In figure the upper wire is made of steel and the lower of copper.
Question: In figure the upper wire is made of steel and the lower of copper. The wires have equal cross-section. Find the ratio of the longitudinal strains developed in the two wires. Solution:...
Read More →A steel wire and a copper wire of equal length and
Question: A steel wire and a copper wire of equal length and equal cross-sectional area are joined end to end and the combination is subjected to a tension. Find the ratio of (a) the stresses developed in the two wires and (b) the strains developed. Y of steel = $2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$, Y of copper $=1.3 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$. Solution:...
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Question: The elastic limit of steel is $8 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}$ and it's Young's modulus $2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$. Find the maximum elongation of a halfmeter steel wire that can be given without exceeding the elastic limit. Solution:...
Read More →A vertical metal cylinder of radius
Question: A vertical metal cylinder of radius $2 \mathrm{~cm}$ and length $2 \mathrm{~m}$ is fixed at the lower end and a load of $100 \mathrm{~kg}$ is put on it. Find (a) the stress (b) the strain and (c) the compression of the cylinder. Young's modulus of the metal $=2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$ Solution:...
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Question: A load of $10 \mathrm{~kg}$ is suspended by a metal wire $3 \mathrm{~m}$ long and having a cross-sectional area $4 \mathrm{~mm}^{2}$. Find (a) the stress (b) the strain and (c) the elongation. Young's modulus of the metal is $2.0 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$. Solution:...
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Question: If $A$ is an invertible matrix and $A^{-1}=\left(\begin{array}{ll}3 4 \\ 5 6\end{array}\right)$ then $A=$ ? A. $\left(\begin{array}{cc}6 -4 \\ -5 3\end{array}\right)$ B. $\left(\begin{array}{cc}\frac{1}{3} \frac{1}{4} \\ \frac{1}{5} \frac{1}{6}\end{array}\right)$ c. $\left(\begin{array}{cc}-3 2 \\ \frac{5}{2} \frac{-3}{2}\end{array}\right)$ D.None of these Solution:...
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Question: If $|A|=3$ and $A^{-1}=\left(\begin{array}{cc}3 -1 \\ \frac{-5}{3} \frac{2}{3}\end{array}\right)$ then adj $A=?$ A. $\left(\begin{array}{cc}9 3 \\ -5 -2\end{array}\right)$ B. $\left(\begin{array}{cc}9 -3 \\ -5 2\end{array}\right)$ c. $\left(\begin{array}{cc}-9 3 \\ 5 -2\end{array}\right)$ D. $\left(\begin{array}{rr}9 -3 \\ 5 -2\end{array}\right)$ Solution:...
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Question: If $A=\left(\begin{array}{cc}2 -1 \\ 1 3\end{array}\right)$, then $A^{-1}=?$ A. $\left(\begin{array}{ll}\frac{3}{7} \frac{-1}{7} \\ \frac{1}{7} \frac{2}{7}\end{array}\right)$ B. $\left(\begin{array}{cc}\frac{3}{7} \frac{1}{7} \\ \frac{-1}{7} \frac{2}{7}\end{array}\right)$ c. $\left(\begin{array}{ll}\frac{1}{3} \frac{1}{7} \\ \frac{1}{7} \frac{2}{7}\end{array}\right)$ D.None of these Solution:...
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Question: If $A$ and $B$ are invertible square matrices of the same order then $(A B)^{-1}=?$ A. $A B^{-1}$ B. $A^{-1} B$ C. $\mathrm{A}^{-1} \mathrm{~B}^{-1}$ D. $B^{-1} A^{-1}$ Solution: $(A B)^{-1}=B^{-1} A^{-1}$...
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Question: If $A=\left(\begin{array}{cc}3 -4 \\ -1 2\end{array}\right)$ and $B$ is a square matrix of order 2 such that $A B=1$ then $B=$ ? A. $\left(\begin{array}{ll}1 2 \\ 2 3\end{array}\right)$ B. $\left(\begin{array}{ll}1 \frac{1}{2} \\ 2 \frac{3}{2}\end{array}\right)$ c. $\left(\begin{array}{cc}1 2 \\ \frac{1}{2} \frac{3}{2}\end{array}\right)$ D.None of these Solution:...
