Work Done as Dot Product

Q. Derive the formula for the range of a projectile thrown with a velocity $u$ at an angle $\theta$ from the horizontal.

Q. In the product

$\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})=q\overrightarrow{v}\times (B\hat{i}+B\hat{j}+B_0\hat{k})$

For $q=1$ and $\overrightarrow{v}=2\hat{i}+4\hat{j}+6\hat{k}$,
$\overrightarrow{F}=4\hat{i}-20\hat{j}+12\hat{k}$

What will be the complete expression for $\overrightarrow{B}$?

1. $6\hat{i}+6\hat{j}-8\hat{k}$
2. $-8\hat{i}-8\hat{j}-6\hat{k}$
3. $-6\hat{i}-6\hat{j}-8\hat{k}$
4. $8\hat{i}+8\hat{j}-6\hat{k}$

Q. A force $F=-k(yi+xj)$ (where $k$ is a positive constant) acts on a particle moving in the $xy$ - plane. Starting from the origin, the particle is taken along the positive $x$ - axis to the point $(a,0)$ and then parallel to the $y$ - axis to the point $(a,a)$. The total work done by the force $F$ on the particle is

1. $-2k{a}^2$
2. $2k{a}^2$
3. $-k{a}^2$
4. $k{a}^2$

Q. In vector diagram shown in fig. R is the resultant of vectors A and B.
If R = $\frac{B}{\sqrt{2}}$, the value of angle $\theta$ is

1. 
2. 
3. 
4. 

Q. A particle moves from a point $(-2\hat{i}+5\hat{j})\text{m}$ to $(4\hat{j}+3\hat{k})\text{m}$ when a force of $(4\hat{i}+3\hat{j})\text{N}$ is applied. Then, work done by the force is

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

Q. A body of mass $3\text{kg}$ is under a constant force has displacement $s$, given by the relation $s=\frac{1}{3}{t}^2$, where $t$ is in seconds, $s$ is in metres. Work done by the force in $2$ seconds is

1. $\frac{19}{5}\text{J}$
2. $\frac{5}{19}\text{J}$
3. $\frac{3}{8}\text{J}$
4. $\frac{8}{3}\text{J}$

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. A force $\overrightarrow{F}=3\hat{i}+c\hat{j}+2\hat{k}$ acting on a particle causes a displacement $\overrightarrow{S}=-4\hat{i}+2\hat{j}-3\hat{k}$ in its own direction. If the work done by the force is $6\text{J}$, then the value of $c$ will be

1. $12$
2. $1$
3. $6$
4. $0$

Q.

The unit vector in $ZOX-$ plane and making angle $45°$ and $60°$, respectively with $\stackrel{\rightharpoonup }{a}=2\hat{i}+2\hat{j}-\hat{k}$ and $\stackrel{\rightharpoonup }{b}=\hat{j}-\hat{k}$is?

1. $\frac{-1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}$

2. $\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{k}$

3. $\frac{1}{3\sqrt{2}}\hat{i}+\frac{4}{3\sqrt{2}}\hat{j}+\frac{1}{3\sqrt{2}}\hat{k}$

4. None of these

Q. A block of mass $0.50\text{kg}$ is moving with a speed of $2.00{\text{ms}}^{-1}$ on a smooth surface. it strikes another mass of $1.00\text{kg}$ and then they move together as a single body. The energy loss during the collision is

1. $0.16\text{J}$
2. $1.00\text{J}$
3. $0.67\text{J}$
4. $0.34\text{J}$

Q. Force acting on a particle is $(2\overline{i}+3\overline{j})$ N. work done by this force is zero, when a particle is moved on the line 3y + kx = 5 Here value of k is

1. 4
2. 3
3. 2
4. 1

Q.

A box is pushed through 4.0 m across a floor offering 100 N resistance. How much work is done by the resisting force ?

Q.

Can you find dot product of a vector with a scalar?

Q. Which of the following is/are conservative force(s)?

1. $\overrightarrow{F}=2r^3\hat{r}$
2. $\overrightarrow{F}=-\frac{5}{r}\hat{r}$
3. $\overrightarrow{F}=\frac{3(x\hat{i}+y\hat{j})}{(x^2+y^2{)}^{\frac{3}{2}}}$
4. $\overrightarrow{F}=\frac{3(y\hat{i}+x\hat{j})}{(x^2+y^2{)}^{\frac{3}{2}}}$

Q. A force $\overrightarrow{F}=(3\hat{i}+4\hat{j})\text{N}$ acts on a particle moving in $x-y$ plane . Starting from origin, the particle first goes along $x$ - axis to the point $(4\text{m, 0)}$ and then parallel to the $y$ - axis to the point $(4\text{m},3\text{m})$. The total work done by the force on the particle is

1. $+12\text{J}$
2. $-6\text{J}$
3. $+24\text{J}$
4. $-12\text{J}$

Q.

