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Question: A neutron initially at rest decays into a proton, an electron and an antineutrino. The ejected electron has a momentum of $1.4 \times 10^{-26} \mathrm{Kg}-\mathrm{m} / \mathrm{s}$ and the antineutrino $6.4 \times 10^{-27} \mathrm{Kg}-\mathrm{m} / \mathrm{s}$. Find the recoil speed of the proton (a) if the electron and the antineutrino are ejected along the same direction and (b) if they are ejected along perpendicular directions. Mass of the proton $=1.67 \times 10^{-27} \mathrm{Kg}$. ...
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Question: A man of mass $50 \mathrm{Kg}$ starts moving on the earth and acquires a speed of $1.8 \mathrm{~m} / \mathrm{s}$. With what speed does the earth recoil? Mass of earth $=6 \times 10^{24} \mathrm{Kg}$. Solution: Conservation of lin. Momentum (C.O.L.M) $0=M_{E} V_{E}+M_{M} V_{M}$ M-Man E-Earth $V_{E}=\frac{-M_{M} V_{M}}{M_{E}}=\frac{-50 \times 1.4 \times 10^{7}}{6 \times 10^{24}}$ $=-15 \times 10^{-24} \mathrm{~m} / \mathrm{s}$...
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Question: A uranium-238 nucleus, initially at rest, emits an alpha particle with a speed of $1.4 \times 10^{7} \mathrm{~m} / \mathrm{s}$. Calculate the recoil speed of the residual nucleus thorium-234. Assume that the mass of a nucleus is proportional to the mass number. Solution: $\Rightarrow 0=M_{X} V_{X}+M_{\alpha}\left(1.4 \times 10^{7}\right)$ $\Longrightarrow V_{X}=\frac{-M_{c}\left(1.4 \times 10^{7}\right)}{M_{X}}$ $=\frac{-4}{234}\left(1.4 \times 10^{7}\right)=2.4 \times 10^{5} \mathrm{~...
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Question: Find the ratio of the linear momenta of two particles of masses $1.0 \mathrm{Kg}$ and $4.0 \mathrm{Kg}$ if their kinetic energies are equal. Solution: $\frac{1}{2} m_{1} v_{1}^{2}=\frac{1}{2} m_{2} v_{2}^{2} \Rightarrow \frac{v_{2}}{v_{1}}=\sqrt{\frac{m_{1}}{m_{2}}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$ $p_{1} v_{1}=p_{2} v_{2} \Rightarrow \frac{p_{1}}{p_{2}}=\frac{v_{2}}{v_{1}}$ $\Rightarrow \frac{p_{1}}{p_{2}}=\frac{1}{2}$...
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Question: The balloon, the light rope and the monkey shown in figure are at rest in air. If the monkey reaches the top of the rope, by what distance does the balloon descend? Mass of the balloon=M, mass of the monkey $=m$ and length of the rope ascended by the monkey $=L$. Solution: Let balloon be at origin. $y_{C O M}=\frac{0 . M-L m}{m+M}=0$ Both monkey and balloon are at origin. $y_{C O M}^{\circ}=\frac{M(0)+m(0)}{M+m}=0$ Shift= $y_{\text {COM }}^{\prime}-y_{\text {COM }}$ $=\frac{-L m}{M+m}$...
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Question: Mr. Verma (50Kg) and Mr. Mathur $(60 \mathrm{Kg})$ are sitting at the two extremes of a $4 \mathrm{~m}$ long boat (40Kg) standing still in water. To discuss a mechanic problem, they come to the middle of the boat. Neglecting friction with water, how far does the boat move on the water during the process? Solution: V-Verma M-Mathur B-Boat $x_{C O M}=\frac{0(50)+2 \times 40+60 \times 4}{150}$ $=\frac{320}{150} \mathrm{~m}$ When they come to center $x_{\text {COM }}=\frac{2(50)+2(40)+60(2...
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Question: If $A$ is singular then $A(\operatorname{adj} A)=$ ? A. A unit matrix B.A null matrix C.A symmetric matrix D. None of these Solution:...
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Question: The matrix $A=\left(\begin{array}{ccc}2 -2 -4 \\ -1 3 4 \\ 1 -2 -3\end{array}\right)$ is A. Nonsingular B. Idempotent C. Nilpotent D. Orthogonal Solution: Here the diagonal value is $2+3-3=1$ So the given matrix is idempotent....
Read More →A smooth sphere of radius R is made to translate
Question: A smooth sphere of radius $\mathrm{R}$ is made to translate in a straight line with a constant acceleration a. A particle kept on the top of the sphere is released from there at zero velocity with respect to the sphere. Find the speed of the particle with respect to the sphere as a function of the angle $\theta$ it slides. Solution:...
Read More →A chain of length 1 and mass m lies on the surface of a smooth
Question: A chain of length 1 and mass $m$ lies on the surface of a smooth sphere of radius $R1$ with one end tied to the top of the sphere. (a) Find the gravitational potential energy of the chain with reference level at the centre of the sphere. (b) Suppose the chain is released and slides down the sphere. Find the kinetic energy of the chain, when it has slid through an angle 9 . (c) Find the tangential acceleration $\mathrm{dv} / \mathrm{dt}$ of the chain when the chain starts sliding down. ...
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Question: The matrix $A=\left(\begin{array}{cc}a b b^{2} \\ -a^{2} -a b\end{array}\right)$ is A. idempotent B. Orthogonal C. Nilpotent D. None of these Solution: Matrix $A$ is said to be nilpotent since there exist a positive integer $k=1$ such that $A k$ is zero matrix....
