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Question: A hemispherical bowl of radius $R$ is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle $\theta$ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is $u$. Find the range of the angular speed for which the block will not slip. Solution: Particle will have tendency to move up. So, frictional force is in downward di...
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Question: In a children's park a heavy rod is pivoted at the centre and is made to rotate about the pivot so that the rod always remains horizontal. Two kids hold the rod near the ends and thus rotate with the rod (figure 7-E2). Let the mass of each kid be $15 \mathrm{~kg}$, the distance between the points of the rod where the two kids hold it be $3.0 \mathrm{~m}$ and suppose that the rod rotates at the rate of 20 revolutions per minute. Find the force of friction exerted by the rod on one of th...
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Question: A track consists of two circular parts $A B C$ and CDE of equal radius $100 \mathrm{~m}$ and joined smoothly as shown in figure (7-E1). Each part subtends a right angle at its centre. A cycle weighing $100 \mathrm{~kg}$ together with the rider travels at a constant speed of $18 \mathrm{~km} / \mathrm{h}$ on the track. (a) Find the normal contact force by the road on the cycle when it is at $B$ and at $D$. (b) Find the force of friction exerted by the track on the tyres when the cycle i...
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Question: A block of mass $m$ is kept on a horizontal ruler. The friction coefficient between the ruler and the block is $\mathrm{g}$. The ruler is fixed at one end and the block is at a distance $\mathrm{L}$ from the fixed end. The ruler is rotated about the fixed end in the horizontal plane through the fixed end. (a) What can the maximum angular speed be for which the block does not slip? (b) If the angular speed of the ruler is uniformly increased from zero at an angular acceleration a, at wh...
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Question: A car goes on a horizontal circular road of radius $\mathrm{R}$, the speed increasing at a constant rate $\mathrm{dv} / \mathrm{dt}=$ a. The friction dt coefficient between the road and the tyre is $\mu$. Find the speed at which the car will skid. Solution: From free body diagram $\mathrm{ff}=\sqrt{\left(F_{t}^{2}+F_{c}^{2}\right)}$ $\mu \mathrm{N}=\sqrt{(m a)^{2}+\left(\frac{m v^{2}}{R}\right)^{2}}$ $(\mu \mathrm{mg})=\sqrt{(m a)^{2}+\left(\frac{m v^{2}}{R}\right)^{2}}$ Squaring and s...
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Question: A motorcycle has to move with a constant speed on an overbridge which is in the form of a circular arc of radius $R$ and has a total length $L$. Suppose the motorcycle starts from the highest point. (a) What can its maximum velocity be for which the contact with the road is not broken at the highest point? (b) If the motorcycle goes at speed 1/42 times the maximum found in part (a), where will it lose the contact with the road? (c) What maximum uniform speed can it maintain on the brid...
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Question: A turn of radius $20 \mathrm{~m}$ is banked for the vehicles going at a speed of $36 \mathrm{~km} / \mathrm{h}$. If the coefficient of static friction between the road and the tyre is $0.4$, what are the possible speeds of a vehicle so that it neither slips down nor skids up? Solution: For banking angle, $\tan \theta=^{\frac{V^{2}}{R g}}$ $=\frac{\left(36 \times \frac{5}{48}\right)^{2}}{(20)(10)}$ $\tan \theta=0.5$ $V \max =\sqrt{\operatorname{Rg} \frac{(\mu+\tan \theta)}{(1-\mu \tan \...
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Question: Suppose the amplitude of a simple pendulum having a bob of mass $m$ is $\theta_{0}$. Find the tension in the string when the bob is at its extreme position. Solution: At extreme position, $\mathrm{T}=\operatorname{mgcos} \theta_{0}$ (No centrifugal force as $v=0$ at extreme position.)...
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Question: Suppose the bob of the previous problem has a speed of $1.4 \mathrm{~m} / \mathrm{s}$ when the string makes an angle of $0.20$ radian with the vertical. Find the tension at this instant. You can use $\cos \theta=1-\theta^{2} / 2$ and $\sin \theta=$ 0 for small $\theta$. Solution: At angle $\theta$ $T=m g \cos \theta+\frac{m v^{2}}{R}$ $\mathrm{T}=\operatorname{mg}\left(1-\frac{\theta^{2}}{2}\right)+\frac{m v^{2}}{R}$ $T=(0.1)(10)\left(1-\frac{(0.2)^{2}}{2}\right)+\frac{(0.1)(1.4)^{2}}{...