A block of mass 2.0 kg kept at rest on an inclined place of inclination ${37}^{\circ }$ is pulled up the plane by applying a constant force of 20 N parallel to the incline. The force acts for on second. (a) Show that the work done by the applied force does not exceed 40 J. (b) Find the work done by the force of gravity in that one second if the work done by the applied force is 40 J.(c) Find the kinetic energy of the block at the instant the force ceases to act. Take g=$10m/s^2$.

Q. A block of mass M is pulled along a horizontal surface by applying a force at an angle with the horizontal. The friction coefficient between the block and the surface is $\mu$. If the block travels at a uniform velocity, then the work done by this applied force during a displacement d of the block is—

1. 
2. 
3. 
4. 

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. 
2. 
3. 
4. 

Q. For the resultant of two vectors to be maximum, the angle between them should be

1. 0 degree
2. 45 degrees
3. 90 degrees
4. 180 degrees

Q.

Let $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{R}$ be such that ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=1$, if $\mathrm{acos\theta }=\mathrm{bcos}\left(\mathrm{\theta }+\frac{2\mathrm{\pi }}{3}\right)=\mathrm{ccos}\left(\mathrm{\theta }+\frac{4\mathrm{\pi }}{3}\right)$, where $\mathrm{\theta }=\frac{\mathrm{\pi }}{9}$,

Then the angle between the vectors $\mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+\mathrm{c}\hat{\mathrm{k}}$ and $\mathrm{b}\hat{\mathrm{i}}+\mathrm{c}\hat{\mathrm{j}}+\mathrm{a}\hat{\mathrm{k}}$ is:

1. $\frac{\mathrm{\pi }}{2}$

2. $\frac{2\mathrm{\pi }}{3}$

3. $\frac{\mathrm{\pi }}{9}$

4. $0$

Q.

Why is the Dot Product a Scalar?

Q. Three block $A,B\text{and}C$ are lying on a smooth horizontal surface, as shown in the figure. $A$ and $B$ have equal masses, $m$ while $C$ has mass $M$. Block $A$ is given an initial speed $v$ towards $B$ due to which it collides with $B$ perfectly inelastically. The combined mass collides with $C$, also perfectly inelastically ${\frac{5}{6}}^{th}$ of the initial kinetic energy is lost in the whole process. What is the value of $\frac{M}{m}?$

1. $5$
2. $4$
3. $2$
4. $3$

Q. A block moving horizontally on a smooth surface with a speed of $20{\text{ms}}^{-1}$ bursts into two equal pieces continuing in the same direction. If one of the points moves at $30{\text{ms}}^{-1}.$ Find the fractional change in kinetic energy.

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $\frac{1}{16}$
4. $\frac{1}{8}$

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

Q. A body having a mass of $100\text{gm}$ is allowed to fall freely (from a very large height) under the action of gravity. Work done by gravity during first $10\text{s}$ of journey is (Take $g=10{\text{m/s}}^2$)

1. $500000\text{J}$
2. $50000\text{J}$
3. $5000\text{J}$
4. $500\text{J}$

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.

Let $\overrightarrow{a}=\hat{i}+2\hat{j}-\widehat{k,}\overrightarrow{b}=\hat{i}-\hat{j}$ and $\overrightarrow{c}=\hat{i}-\hat{j}-\hat{k}$be three given vectors. If $\overrightarrow{r}$is a vector such that $\begin{array}{l}\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{c}\times \overrightarrow{a}\end{array}$and $\begin{array}{l}\overrightarrow{r}.\overrightarrow{b}=0\end{array}$then $\begin{array}{l}\overrightarrow{r}.\overrightarrow{a}\end{array}$is equal to _____

Q. A block of mass $m$ is stationary with respect to a rough wedge as shown in figure. Starting from rest in time $t$.
(Given that $m=1\text{kg},\theta ={30}^{\circ },a=2{\text{m/s}}^2,t=4\text{s}$)
Match Column $I$ with column $II$
$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(a) Work done on block by gravity}& \text{(p)}144\text{J}\\ \text{(b) Work done on block by normal reaction}& \text{(q)}32\text{J}\\ \text{(c) Work done on block by friction}& \text{(r)}56\text{J}\\ \text{(d) Work done on block by all the forces}& \text{(s)}48\text{J}\\ & \text{(t)}\text{None}\end{array}$

1. $a-p,b-t,c-s,d-q$
2. $a-p,b-t,c-q,d-q$
3. $a-q,b-t,c-s,d-p$
4. $a-t,b-p,c-s,d-q$

Q.

Examples for positive, zero and negative work

Q. A force $\overrightarrow{F}=6x\hat{i}+2y\hat{j}$ displaces a body from $\overrightarrow{r_1}=3\hat{i}+8\hat{j}$ to $\overrightarrow{r_2}=5\hat{i}-4\hat{j}$. Find the work done by the force.

1. $64\text{J}$
2. $16\text{J}$
3. $0\text{J}$
4. $75\text{J}$