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Question: Consider a gravity-free hall in which a tray of mass $M$, carrying a cubical block of ice mass $m$ and edge $L$, is at rest in the middle. If the ice melts, by what distance does the center of mass of "the tray plus the ice" system descend? Solution: As ice melts, COM would no shift, as there is no external force (not even gravity)....
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Question: Two blocks of masses $10 \mathrm{Kg}$ and $30 \mathrm{Kg}$ are placed along a vertical line. The first block is raised through a height of $7 \mathrm{~cm}$. By what distance should the second mass be moved to raise the center of mass by $1 \mathrm{~cm}$ ? Solution: $y_{\text {COM }}=\frac{30 y_{2}+10 y_{1}}{40}$ $y_{\text {COM }}^{\prime}=\frac{10\left(y_{1}+7\right)+30\left(y_{2}+\alpha\right)}{40}$ $1+y_{C O M}=y_{C O M}^{\prime}$ $\Rightarrow 40=70+30 \alpha$ $\Rightarrow \alpha=\fr...
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Question: Two blocks of masses $10 \mathrm{Kg}$ and $20 \mathrm{Kg}$ are placed on the X-axis. The first mass is moved on the axis by a distance of $2 \mathrm{~cm}$. By what distance should the second mass be moved to keep the position of the center of mass unchanged? Solution: Initially, let $10 \mathrm{Kg}$ be ${ }^{x_{1}} \mathrm{Cm}$ away $20 \mathrm{Kg} \mathrm{be}{ }^{x_{2}} \mathrm{~cm}$ away $\therefore x_{C O M}=\frac{10 x_{1}+20 x_{2}}{30}$ $\therefore x_{C O M}=\frac{10 x_{1}+20 x_{2}...
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Question: Calculate the velocity of the center of mass of the system of particles shown in figure. Solution: Velocity coordinates of $1 \mathrm{Kg} \rightarrow\left(-1.5 \cos 37^{\circ},-1.5 \sin 37^{\circ}\right)$ For another $1 \mathrm{Kg} \rightarrow\left(2 \cos 37^{\circ},-2 \sin 37^{\circ}\right)$ For $1.2 \mathrm{Kg} \rightarrow(0,0.4 \mathrm{~m} / \mathrm{s})$ For $1.5 \mathrm{Kg} \rightarrow\left(-1 \cos 37^{\circ}, 1 \sin 37^{\circ}\right)$ For $0.5 \mathrm{Ka} \rightarrow(3,0)$ $V_{x(\...
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Question: A square plate of edge $d$ and a circular disc of diameter $d$ are placed touching each other at the midpoint of an edge of the plate as shown in figure. Locate the center of mass of the combination, assuming same mass per unit area for the two plates. Solution: Area density of square $=\rho_{s}=\frac{m_{S}}{d^{2}}$ (Mass/unit area) Mass/unit area of circle $=\rho_{C}=\frac{4 m_{C}}{\pi d^{2}}$ $\rho_{s}=\rho_{C} \Rightarrow \frac{m_{s}}{d^{2}}=\frac{4 m_{C}}{\pi d^{2}}$ $\Rightarrow m...
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Question: A disc of radius $R$ is cut out from a larger disc of radius $2 R$ in such a way that the edge of the hole touches the edge of the disc. Locate the center of mass of the residual disc. Solution: C.O.M of $2 \mathrm{R}$ disc $=(2 \mathrm{R}, 0)$ C.O.M of R disc $=(R, 0)$ $=\frac{m_{n} R_{n} R-m_{R} R_{R}}{m_{n} R-m_{R}}$ $=\frac{\rho 4 \pi l R^{2}(8 R-R)}{\rho 4 \pi l R^{2}(4-1)}=\frac{7 R}{3}$ From C.O.M of $2 R$ disc, it is $\frac{7 R}{3}-2 R=\frac{R}{3}$ distance away....
