Solve the following:

Question: Find the area bounded under the curve $y=3 x^{2}+6 x+7$ and the X-axis with the co-ordinates at $x=5$ and $x=10$. Solution: $y=3 x^{2}+6 x+7$ Area bounded under the curve within $x=5$ and $x=10$ is calculated by the method of integration. Area $=\int_{x=0}^{x-\pi} y d x=\int_{x=0}^{x-\pi} \sin x d x=-[\cos \pi-\cos 0]=2$...

Solve the following :

Question: The electric current in a discharging $\mathrm{R}-\mathrm{C}$ circuit is given by $\mathrm{i}=\mathrm{i}_{0} \mathrm{e}^{-\mathrm{t} R \mathrm{RC}}$ where $\mathrm{i}_{0}, \mathrm{R}$ and $\mathrm{C}$ are constant parameters of the circuit and $\mathrm{t}$ is time. Let $\mathrm{i}_{0}=2.00 \mathrm{~A}, \mathrm{R}=6.00 \times 10^{5} \Omega$ and $\mathrm{C}=0.500 \mu \mathrm{F}$. (a) Find the current at $t=0.3 \mathrm{~s}$. (b) Find the rate of change of current at $t=0.3 \mathrm{~s}$. (...

Solve the following :

Question: The electric current in a charging $R-C$ circuit is given by $i=i_{0} e^{-t / R C}$ where $\mathrm{i}_{0}, R$ and $C$ are constant parameters of the circuit and $t$ is time. Find the rate of change of current at (a) $t=0$, (b) $t=R C$, (c) $\mathrm{t}=10 \mathrm{RC}$. Solution: We have, $i=i_{0} e^{-t / R c}$ Rate of change of current $=\frac{d i}{d x}=\frac{d}{d x}\left(i_{0} e^{-\frac{t}{R C}}\right)=-\frac{i_{0}}{R C} \times e^{-\frac{t}{R C}}$ a) When $t=0, \mathrm{di} / \mathrm{dt...

Solve the following :

Question: A curve is represented by $y=\sin x$. If $x$ is changed from $\frac{\frac{\pi}{3}}{t} t o \frac{\pi}{3}+\frac{\pi}{100}$, find approximately the change in $y$. Solution: $y=\sin (x)$ Let $y 1=\sin (\pi / 3)$ and $y 2=\sin (\pi / 3+\pi / 100)$ Change in $y=y 2-y 1=\sin (\pi / 3+\pi / 100)-\sin (\pi / 3)$ $=\sin (\pi / 3+(\pi / 3+\pi / 100-\pi / 3))-\sin (\pi / 3)$ $=0.0157$...

Solve the following :

Question: Give an example for which $\vec{A} \cdot \vec{B}=\vec{C} \cdot \vec{B}$ but $\vec{A} \neq \vec{C}$. Solution: Let us assume that $B$ is along $Y$ axis, and $A$ is along positive $x$ axis and $C$ is along negative $X$ axis. Now, $A \cdot B=B \cdot C=0$. But $A \neq C$...

Solve the following :

Question: The force on a charged particle due to electric and magnetic fields is given by $\vec{F}=q \vec{E}+q \vec{v} \times \vec{B}$. $v$ should a positively charged particle be sent so that the net force on it is zero? Solution: $F=q(E+v \times B)$ Now, for net force to be 0 , we must have $E=-(v \times B)$ So, the direction of $E$ must be opposite to that of $(v \times B)$, so $v$ must be in $Z$ axis and its magnitude is $\mathrm{E} /(\mathrm{B} \sin \theta)$. For $v$ to be minimum. $\theta=...

Solve the following :

Question: A particle moves on a given straight line with a constant $v$. At a certain time it is at a point $\mathrm{P}$ on its straight line path. $O$ is a fixed point. Show that $\overrightarrow{O P} \times \vec{v}$ is independent of the position $\mathrm{P}$. Solution: The particle moves from PP' in a straight line with a constant speed $v$. From the figure, we see that $\mathrm{OP} \times \mathrm{V}=(\mathrm{OP}) \mathrm{v} \sin \theta \hat{\text { un, where } \hat{u} \text { is a unit vecto...

Solve the Following :

Question: If $\vec{A}=2 \vec{\imath}+3 \vec{\jmath}+4 \vec{k}$ and $\vec{B}=4 \vec{\imath}+3 \vec{\jmath}+2 \vec{k}$, find $\vec{A} \times \vec{B}$ Solution: A, $B$ and $C$ are mutually perpendicular vectors. Now, if we take cross product between any two vectors, the resultant vector will be in parallel to the third vector, as there are only three axis perpendicular to each other. So if we consider $(A \times B)$, then it is parallel to $C$, and so angle between the resultant vector and $C$ is $...

Question: Prove that $\vec{A} \(\vec{A} \times \vec{B})=0$ Solution: $(\mathbf{A} \times \mathbf{B})=\mathrm{AB} \sin \Theta$ , where is a unit vector perpendicular to both $A$ and $B$. Now, $\mathbf{A} .(\mathbf{A} \times \mathbf{B})$ is basically a dot product between two vectors which are perpendicular to each other. Then $\cos 90^{\circ}=0$, and thus A. $(\mathbf{A} \times \mathbf{B})=0$...

