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Question: Water falling from a $50 \mathrm{~m}$ high fall is to be used for generating electric energy. If $1.8 \times 10^{5} \mathrm{~kg}$ of water falls per hour and half the gravitational potential energy can be converted into electric energy, how many 100 W lamps can be lit. Solution: Potential Energy P.E. $=m g h=1.8 \times 10^{5} \times 9.8 \times 50$ P.E. $=882 \times 10^{5} \mathrm{~J} / \mathrm{hr}$ Electrical energy $=\frac{1}{2}$ P.E. $=441 \times 10^{5} \mathrm{~J} / \mathrm{hr}$ Pow...
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Question: A $250 \mathrm{~g}$ block slides on a rough horizontal table. Find the work done by the frictional force in bringing the block to rest if it is initially moving at a speed of $40 \mathrm{~cm} / \mathrm{s}$. If the friction coefficient between the table and the block is $0.1$, how far does the block move before coming to rest? Solution: We know, $\mathrm{ma}=\mu \mathrm{R}$ $a=\frac{(\mu \mathrm{mg})}{m}=\mu \mathrm{g}$ $a=0.1 \times 9.8=0.98 \mathrm{~m} / \mathrm{s}^{2}$ and $v^{2}-u^{...
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Question: A block of mass $2.0 \mathrm{~kg}$ is pushed down an inclined plane of inclination $37^{\circ}$ with a force of $20 \mathrm{~N}$ acting parallel to the incline. It is found that the block moves on the incline with an acceleration of $10 \mathrm{~m} / \mathrm{s}^{2}$. If the block started from rest, find the work done (a) by the applied force in the first second, (b) by the weight of the block in the first second and (c) by the frictional force acting on the block in the first second. T...
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Question: A block of mass $2.0 \mathrm{~kg}$ kept at rest on an inclined plane of inclination $37^{\circ}$ is pulled up the plane by applying a constant force of $20 \mathrm{~N}$ parallel to the incline. The force acts for one second. (a) Show that the work done by the applied force does not exceed $40 \mathrm{~J}$. (b) Find the work done by the force of gravity in that one second if the work done by the applied force is $40 \mathrm{~J}$. (c) Find the kinetic energy of the block at the instant t...
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Question: A particle of mass $m$ moves on a straight line with its velocity varying with the distance travelled according to the equation $v=a \sqrt{x}$, where a is a constant. Find the total work done by all the forces during a displacement from $x=0$ to $x=d$. Solution: $\mathrm{v}=\mathrm{a}^{\sqrt{x}}$ $v_{1}=0$ and $v_{2}=a \sqrt{d}$ $a^{\prime}=\frac{\left(v 2^{2}-v 1^{2}\right)}{2 d}=\frac{a^{2}}{2}$ Force $\mathrm{F}=\mathrm{mal}=\left(\mathrm{ma}^{2}\right) / 2$ Work done $\mathrm{W}=\m...
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Question: Find the average frictional force needed to stop a car weighing $500 \mathrm{~kg}$ in a distance of $25 \mathrm{~m}$ if the initial speed is $72 \mathrm{~km} / \mathrm{h}$. Solution: $v^{2}-u^{2}=-2 a s$ $a=\frac{\left(v^{2}-u^{2}\right)}{-2 s}$ $a=\frac{(0-400)}{-2} \times 25=8 \mathrm{~m} / \mathrm{s}^{2}$ frictional force $F=m a=500 \times 8$ $F=4000 \mathrm{~N}$...
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Question: A block of weight $100 \mathrm{~N}$ is slowly slid up on a smooth incline of inclination $37^{\circ}$ by a person. Calculate the work done by the person in moving the block through a distance of $2.0 \mathrm{~m}$, if the driving force is (a) parallel to the incline and (b) in the horizontal direction. Solution: Force $F=m g \sin \theta$ $F=100 \times \sin 37^{\circ}=60 \mathrm{~N}$ (a) Work done $\mathrm{W}=\mathrm{Fd} \cos \theta$ $W=60 \times 2 \times \cos 0=120 \mathrm{~J}$ (b) In t...
