Work Done as Dot Product

Q.

A particle acted upon by constant forces $F_1=4\hat{i}+\hat{j}-3\hat{k}$ and $F_2=3\hat{i}+\hat{j}-\hat{k}$ is displaced from the point $r_1=\hat{i}+2\hat{j}-3\hat{k}$ to point $r_2=5\hat{i}+4\hat{j}-\hat{k}$. The total work done by the forces in SI unit is

1. $20$

2. $24$

3. $50$

4. $30$

5. $35$

Q. The displacement of body of mass 4 kg varies with time (t) as S = t² + 3t where S is in meters and t is in seconds. The work done by all the forces acting on the body during the time interval t = 0 s to t = 2 s is

4 kg द्रव्यमान के पिण्ड का विस्थापन, समय (t) के साथ S = t² + 3t के अनुसार परिवर्तित होता है, जहाँ S मीटर में है तथा t सेकण्ड में है। समय अन्तराल t = 0 s से t = 2 s के दौरान पिण्ड पर कार्यरत सभी बलों द्वारा किया गया कार्य है

1. 64 J
2. 80 J
3. 40 J
4. 70 J

Q. Two forces $\overrightarrow{F_1}=\hat{i}-2\hat{j}+2\hat{k}$ $\text{N}$ and $\overrightarrow{F_2}=-3\hat{i}+\hat{j}-3\hat{k}$ $\text{N}$ acts on a body and causes it to displace from position $\overrightarrow{s_1}=\hat{i}-\hat{j}$ $\text{m}$ to $\overrightarrow{s_2}=2\hat{j}-a\hat{k}$ $\text{m}$. Find the value of ${}^{\prime }{a}^{\prime }$ if the amount of net work done by the forces is $5\text{J}$ during the displacement.

1. $-4$
2. $8$
3. $6$
4. $-5$

Q. Find the work done in lifting $5$ cement bags, each of mass $25\text{kg}$ to the top of a building of height $50\text{m}$.(consider $g=10m/s^2$)

1. $12.5\text{kJ}$
2. $-25\text{kJ}$
3. $62.5\text{kJ}$
4. $-62.5\text{kJ}$

Q. Blocks of masses $m,2m,4m$ and $8m$ are arranged in a line on a frictionless floor. Another block of mass $m,$ moving with speed $v$ along the same line (as shown) collides with mass $m$ in a perfectly inelastic manner. All the subsequent collisions are also perfectly inelastic. By the time the last block of mass $8m$ starts moving, the total energy loss is $p\%$ of the original energy. The value of $p$ is close to-

1. $37$
2. $94$
3. $77$
4. $87$

Q. A force of $(10\hat{i}-3\hat{j}+6\hat{k})\text{N}$ acts on a body of $5\text{kg}$ and displaces it from $A(6\hat{i}-5\hat{j}+3\hat{k})\text{m}$ to $B(10\hat{i}-2\hat{j}+7\hat{k})\text{m}$. Work done by the force is equal to

1. zero
2. $55\text{J}$
3. $100\text{J}$
4. $221\text{J}$

Q. A cord is used to lower vertically a block of mass M by a distance d with constant downward acceleration Work done by the cord on the block is

1. $Mg\frac{d}{4}$
2. $3Mg\frac{d}{4}$
3. $-3Mg\frac{d}{4}$
4. $Mgd$

Q. A block of mass $(m=1kg)$ is released from rest over a circular curved path, as shown in diagram. The work done (in joules) by normal force, when block is displaced by $\theta ={45}^{\circ }$ is

Q. When an object is stationary but the point of application of force moves on the object, work done is

1. positive
2. zero
3. negative
4. either zero or positive

Q. A helicopter lifts a $72\text{kg}$ astronaut $15\text{m}$ vertically from the ocean by means of a cable. The acceleration of the astronaut is $\frac{g}{10}$. How much work is done on the astronaut by the force from the helicopter ?