Find the angle between them.

Question: Let $\vec{a}=2 \vec{\imath}+3 \vec{\jmath}+4 \vec{k}$ and $\vec{b}=3 \vec{\imath}+4 \vec{\jmath}+5 \vec{k}$. Find the angle between them. Solution: $a=2 i+3 j+4 k$ and $b=3 i+4 j+5 k$ Let angle between them is $\theta$ Then a.b $=2.3+3.4+4.5=38$ $|a|=\sqrt{\left(2^{2}+3^{2}+4^{2}\right)=\sqrt{29}}$ $|a|=\sqrt{\left(2^{2}+3^{2}+4^{2}\right)=\sqrt{29}}$ Now $\cos \theta=a \cdot b /|a||b|=38 / \sqrt{1450}$...

Solve the following

Question: Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{1}$ be a regular hexagon. Write the x-components of the vectors represented by the six sides taken in order. Use the fact that the resultant of these six vectors is zero, to prove that $\cos 0+\cos \pi / 3+\cos 2 \pi / 3+\cos 4 \pi / 3+\cos 5 \pi / 3=0$ Use the known cosine values to verify the result. Solution: From polygon law of vector addition, the resultant of the six vectors can be affirmed to be zero. Here their magnitudes are the same...

Find (a) the scalar product of the two vectors, (b) the magnitude of their vector product?

Question: Two vectors have magnitudes $2 \mathrm{~m}$ and $3 \mathrm{~m}$. The angle between them is $60^{\circ}$. Find (a) the scalar product of the two vectors, (b) the magnitude of their vector product? Solution: We have $a=2 \mathrm{~m}, b=3 \mathrm{~m}$. $\theta=60^{\circ}$ is the angle between the two vectors Scalar product between the two vectors $=a \cdot b=$ $2 \times 3 \times \cos \left(60^{\circ}\right)=3 \mathrm{~m}^{2}$ Vector product between the two vectors $=a \times b=$ $2 \times...

Solve the following

Question: Suppose $\vec{a}$ is a vector of magnitude $4.5$ unit due north. What is the vector(a) $3^{\vec{a}}$, (b) $-4 \vec{a}_{?}$ Solution: $a=4.5 \mathbf{n}$, where $\mathbf{n}$ is unit vector in north direction a) $3 a=4.5 \times 3 n=13.5$ in north direction b) $-4 a=4.5 X-4 n=18$ in south direction...

Solve the following

Question: A mosquito net over a $7 \mathrm{ft}^{\times} 4 \mathrm{ft}$ bed is $3 \mathrm{ft}$ high. The net has a hole at one corner of the bed through which a mosquito enters the bed. It flies and sits at the diagonally opposite upper corner of the net. (a) Find the magnitude of the displacement of the mosquito. (b) Taking the hole as the origin, the length of the bed as the X-axis, its width as the Y-axis, and vertically up as the Z-axis, write the components of the displacement vector. Soluti...

Solve the following

Question: A carrom board ( $4 \mathrm{ft}^{\times} 4 \mathrm{ft}$ square) has the queen at the center. The queen, hit by the striker moves to the front edge, rebounds and goes in the hole behind the striking line. Find the magnitude of displacement of the queen (a) from the center to the front edge, (b) from the front edge to the hole and (c) from the center to the hole. Solution: In $\otimes A B C, \tan l=x / 2$ and in $\otimes D C F, \tan l=(2-x) / 4$, So, $(x / 2)=(2-x) / 4$. Solving, $4-2 x=...

Find the displacement of the car.

Question: A spy report about a suspected car reads as follows. "The car moved $2.00 \mathrm{~km}$ towards east, made a perpendicular left turn, ran for $500 \mathrm{~m}$, made a perpendicular right turn, ran for $4.00 \mathrm{~km}$ and stopped." Find the displacement of the car. Solution: $A B=2 i+0.5 j+4 i=6 i+0.5 j$ As the car went forward, took a left and then a right. So, $A B=\left(6^{2}+0.5^{2}\right)^{1 / 2}=6.02 \mathrm{~km}$ And $\phi=\tan ^{-1}(\mathrm{BE} \backslash \mathrm{AE})=\tan ...

Solve the following

Question: Two vectors have magnitudes 3 unit and 4 unit respectively. What should be the angle between them if the magnitude of the resultant is (a) 1 unit, (b) 5 unit and (c) 7 unit. Solution: Let $\Theta$ be the angle between them. Then, using the relation $\mathrm{R}^{2}=\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB} \cos \theta$, a) We get for $R=1$, $1=9+16+24 \operatorname{Cos} \theta$ Or, $\theta=180^{\circ}$ b) For, $R=5$, we have $25=9+16+24 \operatorname{Cos} \theta$ Or, $\cos \theta=0$ $...

