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Question: Two small bodies of masses $10 \mathrm{~kg}$ and $20 \mathrm{~kg}$ are kept a distance $1.0 \mathrm{~m}$ apart and released, Assuming that only mutual gravitational forces are acting; find the speeds of the particles when the separation decreases to $0.5 \mathrm{~m}$. Solution: $m_{1}=10 \mathrm{~kg}, \quad m_{2}=20 \mathrm{~kg}$ Initially, Finally, $\underset{\gamma_{f}=0.5 \mathrm{~m}}{\stackrel{10 \mathrm{~kg}}{\rightarrow} \mathrm{O}_{1} \quad V_{2} \text { - 20kf }}$ If both bodie...
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Question: Find the acceleration due to gravity of the moon at a point $1000 \mathrm{~km}$ above the moon's surface. The mass of the moon is $7.4 \times 10^{22} \mathrm{~kg}$ and its radius is $1740 \mathrm{~km}$. Solution: The acceleration due to gravity at a point at height ' $h$ ' from the surface of moon of radius $R$ is $g=\frac{\mathrm{GM}}{(\mathrm{R}+\mathrm{h})^{2}}$ $=\frac{6.67 \times 10^{-11} \times 7.4 \times 10^{22}}{(1740+1000)^{2} \times 10^{6}}$ $=0.65 \mathrm{~m} / \mathrm{s}^{2...
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Question: Four particles of equal masses $M$ move along a circle of radius $R$ under the action of their mutual gravitational attraction. Find the speed of each particle. Solution: Here, $\mathrm{r}_{13}=2 \mathrm{R}$ $r_{14}=2 R \cos 45=\sqrt{2} R$ $r_{12}=2 R \sin 45=\sqrt{2} R$ Now, force on particle 1 Due to particle 4: $\mathrm{F}_{14}=\frac{\mathrm{GMM}}{(\sqrt{2} \mathrm{R})^{2}}=\frac{\mathrm{GM}^{2}}{2 \mathrm{R}^{2}}$ Due to particle 2: $\mathrm{F}_{12}=\frac{\mathrm{GMM}}{(\sqrt{2} \m...
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Question: Three uniform spheres each having a mass $M$ and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two. Solution: Let us assume that the center of mass of spheres are located at point $P, Q$ and $R$. So, re-drawing the figure, it is an equilateral triangle. Force on mass $M$ at point $P$ $\mathrm{F}_{\mathrm{PR}}=\mathrm{F}_{\mathrm{PQ}} \frac{G M M}{(2 a)^{2}}=\frac{G M^{2}}{4 a^{2}}=\m...
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Question: Three equal masses $\mathrm{m}$ are placed at the three corners of an equilateral triangle of side a. Find the force exerted by this system on another particle of mass $m$ placed at (a) the mid-point of a side, (b) at the center of the triangle. Solution: Mass ' $m$ ' is at mid-point of side PR. (a) OP $=\mathrm{OR}=\mathrm{a} / 2$ $\overline{F_{o p}}=\frac{G \times m \times m}{\left(a_{2}\right)^{2}} \quad \mathrm{~K}=\frac{\mathrm{Gm}^{2}}{a^{2}}$ $\overline{F_{O P}}=\frac{4 \mathrm{...
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Question: Four particles having masses $\mathrm{m}, 2 \mathrm{~m}, 3 \mathrm{~m}$ and $4 \mathrm{~m}$ are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass $\mathrm{m}$ placed at the center. Solution: $r=a^{a} / \sqrt{2}$ Here $\overrightarrow{F_{1}}=\frac{G \cdot m \cdot m}{r^{2}}$ $=\frac{\text { G.mm }}{a^{2}}(2)=2 k$ $\mathrm{K}=\frac{G m^{2}}{a^{2}}$ $\overrightarrow{F_{2}}=\frac{G \cdot 2 m \cdot m}{r^{2}}=4 \mathrm{~K}$ $\overright...
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Question: Two spherical balls of mass $10 \mathrm{~kg}$ each are placed $10 \mathrm{~cm}$ apart. Find the gravitational force of attraction between them. Solution: $\mathrm{F}=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}$ $\mathrm{~F}=$ Gravitational force between two bodies $\mathrm{G}=$ Gravitational constant $\mathrm{m}_{1}, \mathrm{~m}_{2}=$ Masses of the two bodies $\mathrm{r}=$ Distance between the bodies Here, $\mathrm{m}_{1}=\mathrm{m}_{2}=10 \mathrm{~kg}$ $\mathrm{Nm} \mathrm{m}^{2} / \mathrm{k...