1. $1188\text{J}$
2. $11880\text{J}$
3. $118.8\text{J}$
4. $9720\text{J}$

Q.

Why is cross product useful?

Q. A body of mass $2\text{kg}$ is thrown vertically upwards with an initial speed of $10\text{m/s}$. The work done by the force of gravity during the time the body reaches its maximum height. (Consider $g=10{\text{m/s}}^2$)

1. $100\text{J}$
2. $0\text{J}$
3. $-100\text{J}$
4. $20\text{J}$

Q. A particle moves along the $x-$axis from $x=1\text{m}$ to $x=3\text{m}$ under the influence of a force $F=3x^2-2x+5$ newton acting along $x-$ axis. The work done in this process is

1. $9\text{J}$
2. $28\text{J}$
3. $27\text{J}$
4. zero

Q. Figure shows three forces applied to a trunk that moves leftward by $3$ m over a smooth floor. The force magnitudes are $F_1=5N,F_2=9N,$ and $F_3=3N$. The net work done on the trunk by the three forces

1. $1.50\text{J}$
2. $2.40\text{J}$
3. $3.00\text{J}$
4. $6.00\text{J}$

Q.

A force of 5 N acts on a 15 kg body initially at rest. The work done by the force during the first second of motion of the body is

1. 5 J

2. $\frac{5}{6}J$

3. 6 J

4. 75 J

Q. A force $\overrightarrow{F}=b\frac{-yi+xj}{x^2+y^2}\text{N}$ ($b$ is a constant) acts on a particle as it undergoes counterclockwise circular motion in a circle $x^2+y^2=16$.
The work done by the force when the particle undergoes one complete revolution is ($x,y$ are in $m$)

1. Zero
2. $2\pi b\text{J}$
3. $2b\text{J}$
4. None of these

Q. A particle projected with an initial velocity $u$ at angle $\theta$ from the ground. What is the work done by gravity during the time it reaches the highest point $\left(P\right)$ in its path

1. $\frac{-m{u}^2{\sin }^2\theta }{2}$
2. $+\frac{m{u}^2{\sin }^2\theta }{2}$
3. $0$
4. $+m{u}^2\sin \theta$

Q. An object is displaced from point $A(0,1,2)m$ to a point $B(1,2,3)m$ under the influence of a force $\overrightarrow{F}=2x\hat{i}+3y^2\hat{j}+\hat{k}.$ The workdone by this force is

1. $-10\text{J}$
2. $12\text{J}$
3. $9\text{J}$
4. $0\text{J}$

Q. A particle of mass $m$ is projected with an initial velocity $u$ at an angle $\theta$ from the ground. What is the work done by gravity during the time it reaches the highest point P?

1. $\frac{-m{u}^2{\sin }^2\theta }{2}$
2. $+\frac{m{u}^2{\sin }^2\theta }{2}$
3. \N
4. $+m{u}^2\sin \theta$

Q. A constant force $\overrightarrow{F}=2\hat{i}-3\hat{j}+\hat{k}$ is applied on a ball to displace it from $\overrightarrow{r_1}=\hat{i}+\hat{j}$ to $\overrightarrow{r_2}=4\hat{i}+3\hat{j}$, which of the following statements is true?

1. Force and displacement are perpendicular to each other and work done by force is equal to zero
2. Force and displacement are parallel to each other and work done by force is equal to zero
3. Force and displacement are anti parallel to each other and work done by force is equal to zero
4. Force and displacement are perpendicular to each other and work done by force is not equal to zero

Q.

