moment of inertia; $\vec{F} =$ force; $t =$ time)

Choose the correct answer from the options given below:

A square Lamina OABC of length 10 cm is pivoted at ' $\text{O}$ '. Forces act at Lamina as shown in figure. If Lamina remains stationary, then the magnitude of F is:

A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at 40 cm mark. A mass of 400 g is suspended at 10 cm mark. To maintain the balance of the rod, the mass to be suspended at 90 cm mark, is

A uniform rod of mass 20 kg and length 5 m leans against a smooth vertical wall making an angle of $60^\circ$ with it. The other end rests on a rough horizontal floor. The friction force that the floor exerts on the rod is (Take $g = 10\text{ m/s}^{2}$)

In case of rotational dynamics, which one of the following statements is correct? [$\vec{\omega} =$ angular velocity,$\vec{v} =$ linear velocity

$\vec{r} =$ radius vector, $\vec{\alpha} =$ angular acceleration

$\vec{a} =$ linear acceleration,$\vec{L} =$ angular momentum

$\vec{p} =$ linear momentum, $\vec{\tau} =$ torque,

$\vec{f} =$ centripetal force]

A heavy iron bar, of weight $W$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is:

A heavy iron bar of weight $12\text{ kg}$ is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle $60^\circ$ with the horizontal, the weight experienced by the man is:

Given below are two statements: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$

Assertion A: An electric fan continues to rotate for some time after the current is switched off. Reason R: Fan continues to rotate due to inertia of motion.

In the light of above statements, choose the most appropriate answer from the options given below.

A force of $-P\hat{k}$ acts on the origin of the coordinate system. The torque about the point $(2, -3)$ is $P(a\hat{i} + b\hat{j})$. The ratio of $\frac{a}{b}$ is $\frac{x}{2}$. The value of $x$ is ___________.

An object of mass 8 kg is hanging from one end of a uniform rod CD of mass 2 kg and length 1 m pivoted at its end C on a vertical wall as shown in figure. It is supported by a cable AB such that the system is in equilibrium. The tension in the cable is (Take g = 10 m/s$2$)

The torque of a force $5\hat{i}+3\hat{j}-7\hat{k}$ about the origin is $\tau$. If the force acts on a particle whose position vector is $2\hat{i}+2\hat{j}+\hat{k}$, then the value of $\tau$ will be

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

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be x $\times$ 10$-$2 kg. The value of x is ___________.

A $\sqrt{34}$ m long ladder weighing 10 kg leans on a frictionless wall. Its feet rest on the floor 3 m away from the wall as shown in the figure. If Ff and Fw are the reaction forces of the floor and the wall, then ratio of $F_w/F_f$ will be:

(Use g = 10 m/s2.)

A triangular plate is shown. A force $\vec{F}$= 4$\hat{i} -$ 3$\hat{j}$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are:

A force $\vec{F} = 4\hat{i} + 3\hat{j} + 4\hat{k}$ is applied on an intersection point of x = 2 plane and x-axis. The magnitude of torque of this force about a point (2, 3, 4) is ___________. (Round off to the Nearest Integer)

A metre scale is supported on a wedge at its centre of gravity. A body of weight 'w' is suspended from the $20\text{ cm}$ mark and another weight of 25 gram is suspended from $74\text{ cm}$ mark to balance it and the metre scale remains perfectly horizontal. Neglecting the weight of the metre scale, the weight of the body is

A uniform rod of length 200 cm and mass 500 g is balanced on a wedge placed at 40 cm mark. A mass of 2 kg is suspended from the rod at 20 cm and another unknown mass 'm' is suspended from the rod at 160 cm mark as shown in the figure. Find the value of 'm' such that the rod is in equilibrium. (g = 10 m/s2)

Consider a frame that is made up of two thin massless rods AB and AC as shown in the figure. A vertical force $\vec{P}$of magnitude 100 N is applied at point A of the frame.

Suppose the force $\vec{P}$is resolved parallel to the arms AB and AC of the frame.