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Question: The bob of a simple pendulum of length $1 \mathrm{~m}$ has mass $100 \mathrm{~g}$ and a speed of $1.4 \mathrm{~m} / \mathrm{s}$ at the lowest point in its path. Find the tension in the string at this instant. Solution: At lowest point $T=m g+\frac{m v^{2}}{R}$ $=(0.1)(10)+\frac{(0.1)(1.4)^{2}}{(1)}$ $\mathrm{T}=1.2 \mathrm{~N}$...
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Question: A simple pendulum is suspended from the ceiling of a car taking a turn of radius $10 \mathrm{~m}$ at a speed of $36 \mathrm{~km} / \mathrm{h}$. Find the angle made by the string of the pendulum with the vertical if this angle does not change during the turn. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: $\mathrm{T} \sin \theta=\frac{m v^{2}}{R}$ $T \cos \theta=m g$ $\tan \theta=\frac{v 2}{R g}$ $\tan \theta=\frac{\left(36 \times \frac{5}{38}\right)^{2}}{10 \times 10}$ $\theta=45^...
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Question: A mosquito is sitting on an L.P. record disc rotating on a turn table at 333 per minute. The distance 3 of the mosquito from thecentre of the turn table is $10 \mathrm{~cm}$. Show that the friction coefficient between the record and the mosquito is greater than $\mathrm{It} \pi^{2} / 81$. Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: $\omega=33^{\frac{1}{3}} \mathrm{rpm}=\frac{100}{3} \mathrm{rpm}$ $=\left(\frac{100}{3}\right) \times \frac{(2 \pi)}{60}$ $=\frac{10 \pi}{9...
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Question: A ceiling fan has a diameter (of the circle through the outer edges of the three blades) of $120 \mathrm{~cm}$ and rpm 1500 at full speed. Consider a particle of mass $1 \mathrm{~g}$ sticking at the outer end of a blade. How much force does it experience when the fan runs at full speed? Who experts this force on the particle? How much force does the particle exert on the blade along its surface? Solution: Radius of the circle $=\frac{120}{2} \mathrm{~cm}=60 \mathrm{~cm}$ Angular speed=...
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Question: A stone is fastened to one end of a string and is whirled in a vertical circle of radius $\mathrm{R}$. Find the minimum speed the stone can have at the highest point of the circle. Solution: At highest point $\mathrm{T}+\mathrm{mg}=\frac{m v^{2}}{R}$ For minimum speed, $\mathrm{T}=0$ $m g=\frac{m v^{2}}{R}$ $V=\sqrt{\mathrm{Rg}}$...
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Question: In the Bohr model of hydrogen atom, the electron is treated as a particle going in a circle with the centre at the proton. The proton itself is assumed to be fixed in an inertial frame. The centripetal force is provided by the Coulomb attraction. In the ground state, the electron goes round the proton in a circle of radius $5.3 \times 10^{-11} \mathrm{~m}$. Find the speed of the electron in the ground state. Mass of the electron $=9.1 \times 10^{-31} \mathrm{~kg}$ and charge of the ele...
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Question: A circular road of radius $50 \mathrm{~m}$ has the angle of banking equal to $30^{\circ}$. At what speed should a vehicle go on this road so that the friction is not used? Solution: $\tan \theta=\frac{v^{2}}{R g}$ $\tan 30^{\circ}=\frac{v^{2}}{(50)(10)}$ $\mathrm{v} \cong 17 \mathrm{~m} / \mathrm{s}$...
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Question: If the road of the previous problem is horizontal (no banking), what should be the minimum friction coefficient so that a scooter going at $18 \mathrm{~km} / \mathrm{hr}$. does not skid? Solution: Centrifugal force=Frictional force $\frac{m v^{2}}{R}=\mu \mathrm{N}$ $\frac{m v^{2}}{R}=\mu \mathrm{mg}$ $\mu=\frac{v^{2}}{R g}=\frac{\left(18 \times \frac{5}{48}\right)^{2}}{10 \times 10}$. $\mu=0.25$...