Read More →Figure (8-E17) shows a smooth track which consists of a straight inclined
Question: Figure (8-E17) shows a smooth track which consists of a straight inclined part of length 1 joining smoothly with the circular part. A particle of mass $\mathrm{m}$ is projected up the incline from its bottom. (a) Find the minimum projection-speed $v$, for which the particle reaches the top of the track. (b) Assuming that the projection-speed is $2 v_{0}$ and that the block does not lose contact with the track before reaching its top, find the force acting on it when it reaches the top....
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Question: If $A=\left(\begin{array}{cc}\cos \theta -\sin \theta \\ \sin \theta \cos \theta\end{array}\right)$ then $A^{-1}=?$ A. A B. $-\mathrm{A}$ C. Adj $A$ D. $-\operatorname{adj} \mathrm{A}$ Solution:...
Read More →A particle of mass m is kept on the top of a smooth sphere of radius R.
Question: A particle of mass $m$ is kept on the top of a smooth sphere of radius $R$. It is given a sharp impulse which imparts it a horizontal speed v. (a) Find the normal force between the sphere and the particle just after the impulse. (b) What should be the minimum value of $v$ for which the particle does not slip on the sphere? (c) Assuming the velocity $v$ to be half the minimum calculated in part, (b) Find the angle made by the radius through the particle with the vertical when it leaves ...
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Question: If $A=\left(\begin{array}{lll}1 \lambda 2 \\ 1 2 5 \\ 2 1 1\end{array}\right)$ is not invertible then $\lambda=$ ? A. 2 B. 1 C. $-1$ D. 0 Solution:...
Read More →A particle of mass m is kept on a fixed,
Question: A particle of mass $\mathrm{m}$ is kept on a fixed, smooth sphere of radius $\mathrm{R}$ at a position, where the radius through the particle makes an angle of $30^{\circ}$ with the vertical. The particle is released from this position. (a) What is the force exerted by the sphere on the particle just after the release? (b) Find the distance travelled by the particle before it leaves contact with the sphere. Solution:...
Read More →A particle slides on the surface of a fixed smooth sphere starting
Question: A particle slides on the surface of a fixed smooth sphere starting from the topmost point. Find the angle rotated by the radius through the particle, when it leaves contact with the sphere. Solution:...
Read More →If A is a square matrix such that
Question: If $A$ is a square matrix such that $|A| \neq 0$ and $A^{2}-A+2 \mid=0$ then $A^{-1}=$ ? A. (I-A) B. $(1+A)$ C. $\frac{1}{2}(\mathrm{I}-\mathrm{A})$ D. $\frac{1}{2}(I+A)$ Solution:...
Read More →A simple pendulum of length L having a bob of mass m
Question: A simple pendulum of length $L$ having a bob of mass $m$ is deflected from its rest position by an angle 9 and released (figure 8-E16). The string hits a peg which is fixed at a distance $x$ below the point of suspension and the bob starts going in a circle centred at the peg. (a) Assuming that initially the bob has a height less than the peg, show that the maximum height reached by the bob equals its initial height. (b) If the pendulum is released with $\theta=90^{\circ}$ and $x=L / 2...
Read More →If A and B are two nonzero square matrices
Question: If $A$ and $B$ are two nonzero square matrices of the same order such that $A B=0$ then A. $|A|=0$ or $|B|=0$ B. $|A|=0$ and $|B|=0$ C. $|A| \neq 0$ and $|B| \neq 0$ D.None of these Solution:...
Read More →If A and B are invertible matrices of the same order then
Question: If $A$ and $B$ are invertible matrices of the same order then $(A B)^{-1}=?$ A. $\left(A^{-1} \times B^{-1}\right)$ B. $\left(A \times B^{-1}\right)$ C. $\left(A^{-1} \times B\right)$ D. $\left(B^{-1} \times A^{-1}\right)$ Solution: $(A B)(A B)^{-1}=1$...
Read More →A heavy particle is suspended by a 1.5 m long string
Question: A heavy particle is suspended by a $1.5 \mathrm{~m}$ long string. It is given a horizontal velocity of $\sqrt{57} \mathrm{~m} / \mathrm{s}$. (a) Find the angle made by the string with the upward vertical, when it becomes slack. (b) Find the speed of the particle at this instant. (c) Find the maximum height reached by the particle over the point of suspension. Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →The bob of a stationary pendulum is given a sharp
Question: The bob of a stationary pendulum is given a sharp hit to impart it a horizontal speed of $\sqrt{3} g \mathrm{~g}$. Find the angle -rotated by the string before it becomes slack. Solution:...
Read More →If A is an invertible square matrix then
Question: If $A$ is an invertible square matrix then $\left|A^{-1}\right|=?$ A. $|\mathrm{A}|$ B. $\frac{1}{|\mathrm{~A}|}$ C. 1 D. 0 Solution:...
Read More →Figure (8-E15) shows a smooth track, a part of
Question: Figure (8-E15) shows a smooth track, a part of which is a circle of radius $\mathrm{R}$. A block of mass $m$ is pushed against a spring of spring constant $\mathrm{k}$ fixed at the left end and is then released. Find the initial compression of the spring so that the block presses the track with a force mg when it reaches the point $\mathrm{P}$, where the radius of the track is horizontal. Solution:...
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