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Question: A uniform disc of radius $\mathrm{R}$ is put over another uniform disc of radius $2 \mathrm{R}$ of the same thickness and density. The peripheries of the two discs touch each other. Locate the center of mass of the system. Solution: C.O.M of $2 \mathrm{R}$ disc $=(2 \mathrm{R}, 0)$ C.O.M of $R$ disc $=(R, 0)$ $=\frac{m_{2 R^{x}}{ }^{x}+m_{R^{x} R}}{m_{n R}+m_{R}}$ $\left\{m_{2 R}=\rho .4 \pi(2 R)^{2}\right\}$ $\left\{m_{R}=\rho .4 \pi R^{2} l\right\}$ $=\frac{\left(\rho 4 \pi l R^{2}\r...
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Question: Seven homogeneous bricks, each of length $L$, are arranged in shown in figure. Each brick is displaced with respect to the one in contact by $L / 10$. Find the $x$-coordinate of the center of mass relative to the origin shown. Solution: take $O$ as $(0,0)$ brick $A$ and $E$ is from $\left(\frac{L}{10}\right.$ to $\left.L+\frac{L}{10}\right)$ brick $B$ and $D$ is from $\left(\frac{2 L}{10}\right.$ to $\left.L+\frac{2 L}{10}\right)$ brick $C$ is from $\left(\frac{3 L}{10}\right.$ to $\le...
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Question: The structure of the water molecule in figure. Find the distance of the center of mass of the molecule from the center of the oxygen atom. Solution: $\mathrm{O}^{H_{1}}=\alpha=O H_{2}=0.92 \mathrm{~A}^{\circ}$ Take $O^{\prime}$ as oriqin. then $00^{\prime}=\alpha \cos 52^{\circ}$ $H_{1}=\left(-\alpha \sin 52^{\circ}, 0\right)$ $H_{2}=\left(\alpha \sin 52^{\circ}, 0\right)$ $0=\left(0, \alpha \cos 52^{\circ}\right)=0.59 A^{\circ}$ $x_{C O M}=\frac{m_{H_{1}} x_{H_{1}}+m_{H_{2}} x_{H_{2}}...
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Question: Three particles of masses $1.0 \mathrm{~kg}, 2.0 \mathrm{~kg}$ and $3.0 \mathrm{~kg}$ are placed at the corners $A, B$ and $C$ respectively of an equilateral triangle $A B C$ of edge $1 \mathrm{~m}$. Locate the center of mass of the system. Solution: Take $A$ as origin $(0,0)$ then $C=(1,0)$ $B=\left(\frac{1}{2}, \sqrt{1-\left(\frac{1}{2}\right)^{2}}\right)=\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ $x_{C O M}=\frac{x_{A} m_{A}+m_{B} x_{B}+m_{C} x_{C}}{m_{A}+m_{B}+m_{C}}=\frac{1 .(0...
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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: $m^{\frac{d v}{d t}}=m a \cos \theta+m g \sin \theta$ $V d v=a R \cos \theta d \theta+g R \sin \theta d \Theta[v=R d \Theta / d t]$ Integ $V=[2 R(a \sin \t...
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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 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 Solution: (a) G.P.E $=d m g R \cos \theta$ $=(\mathrm{mgR} \cos \theta \...
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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 $m$ 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 $2 \mathrm{v}_{0}$ 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: A particle of mass $m$ is kept on the top of a smooth sphere of radius $\mathrm{R}$. 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 i...
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Question: A particle of mass $m$ is kept on a fixed, smooth sphere of radius $R$ 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: (a) Net force $=\operatorname{mg} \cos \theta=\sqrt{3} / 2 \mathrm{mg}$ (b) $\mathrm{N}...
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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: From figure, $\left(m v^{2}\right) / R=m g \cos \theta$ $v=(\sqrt{g} R \cos \theta) \ldots \ldots 1$ and Change in K.E. = Work done $\frac{1}{2} m v^{2}-0=m g(R-R \cos \theta)$ $v=(\sqrt{2} g R[1-\cos \theta]) \ldots . .2$ from 1 and 2 $g R \cos \theta=2 g R(1-\cos \theta)$ $\theta=\cos ^{...