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Question: If $A=\left(\begin{array}{ll}2 5 \\ 1 3\end{array}\right)$ then adj $A=$ ? A. $\left(\begin{array}{cc}3 -5 \\ -1 2\end{array}\right)$ B. $\left(\begin{array}{cc}3 -1 \\ -5 2\end{array}\right)$ c. $\left(\begin{array}{cc}-1 2 \\ 3 -5\end{array}\right)$ D.None of these Solution:...
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Question: If matrices $\mathrm{A}$ and $\mathrm{B}$ anticommute then A. $\mathrm{AB}=\mathrm{BA}$ B. $\mathrm{AB}=-\mathrm{BA}$ C. $(A B)=\left(B A^{-1}\right)$ D. None of these Solution: If $A$ and $B$ anticommute then $A B=-B A$...
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Question: If $A=\left(\begin{array}{ll}3 1 \\ 7 5\end{array}\right)$ and $A^{2}+x \mid=y A$ then the values of $x$ and $y$ are A. $X=6, y=6$ B. $X=8, y=8$ C. $x=5, y=8$ D. $x=6, y=8$ Solution:...
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Question: If $A=\left(\begin{array}{cc}-2 3 \\ 1 1\end{array}\right)$ then $\left|A^{-1}\right|=?$ A. $-5$ B. $\frac{-1}{5}$ C. $\frac{1}{25}$ D. 25 Solution:...
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Question: For any 2-rowed square matrix $\mathrm{A}$, if $\mathrm{A}(\operatorname{adj} \mathrm{A})=\left(\begin{array}{ll}8 0 \\ 0 8\end{array}\right)$ then the value of $|\mathrm{A}|$ is A. 0 B. 8 C. 64 D. 4 Solution: $|A|=8$...
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Question: The track shown in figure is frictionless. The block $B$ of mass $2 \mathrm{~m}$ is lying at rest and the block $A$ of mass $\mathrm{m}$ is pushed along the track with speed. The collision between $\mathrm{A}$ and $\mathrm{B}$ is perfectly elastic. With what velocity should the block $A$ be started to get the sleeping man awakened? Solution: Use C.O.L.M, $m \cdot V_{1}=2 m(0)=m \cdot V_{2}+(2 m)\left(V_{3}\right)$ When B reaches man, Use C.O.E.L, $V_{3}=\sqrt{2 g h}$ $\Rightarrow V_{1}...
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Question: Consider a gravity-free hall in which a experimenter of mass $50 \mathrm{~kg}$ is resting on a $5 \mathrm{~kg}$ pillow, $8 \mathrm{ft}$ above the floor of the hall. He pushes the pillow down so that it starts falling at the speed of $8 \mathrm{fts} / \mathrm{s}$. The pillow makes a perfectly elastic collision with the floor, rebounds and reaches the experimenter's head. Find the time elapsed in the process. Solution: $V_{p m}=8$ (velocity of pillow w.r.t man) $V_{p m}=\overline{V_{p}}-...
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Question: Consider the situation of the previous problem. Suppose the block of mass $m_{1}$ is pulled by a constant force $F_{1}$ and the other block is pulled by a constant force $F_{2}$. Find the maximum elongation that the spring will suffer. Solution: $F_{1} \neq F_{2}$ $a_{m 1}=\frac{F_{1}-F_{2}}{m_{1}+m_{2}} a_{m 2}=\frac{F_{2}-F_{1}}{m_{1}+m_{2}}$ Net force of $m_{1}=F_{1}-m_{1} a_{m 1}$ $F_{1}^{\prime}=\frac{m_{2} F_{1}+m_{1} F_{2}}{m_{1}+m_{2}}$ Net force of $m_{2}=F_{2}-m_{2} a_{m 2}$ ...
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Question: Consider the situation of the previous problem. Suppose each of the blocks is pulled by a constant force $F$ instead of any impulse. Find the maximum elongation that the spring will suffer and the distances moved by the two blocks in the process. Solution: If ${ }^{x_{1}}$ and ${ }^{x_{2}}$ is travelled by $m_{1}, m_{2}$ under $F$ Work done $=F\left(x_{1}+x_{2}\right)$ Use C.O.E.L $\Rightarrow \frac{1}{2} k\left(x_{1}+x_{2}\right)^{2}=F\left(x_{1}+x_{2}\right)$ $\Rightarrow x_{1}+x_{2}...