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Question: A solid sphere rolling on a rough horizontal surface with a linear speed v collides elastically with a fixed, smooth, vertical wall. Find the speed of the sphere after it has started pure rolling in the backward direction. Solution: $L_{i}=L_{f}$ $m v R-I \omega=m v^{\prime} R+I \omega^{\prime}$ $m v R-\left(\frac{2}{5} m R^{2}\right)\left(\frac{v}{R}\right)=m v^{\prime} R+\left(\frac{2}{5} m R^{2}\right)\left(\frac{v !}{R}\right)$ $v^{\prime}=\frac{3 v}{7}$...
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Question: A solid sphere is set into motion on a rough horizontal surface with a linear speed $v$ in the forward direction and an angular speed $v / R$ in the anticlockwise direction as shown in figure. Find the linear speed of the sphere (a) when it stops rotating and (b) when slipping finally ceases and pure rolling starts. Solution: Conserving angular momentum about bottom point $L_{i}=L_{f}$ $m v R-I \omega=m v^{\prime} R$ $m v R-\left(\frac{2}{5} m R^{2}\right)\left(\frac{v}{R}\right)=m v^{...
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Question: A solid sphere of mass $0.50 \mathrm{~kg}$ is kept on a horizontal surface. The coefficient of static friction between the surfaces in contact is $2 / 7$. What maximum force can be applied at the highest point in the horizontal direction so that the sphere does not slip on the surface? Solution: $F-f f=m a$ $\tau=I \alpha$ $(F R+f f R)=\left(\frac{2}{5} m R^{2}\right)\left(\frac{a}{R}\right)$ $F+f f=\frac{2}{5} m a$ $f f=\mu \mathrm{ii})$ Solving (i),(ii) and (iii) $\mathrm{F}=3.3 \mat...
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Question: A hollow sphere of radius $R$ lies on smooth horizontal surface. It is pulled by a horizontal force acting tangentially from the highest point. Find the distance travelled by the sphere during the time it makes one full rotation. Solution: $\tau=I \alpha$ $F R=\left(\frac{2}{3} M R^{2}\right)_{\alpha}$ $\alpha=\frac{3 F}{2 M R}$ Time to complete one rotation $\quad \theta=\omega_{0} t+\frac{1}{2} \alpha t^{2}$ By, $2 \pi=\frac{1}{2}\left(\frac{3 F}{2 M R}\right) t^{2}$...
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Question: A thin spherical shell lying on a rough horizontal surface is hit by cue in such a way that the line of action passes through the centre of the shell. As a result the shell starts moving with the linear speed v without any initial angular velocity. Find the linear speed of the shell after it starts pure rolling on the surface. Solution: By angular momentum conservation at bottom most point $L_{i}=L_{f}$ $m v R=m v^{\prime} R+I \omega^{\prime}$ $m v R=\left(\frac{2}{3} m R^{2}\right)\le...
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Question: A uniform wheel of radius $R$ is set into rotation about its axis at an angular speed $\omega$. This rotating wheel is now placed on a rough horizontal surface with its axis horizontal. Because of friction at the contact, the wheel accelerates forward and its rotation decelerates till the wheel starts pure rolling on the surface. Find the linear speed of the wheel after it starts pure rolling. Solution: By angular momentum conservation at bottom point since ${ }^{\tau_{e x t}}=0$ $L_{i...
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Question: A thin spherical shell of radius $R$ lying on a rough horizontal surface is hit sharply and horizontally by a cue. Where should it be hit so that the shell does not slip on the surface? Solution: $v_{C}=R \omega$ (Pure Rolling) By angular momentum conservation $m v_{C} h=I \omega$ $m v_{C} h=\left(\frac{2}{3} m R^{2}\right)\left(\frac{v}{R}\right)$ $h=\frac{2 R}{3}$...
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Question: Figure shows a small spherical ball of mass $m$ rolling down the loop track. The ball is released on the linear portion at a vertical height $\mathrm{H}$ from the lowest point. The circular part shown has a radius $\mathrm{R}$. (a) Find the kinetic energy of the ball when it is at a point $\mathrm{A}$ where the radius makes an angle $\theta$ with the horizontal. (b) Find the radial and the tangential accelerations of the centre when the ball is at $A$. (c) Find the normal force and the...