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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...
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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: (a) $m g \cos \theta=\left(m v^{2}\right) / l$ $v=(\sqrt{g l} \cos ...
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Question: The bob of a stationary pendulum is given a sharp hit to impart it a horizontal speed of $\sqrt{3} \mathrm{gl}$. Find the angle -rotated by the string before it becomes slack. Solution: $\frac{1}{2} \mathrm{~m} \mathrm{v}^{2}-\frac{1}{2} \mathrm{mu}^{2}=-\mathrm{mgh}$ $v^{2}=3 g l-2 g l(1+\cos \theta) \ldots \ldots \ldots 1$ and $\left(m v^{2}\right) / l=m g \cos \theta$ $\mathrm{v}^{2}=\mathrm{gl} \cos \theta \ldots \ldots \ldots 2$ from equation 1 and 2 $3 g l-2 g l-2 g l \cos \theta...
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Question: Figure (8-E15) shows a smooth track, a part of which is a circle of radius $\mathrm{R}$. A block of mass $n$ pushed against a spring of spring constant $k$ fixed at the left end and is then released. Find the in 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: According to figure, $\left(m v^{2}\right) / R=m g$ $\mathrm{v}=\sqrt{g R}$ Energy is same at $A$ and $P$...
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Question: A simple pendulum consists of a $50 \mathrm{~cm}$ long string connected to a $100 \mathrm{~g}$ ball. The ball is pulled aside so that the string makes an angle of $37^{\circ}$ with the vertical and is then released. Find the tension in the string when the bob is at its lowest position. Solution: $\cos \theta=\frac{A C}{A B}$ and $\mathrm{AC}=\mathrm{AB} \cos \theta$ $A C=0.4$ and $C D=A D-A C$ $C D=0.1 \mathrm{~m}$ Energy is same at $B$ and $D$ $\frac{1}{2} m v^{2}=m g h$ $\frac{\frac{...
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Question: The bob of a pendulum at rest is given a sharp hit to impart a horizontal velocity . $V 10 \mathrm{gl}$, where 1 is the length of the pendulum. Find the tension in the string when (a) the string is horizontal, (b) the bob is at its highest point and (c) the string makes an angle of $60^{\circ}$ with the upward vertical. Solution: (a) By law of conservation of energy, $\frac{1}{2} \mathrm{mv}_{1}^{2}=\frac{1}{2} \mathrm{mv} \mathrm{v}_{2}^{2}+\mathrm{mgl}$ $\frac{1}{2} \times m \times(\...
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Question: Figure (8-E14) shows a light rod of length 1 rigidly attached to a small heavy block at one end and a hook at the other end. The system is released from rest with the rod in a horizontal position. There is a fixed smooth ring at a depth $\mathrm{h}$ below the initial position of the hook and the hook gets into the ring as it reaches there. What should be the minimum value of $h$ so that the block moves in a complete circle about the ring? Solution: $v_{\min }=\sqrt{2 g l}$ Total energy...
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Question: One end of a spring of natural length $\mathrm{h}$ and spring constant $\mathrm{k}$ is fixed at the ground and the other is fitted with a smooth ring of mass $m$ which is allowed to slide on a horizontal rod fixed at a height $\mathrm{h}$ (figure 8-E13). Initially, the spring makes an angle of $37^{\circ}$ with the vertical when the system is released from rest. Find the speed of the ring when the spring becomes vertical. Solution: $\cos \theta=\cos 37^{\circ}=\mathrm{BC} / \mathrm{AC}...
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Question: Figure (8-E12) shows two blocks $A$ and $B$, each having a mass of $320 \mathrm{~g}$ connected by a light string passing over a smooth light pulley. The horizontal surface on which the block $A$ can slide is smooth. The block $A$ is attached to a spring of spring constant $40 \mathrm{~N} / \mathrm{m}$ whose other end is fixed to a support 40 $\mathrm{cm}$ above the horizontal surface. Initially, the spring is vertical and upstretched when the system is released to move. Find the veloci...
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