Solve the following

Question: Let $\vec{a}=4 \hat{\imath}+3 \hat{\jmath}$ and $\vec{b}=3 \hat{\imath}+4 \hat{\jmath}$. (a) Find the magnitudes of (a) $\vec{a}$, (b) $\vec{b}$, (c) $\vec{a}+\vec{b}$ and (d) $\vec{a}-\vec{b}$. Solution: $a=4 i+3 j, b=3 i+4 j$ $|a|=|b|=\sqrt{\left(3^{2}+4^{2}\right)}=5$ $a+b=7 i+7 j$ and $a-b=i-j$ $|a+b|$ $=\sqrt{\left(7^{2}+7^{2}\right)}=7 \sqrt{2}$ and $|a-b|=\sqrt{\left(1^{2}+1^{2}\right)}=\sqrt{2}$...

Solve the following

Question: Add vectors $A, B$ and $C$ each having magnitude of 100 unit and inclined to the $X$-axis at angles $45^{\circ}, 135^{\circ}$ and $315^{\circ}$ respectively. Solution: Vectors $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are oriented at $45^{\circ}, 135^{\circ}$ and $315^{\circ}$ respectively. $|A|=|B|=|C|=100$ units Let $A=A_{x} \mathbf{i}+A_{y} \mathbf{j}+A_{z} \mathbf{k}, B=B_{x} \mathbf{i}+B_{y} \mathbf{j}+B_{z} \mathbf{k}$, and $C=C_{x} \mathbf{i}+C_{y} \mathbf{j}+C_{z} \mathbf{k}$, ...

Find the resultant.

Question: Let $\vec{A}$ and $\vec{B}$ be the two vectors of magnitude 10 unit each. If they are inclined to the $\mathrm{X}$-axis at angles $30^{\circ}$ and $60^{\circ}$ respectively, find the resultant. Solution: A and $B$ are inclined at angles of 30 degrees and 60 degrees with respect to the $x$ axis Angle between them $=(60-30)=90$ degrees Given that $|\mathrm{A}|=|\mathrm{B}|=10$ units, we get $\mathrm{R}^{2}=\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB} \cos \theta$ $=10^{2}+10^{2}+2.10 .10 ...

Find the resultant.

Question: A vector $\vec{A}$ makes an angle of $20^{\circ}$ and $\vec{B}$ makes an angle of $110^{\circ}$ with the X-axis. The magnitudes of these vectors are $3 \mathrm{~m}$ and $4 \mathrm{~m}$ respectively. Find the resultant. Solution: The angle between $\mathbf{A}$ and $\mathbf{B}$ from the $x$-axis are $20^{\circ}$ and $110^{\circ}$ respectively. Their magnitudes are 3 units and 4 units respectively. Thus the angle between $\mathbf{A}$ and $\mathbf{B}$ is $=110-20=90^{\circ}$ Now, $R^{2}=A^...

Solve the following:

Question: Let $x$ and a stand for distance. Is $\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\frac{1}{a} \sin ^{-1} \frac{a}{x}$ dimensionally correct? Solution: Dimension of the Integral $=\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\int \frac{L}{\sqrt{L^{2}-L^{2}}}=L^{0}$. But for $\frac{1}{a} \sin ^{-1} \frac{a}{x}$, the dimension is [L-1]. So this expression is dimensionally incorrect....

Test if the following are dimensionally correct:

Question: Test if the following are dimensionally correct: a) $h=\frac{2 \sec \theta}{\rho r g}, b=\sqrt{\frac{p}{\rho}}$, c $\left.)^{\frac{\pi P_{t r r}^{4}}{8 \eta l}}, \mathrm{~d}\right)^{\frac{1}{2 \pi} \sqrt{\frac{m g l}{I}}}$ Solution: a) $h=\frac{2 \sec \theta}{\operatorname{prg}}$ Here, $h=[L], \mathrm{S}=\mathrm{F} / \mathrm{L}=\left[\mathrm{MT}^{-2}\right], \rho=\left[\mathrm{ML}^{-3}\right], \mathrm{r}=[\mathrm{L}], \mathrm{g}=\left[\mathrm{LT}^{-2}\right]$ $2 \operatorname{seos} \th...

Guess the expression for its frequency from dimensional analysis

Question: The frequency of vibration of a string depends on the length $L$ between the nodes, the tension $F$ in the string and its mass per unit length $\mathrm{m}$. Guess the expression for its frequency from dimensional analysis. Solution: Let, frequency $\mathrm{v}=\mathrm{F}^{\mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{m}^{\mathrm{c}}$ or $[\mathrm{T}-1]=\left[\mathrm{ML}^{-2}\right]^{\mathrm{a}}\left[\mathrm{L}^{\mathrm{b}}\right]\left[\mathrm{M}^{\mathrm{c}}\right]$ Equating the terms, we...

Obtain Ohm's law from dimensional analysis

Question: Let $I=$ current through a conductor, $R=$ its resistance and $V=$ potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for $R$ and $V$ are $\left[\mathrm{ML}^{2} \mathrm{I}^{-2} \mathrm{~T}^{-3}\right]$ and $\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~J}^{-1}\right]$ respectively Solution: Dimension of $\mathrm{R}=\left[\mathrm{ML}^{2} \mathrm{I}^{-2} \mat...