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Question: Figure shows a rough track, a portion of which is in the form of a cylinder of radius $\mathrm{R}$. With what minimum linear speed should a sphere of radius $r$ be set rolling on the horizontal part so that it completely goes round the circle on the cylindrical part. Solution: For minimum u, the ball will just be in contact with surface at top. So, $\mathrm{N}=0$ $m g=\frac{m v^{2}}{(R-r)}$ $v^{2}=g(R-r)$ Now, by energy conservation $\frac{1}{2} m u^{2}+\frac{1}{2} I \omega_{0}^{2}=\fr...
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Question: A solid sphere of mass $m$ is released from rest from the rim of a hemispherical cup so that it rolls along the surface. If the rim of the hemisphere is kept horizontal, find the normal force exerted by the cup on the ball when the ball reaches the bottom of the cup. Solution: By conservation of energy $m g R=\frac{1}{2} I \omega^{2}+\frac{1}{2} m v^{2}$ $m g R=\frac{1}{2}\left(\frac{2}{5} m R^{2}\right)\left(\frac{v^{2}}{R^{2}}\right)+\frac{1}{2} m v^{2}$ Normal contact at bottom $N-m...
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Question: A hollow sphere is released from the top of an inclined plane of inclination $\theta$. (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding? (b) Find the kinetic energy of the ball as it moves a length I on the incline if the friction coefficient is half the value calculated in part (a). Solution: (a) $m g \sin \theta-f f=m a$ $f f=\mu N=\mu m g \cos \theta$ $\tau=I \alpha$ -(ii) $f f=\left(\frac{-}{5} m R^{2}\right)\left(\frac{a}{R...
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Question: A sphere starts rolling down an incline of inclination $\theta$. Find the speed of its centre when it has covered a distance $\mathrm{I}$. Solution: By energy conservation, $m g(l \sin \theta)=\frac{1}{2} I \omega^{2}+\frac{1}{2} m v^{2}$ $m g l \sin \theta=\frac{1}{2}\left(\frac{2}{5} m R^{2}\right)\left(\frac{v^{2}}{R^{2}}\right)+\frac{1}{2} m v^{2}$ $v=\sqrt{\frac{10}{7} g l \sin \theta}$...
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Question: A small disc is set rolling with a speed $v$ on the horizontal part of the track of the previous problem from right to left. To what height will it climb up the curved part? Solution: By energy conservation, $\frac{1}{2} I \omega^{2}+\frac{1}{2} m v^{2}=m g h$ $\frac{1}{2} m v^{2}+\frac{1}{2}\left(\frac{m R^{2}}{2}\right)\left(\frac{v^{2}}{R^{2}}\right)=m g h$ $h=\frac{3 v^{2}}{4 g}$...
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Question: A small spherical ball is released from a point at a height $h$ on a rough track shown in figure. Assuming that it does not slip anywhere, find its linear speed when it rolls on the horizontal part of the track. Solution: By energy conservation, $m q h=\frac{1}{2} I \omega^{2}+\frac{1}{2} m v^{2}$ $\mathrm{mgh}=\frac{1}{2}\left(\frac{2}{5} m R^{2}\right)\left(\frac{v^{2}}{R^{2}}\right)+\frac{1}{2} m v^{2}$ $v=\sqrt{\frac{10 h}{7}}$...
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Question: If $x=a \cos \theta+b \sin \theta$ and $y=a \sin \theta-b \cos \theta$, show that $y^{2} \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+y=0$ Solution:...
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Question: If $x=a(\cos \theta+\theta \sin \theta)$ and $y=a(\sin \theta-\theta \cos \theta)$, show that $\frac{d^{2} y}{d x^{2}}=\frac{1}{a}\left(\frac{\sec ^{3} \theta}{\theta}\right)$ Solution:...
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Question: If $y=\log \left[x+\left[x+\sqrt{x^{2}+a^{2}}\right]\right.$, then prove that $\left(x^{2}+a^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=0$ Solution:...
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Question: If $y=\left\{x+\sqrt{x^{2}+1}\right\}^{m}$, then show that $\left(x^{2}+1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-m^{2} y=0$ Solution:...
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