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Question: If the horizontal force needed for the turn in the previous problem is to be supplied by the normal force by the road, what should be the proper angle of banking? Solution: Let banking angle be $\theta$ $\tan \theta=\frac{v^{2}}{R g}$ $\tan \theta=\frac{\left(36 \times \frac{5}{18}\right)^{2}}{30 \times 10}=\frac{1}{3}$ $\theta=\tan -1\left(\frac{1}{3}\right)$...
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Question: A particle moves in a circle of radius $1.0 \mathrm{~cm}$ at a speed given by $\mathrm{v}=2.0 \mathrm{t}$ where $\mathrm{v}$ is in $\mathrm{cm} / \mathrm{s}$ and $\mathrm{t}$ in seconds. (a) Find the radial acceleration of the particle at $t=1 \mathrm{~s}$. (b) Find the tangential acceleration at $t=1 \mathrm{~s}$. (c) Find the magnitude of the acceleration at $t=1 \mathrm{~s}$. Solution: (a) Velocity of particle at $t=1 \mathrm{sec}$ $V=2 t$ $\mathrm{V}=2(1)$ $\mathrm{V}=2 \mathrm{~cm...
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Question: Find the acceleration of a particle placed on the surface of the earth at the equator due to earth's rotation. The diameter of earth $=12800 \mathrm{~km}$ and it takes 24 hours for the earth to complete one revolution about its axis. Solution: Speed of particle at equator=distance/time $=\frac{2 \pi R}{T}$ $=\frac{2 \times 3.14 \times 6400 \times 10^{3}}{24 \times 60 \times 60}$ $=465.1 \mathrm{~m} / \mathrm{s}$ Acceleration of particle $=$ $\mathrm{A}_{\Gamma}=\frac{v^{2}}{R}$ $A_{r}=...
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Question: Find the acceleration of the moon with respect to the earth from the following data: Distance between the earth and the moon $=3.85 \times 10^{5} \mathrm{~km}$ and the time taken by the moon to complete one revolution around the earth $=27.3$ days. Solution: Speed of the moon=distance/time $=\frac{2 \pi T}{T}$ $=\frac{2 \times 3.14 \times\left(3.85 \times 10^{8}\right)}{27.3 \times 86400}$ $=1025.4 \mathrm{~m} / \mathrm{s}$ Acceleration of moon $=$ $\mathrm{A}_{\mathrm{r}}=\frac{v^{2}}...
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Question: Figure shows a small block of mass $\mathrm{m}$ kept at the left end of a larger block of mass Mand length I. The system can slide on a horizontal road. The system is started towards right with an initial velocity v. The friction coefficient between the road and the bigger block is $\mu$ and that between the block is $\mu / 2$. Find the time elapsed before the smaller blocks separates from the bigger block. Solution: Initial velocity of both blocks is same. So, $U_{r e l}=0$ When $\mat...
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Question: A person $(40 \mathrm{~kg})$ is managing to be at rest between two vertical walls by pressing one wall A by his hands and feet and the other wall Bby his back. Assume that the friction coefficient between his body and the walls is $0.8$ and that limiting friction acts at all the contacts. (a) Show that the person pushes the two walls with equal force. (b) Find the normal force exerted by either wall on the person. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: (a) Since the man is...
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Question: A block of mass $2 \mathrm{~kg}$ is pushed against a rough vertical wall with a force of $40 \mathrm{~N}$, coefficient of static friction being $0.5$. Another horizontal force of $15 \mathrm{~N}$, is applied on the block in a direction parallel to the wall. Will the block move? If yes, in which direction? If no, find the frictional force exerted by the wall on the block. Solution: Net driving force on the block $F_{\text {driving }}=\sqrt{(m g)^{2}+(15)^{2}}$ Limiting friction force $=...
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Question: Find the acceleration of the block of mass $M$ in the situation of figure. The coefficient of friction between the two blocks is $\mu 1$, and that between the bigger block and the ground is $\mu 2$. Solution: When a block M, moves with acceleration a towards right block $\mathrm{m}$ moves downwards and rightwards with acceleration $2 a$ and a respectively. $N^{\prime}=M g+\mu_{1} N+T$ (Vertical) -(i) $(T+T)-N-\mu_{2} N^{\prime}=M a$ (Horizontal) -(ii) FBD of mass $\mathrm{m}$ N=ma (Hor...
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