The magnitude of the resolved component along the arm AC is xN.

The value of x, to the nearest integer, is ___________.

[Given: sin(35$^\circ$) = 0.573, cos(35$^\circ$) = 0.819

sin(110$^\circ$) = 0.939, cos(110$^\circ$) = $-$ 0.342]

Find the torque about the origin when a force of 3 $\hat{j}$ N acts on a particle whose position vector is 2 $\hat{k}$ m.

A force $\vec{F} = \left(\hat{i} + 2\hat{j} + 3\hat{k}\right)$ N acts at a point

$\left(4\hat{i} + 3\hat{j} - \hat{k}\right)$ m. Then the magnitude of torque

about the point $\left(\hat{i} + 2\hat{j} + \hat{k}\right)$ m will be $\sqrt{x}$ N m.

The value of x is _______.

A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre ‘O’ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is:

Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is $\mu$ = 0.4, the maximum possible value of 100 × $\frac{b}{a}$ for box not to topple before moving is .......

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5m, the angle between the force and the position vector is (in radians):

A rod $l\text{ m}$ long is acted upon by a couple as shown in the figure. The moment of couple is $\tau\text{ N m}$. If the force acts at each end of the rod, then magnitude of each force is

$(\sin 30^\circ = \cos 60^\circ = 0.5)$

The dimensions of torque are same as that of

A slab is subjected to two forces $\vec{F_1}$ and $\vec{F_2}$ of same magnitude F as shown in the figure. Force $\vec{F_2}$ is in XY-plane while force $\vec{F_1}$ acts along z = axis at the point $\left(2\vec{i} + 3\vec{j}\right).$. The moment of these forces about point O will be:

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is $\theta$, then:

The moment of the force, $\vec{F} = 4\hat{i}+5\hat{j}-6\hat{k}$ at

(2, 0, –3), about the point (2, –2, –2), is given by

A force of $40\text{ N}$ acts on a point $B$ at the end of an $L$-shaped object, as shown in the figure. The angle $\theta$ that will produce maximum moment of the force about point $A$ is given by:

A uniform rod $AB$ is suspended from a point $X,$ at a variable distance $x$ from $A$, as shown. To make the rod horizontal, a mass $m$ is suspended from its end $A.$ A set of $(m, x)$ values is recorded. The appropriate variables that give a straight line, when plotted, are:

In a physical balance working on the principle of moments, when 5 mg weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct?

The machine as shown has 2 rods of length1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a:

A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is

A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is

ABC is an equilateral triangle with O as its centre. $\vec{F_1}, \vec{F_2}$ and $\vec{F_3}$ represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then magnitude of $\vec{F_3}$ is

If $\vec{F}$is the force acting on a particle having position vector$\vec{r}$and $\vec{\tau}$be the torque of this force about the origin, then

A force of $-F\hat{k}$ acts on $O,$ the origin of the coordinate system. The torque about the point $(1,-1)$ is

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

$O$ is the centre of an equilateral triangle $ABC.\ F_1,\ F_2$ and $F_3$ are three forces acting along the sides $AB,\ BC$ and $AC$ as shown in figure. What should be the magnitude of $F_3$ so that the total torque about $O$ is zero?

A weightless ladder $20\text{ ft}$ long rests against a frictionless wall at an angle of $60^\circ$ from the horizontal. A $150$ pound man is $4\text{ ft}$ from the top of the ladder. A horizontal force is needed to keep it from slipping. Choose the correct magnitude of force from the following:

Find the torque of a force $\vec{F} = -3\hat{i}+\hat{j}+5\hat{k}$ acting at the point $\vec{r} = 7\hat{i}+3\hat{j}+\hat{k}$.

A couple produces

What is the torque of the force $\vec{F} = 2\hat{i}-3\hat{j}+4\hat{k}\text{ N}$ acting at the point $\vec{r} = 3\hat{i}+2\hat{j}+3\hat{k}\text{ m}$ about origin?