A particle of mass 0.01 kg travels along a curve with velocity given by $4\hat{i}+16\hat{k}m{s}^{-1}$. After some time, its velocity becomes $8\hat{i}+20\hat{j}m{s}^{-1}$due to the action of a conservative force. The work done on particle during this interval of time is

1. 0.32 J

2. 6.9 J

3. 9.6 J

4. 0.96 J

Q. A man pushes a $15\text{kg}$ box of books $2.0\text{m}$ up a ${37}^{\circ }$ incline into the back of a moving van. The box moves at a constant velocity if you push it with a force of $95\text{N}$. Find the work done by gravity on the box. Take $(g=10{\text{m/s}}^2)$

1. $-180\text{J}$
2. $-120\text{J}$
3. $180\text{J}$
4. Zero

Q. Three forces $\overrightarrow{F_1}=2\hat{i}-4\hat{j}+5\hat{k}$, $\overrightarrow{F_2}=\hat{i}+3\hat{j}-8\hat{k}$ and $\overrightarrow{F_3}=-6\hat{i}-10\hat{j}+7\hat{k}$ are acting on a particle present on the positive $x$ axis, $5\text{units}$ away from the origin. The three forces combined displace the body to a point $6\hat{i}-3\hat{j}+10\hat{k}$. What is the net work done by the three forces ?

1. $76\text{J}$
2. $70\text{J}$
3. $-76\text{J}$
4. $-70\text{J}$

Q. A uniform rod of length L and mass m is placed on rough horizontal surface.If coefficient of friction is $\mu =kx$, where k is constant and $\mu$ exist for $0\le x\le L$ (i.e for $x>L$ friction is zero.). If rod moves towards the right, magnitude of work done by friction is:

1. $\frac{kmg{L}^2}{9}$
2. $\frac{kmg{L}^2}{6}$
3. $\frac{kmg{L}^2}{3}$
4. $\frac{2kmg{L}^2}{3}$

Q. A particle moves from a point $(2\hat{i}+\hat{k})\text{m}$ to $(4\hat{i}+3\hat{j}-\hat{k})\text{m}$ when a force of $(3\hat{i}+\hat{j})\text{N}$ is applied. The work done by the force is

1. $8\text{J}$
2. $11\text{J}$
3. $5\text{J}$
4. $9\text{J}$

Q. Assume that a body is displaced from $\overrightarrow{r_A}=(2\text{m},4\text{m},-6\text{m})$ to $\overrightarrow{r_B}=(6\hat{i}-4\hat{j}+2\hat{k})\text{m}$, under a constant force $\overrightarrow{F}=(2\hat{i}+3\hat{j}-\hat{k})\text{N}$. Calculate work done by the force.

1. $-12\text{J}$
2. $+12\text{J}$
3. $-24\text{J}$
4. $+24\text{J}$

Q. A particle of mass 'm' is projected with velocity 'u' at an angle $\theta$ with horizontal. During the period when the particle descends from highest point to the position where its velocity vector makes an angle $\frac{\theta }{2}$ with horizontal, work done by the gravity force is

1. $\frac{1}{2}m{u}^{2}ta{n}^2\frac{\theta }{2}$
2. $\frac{1}{2}m{u}^{2}ta{n}^2\theta$
3. $\frac{1}{2}m{u}^{2}co{s}^2\theta ta{n}^2\frac{\theta }{2}$
4. $\frac{1}{2}m{u}^{2}co{s}^2\frac{\theta }{2}si{n}^2\theta$

Q. The graph shown below represents the variation of the force as a function of position for a body.


Find out the work done by the force in moving the body from $x=3\text{m}$ to $x=7\text{m}$.

1. $12\text{J}$
2. $20\text{J}$
3. $10\text{J}$
4. $14\text{J}$

Q. A particle projected with an initial velocity $u$ at angle $\theta$ from the ground. What is the work done by gravity during the time it reaches the highest point $\left(P\right)$ in its path

1. $\frac{-m{u}^2{\sin }^2\theta }{2}$
2. $+\frac{m{u}^2{\sin }^2\theta }{2}$
3. $0$
4. $+m{u}^2\sin \theta$

Q. A force $\overrightarrow{F}=[y\hat{i}+x\hat{j}]\text{N}$ acts on a particle moving in $x-y$ plane starting from the point $(3,5)\text{m}$, the particle is taken along a straight line to $(5,7)\text{m}$. The work done by the force( in $\text{J}$ upto two decimals) is: