Solve the following :
Question: A body of mass $m$ makes an elastic collision with another identical body at rest. Show that if the collision is not head-on, the bodies go right angle to each other after the collision. Solution: If $m$ is not colliding head-on, along $x$-axis, use C.O.L.M $m v \cos \theta=m\left(v_{1} \cos \alpha+v_{2} \cos \beta\right)-(1)$ Along y-axis, $-m v \sin \theta=m\left(v_{1} \sin \alpha-v_{2} \sin \beta\right)-(2)$ Use C.O.E.L $\frac{1}{2} m v^{2}=\frac{1}{2} m\left(v_{1}^{2}+v_{2}^{2}\rig...
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Question: A small block of superdense material has a mass of $3 \times 10^{24} \mathrm{~kg}$. It is situated at a height $h$ (much smaller than the earth's surface) from where it falls on the earth's surface. Find its speed when its height from the earth's surface has reduced to $\mathrm{h} / 2$. The mass of the earth is $3 \times 10^{24} \mathrm{~kg}$. Solution: b-block e-earth Use C.O.L.M $M_{e} V_{e}=m_{b} V_{b}$ $M_{e} V_{e}=m_{b} V_{b}$ $V_{e}=\frac{m_{b} V_{B}}{M}$ Use C.O.E.L $G M_{e} m_{...
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Question: Figure shows a small body of mass $\mathrm{m}$ placed over a larger mass $\mathrm{M}$ whose surface is horizontal near the smaller mass and gradually curves to become vertical. The smaller mass is pushed on the larger ones at a speed $v$ and the system is left to itself. Assume that all the surfaces are frictionless. (a) Find the speed of the larger block when the smaller block is sliding on the vertical part. (b) Find the speed of the smaller mass when it breaks off the larger mass at...
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Question: A block of mass $\mathrm{m}$ is placed on a triangular block of mass $M$, which in turn is placed on a horizontal surface as shown in figure. Assuming frictionless surfaces between the velocities of the triangular block when the smaller block reaches the bottom end. Solution: Along x-direction ${ }^{a_{C O M}}=0$ $\Rightarrow \frac{m a_{x}-(M+m)\left(a_{2}\right)}{(M+m+m)}=0$ $\Rightarrow a_{2}=\frac{m a_{x}}{m+M}$ $m a=m g \sin \alpha$ $a_{x}=a \cos \alpha=g \sin \alpha \cos \alpha$ $...
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Question: The friction coefficient between the horizontal surface and each of the blocks is shown in figure is $0.20$. The collision between the blocks is perfectly elastic. Find the separation between the two blocks when they come to rest. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: Using work energy principle $\mathrm{V}=$ velocity of $2 \mathrm{~kg}$ near collision $\mathrm{V}=$ velocity of $2 \mathrm{~kg}$ near collision $\frac{1}{2} m\left(V^{2}-1^{2}\right)=-m \times g \mu \times 1...
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Question: The blocks shown in figure have equal masses. The surface of $A$ is smooth but that of $B$ has a friction coefficient of $0.10$ with the floor. Block $A$ is moving at a speed of $10 \mathrm{~m} / \mathrm{s}$ towards $B$ which is kept at rest. Find the distance travelled by $B$ if (a) the collision is perfectly elastic and (b) the collision is perfectly inelastic. Take $g=10^{m} / \mathrm{s}^{2}$. Solution: (a) Use C.O.L.M $m_{A}(10)=m_{A} V_{1}+m_{B} V_{2} \quad\left\{m_{A}=m_{B}\right...
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Question: A uniform chain of mass $M$ and length $L$ is held vertically in such a way that its lower end just touches the horizontal floor. The chain is released from rest in this position. Any portion that strikes the floor comes to rest. Assuming that the chain does not form a heap on the floor, calculate the force exerted by it on the floor when a length $x$ has reached the floor. Solution: $\underset{M}{\text { Consider }} \lambda=$ Mass per unit length of chain $=\frac{M}{L}$ After chain's ...
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Question: Two balls having masses $\mathrm{m}$ and $2 \mathrm{~m}$ are fastened to two light strings of same length I. The other ends of the strings are fixed at $O$. The strings are kept in the same horizontal line and the system is released from rest. The collision between the balls is elastic. (a) Find the velocities of the balls just after their collision. (b) How high will the balls rise after the collision? Solution: (a) For initial velocities, Use C.O.E.L $\frac{1}{2} m V_{1}^{2}=m g l$ $...
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Question: A bullet of mass $10 \mathrm{~g}$ moving horizontally at a speed of $50 \sqrt{7} \mathrm{~m} / \mathrm{s}$ strikes a block of mass $490 \mathrm{~g}$ kept on a frictionless track as shown in figure. The bullet remains inside the block and the system proceeds towards the semicircular track of radius $0.2 \mathrm{~m}$. Where will the block strike the horizontal part after leaving the semicircular track? Solution: Use C.O.L.M, $\left(10 \times 10^{-3} \times 50 \sqrt{7}\right)+0=(490+10) \...
Read More →Find the angle of minimum deviation for an equilateral
Question: Find the angle of minimum deviation for an equilateral prism made of a material of refractive index $1.732$ What is the angle of incidence for this deviation? Solution:...
Read More →A container contains water up to a height
Question: A container contains water up to a height of $20 \mathrm{~cm}$ and there is a point source at the center of the bottom of the container. A rubber ring of radius $r$ floats centrally on the water surface. (a) Find the radius of the shadow of the ring formed on the ceiling if $r=15 \mathrm{~cm}$ (b) Find the maximum value of $r$ which the shadow of the ring is formed on the ceiling refractive index of water $=4 / 3$. Solution: $R=22.67 \mathrm{~cm}$...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left(\sqrt{\frac{1-x}{2}}\right)$ Solution: $A n s)=\frac{1}{2 \sqrt{1-x^{2}}}$...
Read More →A point source is placed at a depth
Question: A point source is placed at a depth $\mathrm{h}$ below the surface of water (refractive index= $\mu$ ). (a) Show that light escape through a circular area $n$ the water surface with its center directly above the point source. (b) Find the angle subtended by a radius of the area on the source. Solution:...
Read More →Light falls from glass (????=1.50) to air.
Question: Light falls from glass $(\mu=1.50)$ to air. Find the angle of incidence for which the angle of deviation of $90^{\circ}$. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left\{\sqrt{1-x^{2}}\right\}$ Solution:...
Read More →Light is incident from glass (????=1.50) to water (????=1.33)
Question: Light is incident from glass $(\mu=1.50)$ to water $(\mu=1.33)$ Find the range of the angle of deviation for which there are two angles of incidence. Solution:...
Read More →Light is incident from glass (????=1.50) to air.
Question: Light is incident from glass $(\mu=1.50)$ to air. Sketch the variation of the angle of deviation $\delta$ with the angle of incident i for $01$ $90^{\circ}$. Solution:...
Read More →Find the maximum angle of refractive index
Question: Find the maximum angle of refractive index when a light ray is refracted from glass $(\mu=1.50)$ to air. Solution: When light ray travels from glass to air then total internal reflection takes place. In this refracted angle is more than $90^{\circ}$ for reflection to occur. Thus, maximum angle is $90^{\circ}$....
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}(\cot x)+\cot ^{-1}(\tan x)$ Solution: $\Rightarrow \frac{d \pi}{d x}-\frac{d 2 x}{d x}$ Ans) $-2$...
Read More →A light ray is incident normally on the face AB
Question: A light ray is incident normally on the face AB of a right angled prism $\mathrm{ABC}(\mu=1.50)$ as shown in fig. What is the largest angle $\phi$ for which the light ray is totally reflected at the surface AC? Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cot ^{-1}(\operatorname{cosec} x+\cot x)$ Solution:...
Read More →An optical fibre (????=1.72) is surrounded by a glass
Question: An optical fibre $(\mu=1.72)$ is surrounded by a glass coating $(\mu=1.50)$ fing the critical angle for total internal reflection at the fiber glass interface. Solution:...
Read More →A light ray is incident at an angle of
Question: A light ray is incident at an angle of $45^{\circ}$ with the normal to $\mathrm{a} \sqrt{2} \mathrm{~cm}$ thick plate (refractive index $=2$ ). Find the shift in the path of the light as it emerges out from the plate. Solution: According to fig,...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\operatorname{cosec}^{-1}\left(\frac{1+\tan ^{2} x}{2 \tan x}\right)$ Solution:...
Read More →A cylindrical vessel, whose diameter and height both
Question: A cylindrical vessel, whose diameter and height both are equal to $30 \mathrm{~cm}$ is placed on a horizontal surface and a small particle in it at a distance of $5 \mathrm{~cm}$ from the center. An eye is placed at a position such that the edge of the bottom is just visible (fig) The particle P is int the plane of drawing. Up to what minimum height should water be poured in the vessel to make the particle P visible? Solution:...
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