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Question: Two blocks of masses $m_{1}$ and $m_{2}$ are connected by a spring of spring constant $k$. The block of mass $m_{2}$ is given a sharp impulse so that it acquires a velocity $v_{0}$ towards right. Find (a) the velocity of the center of mass, (b) the maximum elongation that the spring will suffer. Solution: (a) $V_{C O M}=\frac{m_{2} \times V_{0}+m_{1}(0)}{m_{1}+m_{2}}=\frac{m_{2} V_{0}}{m_{1}+m_{2}}$ (b) Use C.O.E.L $\frac{1}{2} m_{2} V_{0}^{2}-\frac{1}{2}\left(m_{1}+m_{2}\right) V_{C O...
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Question: Two masses $m_{1}$ and $m_{2}$ are connected by a spring of spring constant $k$ and are placed on a frictionless horizontal surface. Initially the spring is stretched through a distance ${ }^{x_{0}}$ when the distance is released from rest. Find the distance moved by the two masses before they again come to rest. Solution: Let ${ }^{x_{1}}$ and ${ }^{x_{2}}$ be travelled by ${ }^{m_{1}}$ and ${ }^{m_{2}}$ $\therefore$ C.O.L. $M \Rightarrow m_{1} x_{1}=m_{1} x_{2}$ $\Rightarrow m_{1} x_...
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Question: A bullet of mass $20 \mathrm{~g}$ moving horizontally at a speed of $300 \mathrm{~m} / \mathrm{s}$ is fired into a wooden block of mass $500 \mathrm{~g}$ suspended by a long string. The bullet crosses the block and the block rises through a height of $20.0 \mathrm{~cm}$, find the speed of the bullet as it emerges from the block. Solution: Use C.O.L.M $300(0.02)=0.5 . V^{\prime}+0.02 V$ From C.O.E.L, $\frac{-1}{2} 0.5 V^{\prime^{2}}=-0.5 g(0.2)$ $V=\frac{300 \times 0.2-0.5 \times 2}{0.0...
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Question: A bullet of mass $25 \mathrm{~g}$ is fired horizontally into a ballistic pendulum of mass $5.0 \mathrm{~kg}$ and gets embedded in it. If the center of the pendulum rises by a distance of $10 \mathrm{~cm}$, find the speed of the bullet. Solution: Use C.O.L.M $0.025 \times u=(0.025+5) V$ $V=\frac{u \times 0.025}{5.025}=\frac{u}{201}$ From C.O.E.L (conservation of energy law) $\frac{-1}{2}(0.025+5) \cdot V^{2}=-(0.025+5) \times g \times h$ $\Rightarrow \frac{u^{2}}{(201)^{2}}=2$ $\Rightar...
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Question: Solve the previous problem if the coefficient of restitution is e. Use $\theta=45^{\circ}, e=\frac{9}{4}$ and $h=5 \mathrm{~m}$. Solution: $\theta=45, e=\frac{3}{4}$ and $h=5 \mathrm{~m}$ $v=\sqrt{2 g h}$ $=\sqrt{20 \times 5}$ $=10 \mathrm{~m} / \mathrm{s}$ $V_{y}=e v \sin 45^{\circ}$ $V_{y}=\frac{3}{4} \times 10 \times \frac{1}{\sqrt{2}}=\frac{15}{2 \sqrt{2}}$ $V_{x}=v \cos 45^{\circ}$ $=\frac{10}{\sqrt{2}}$ $V=\sqrt{V_{x}^{2}+V_{y}^{2}}=\sqrt{\left(\frac{15}{2 \sqrt{2}}\right)^{2}+\l...
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Question: A ball falls on the ground from a height of $2.0 \mathrm{~m}$ and rebounds up to a height of $1.5 \mathrm{~m}$. Find the coefficient of restitution. Solution: $v=\sqrt{2 g h}=\sqrt{2 \times 9.8 \times 2}$ $v_{1}+e v=0$ $v_{1}=-\sqrt{2 \times 9.8 \times 1.5}$ $e=\frac{-v_{1}}{v}$ $=\frac{+\sqrt{2 \times 9.8 \times 1.5}}{\sqrt{2 \times 9.8 \times 2}}$ $=\frac{\sqrt{3}}{2}$...
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