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Question: A box weighing $2000 \mathrm{~N}$ is to be slowly slid through $20 \mathrm{~m}$ on a straight track having friction coefficient $0.2$ with the box. (a) Find the work done by the person pulling the box with a chain at an angle 9 with the horizontal. (b) Find the work when the person has chosen a value of 0 which ensures him the minimum magnitude of the force. Solution: (a) $R+P \sin \theta=2000 \ldots \ldots .1$ $P \cos \theta-0.2 R=0$ Solving 1 and 2 equations we get, $P=\frac{400}{(\c...
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Question: A block of mass $m$ is kept over another block of mass $M$ and the system rests on a horizontal surface (figure 8-E1). A constant horizontal force $F$ acting on the lower block produces an acceleration $F / 2$ $(m+M)$ in the system, the two blocks always move together. (a) Find the coefficient of kinetic friction between the bigger block and the horizontal surface. (b) Find the frictional force acting on the smaller block. (c) Find the work done by the force of friction on the smaller ...
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Question: A block of mass $250 \mathrm{~g}$ slides down an incline of inclination $37^{\circ}$ with a uniform speed. Find the work done against the friction as the block slides through $1 \mathrm{~m}$. Solution: Force $=\mu \mathrm{R}$ and $\mathrm{F}=\mathrm{mg} \sin \theta$ $m g \sin \theta=\mu R$ Work done $W=\mu R \cos \theta$ $W=m g \sin \theta \times \cos 0 \times S$ $W=0.25 \times 9.8 \times 0.60 \times 1 \times 1$ $W=1.5 \mathrm{~J}$...
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Question: A force $F=a+b x$ acts on a particle in the $x$-direction, where $a$ and $b$ are constants. Find the work done by this force during a displacement from $x=0$ to $x=d$. Solution: Work done, $W={ }_{0}^{d d}$ $F d x(F=a+b x)$ $W=\left(a+\frac{1}{2} b d\right) d$...
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Question: A man moves on a straight horizontal road with a block of mass $2 \mathrm{~kg}$ in his hand. If he covers a distance of 40 in with an acceleration of $0.5 \mathrm{~m} / \mathrm{s}^{2}$, find the work done by the man on the block during the motion. Solution: $F=m a$ $W=F . d$ $W=1 \times 40$ $W=40 \mathrm{~J}$...
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Question: A particle moves from a point $r=(2 \mathrm{~m})^{\hat{\imath}}+(3 \mathrm{~m})^{\hat{\jmath}}$ to another point $r_{2}=(3 \mathrm{~m})^{\hat{\imath}}+(2 \mathrm{~m})^{\hat{\jmath}}$ during which a certain force $F=(5 \mathrm{~N})^{\hat{l}}+(5 \mathrm{~N})^{\hat{J}}$ acts on it. Find the work done by the force on the particle during the displacement. Solution: $r=r_{2}-r_{1}$ $=\left(3^{\hat{2}}+2^{\hat{j}}\right)-\left(2^{\hat{2}}+3^{\hat{j}}\right)$ $r=\hat{\imath}_{-} \hat{\jmath}$ ...
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Question: A constant force of $2.50 \mathrm{~N}$ accelerates a stationary particle of mass $15 \mathrm{~g}$ through a displacement of $2.50 \mathrm{~m}$. Find the work done and the average power delivered? Solution: $W=F \cdot d$ $W=F d \cos \theta$ $W=2.5 \times 2.5 \times \cos \theta$ $W=6.25 \mathrm{~J}$ Now, $W=\frac{1}{2} m v^{2}-\frac{1}{2} m v^{2}$ $\frac{1}{2} m v^{2}-0=6.25$ $v=28.8 \mathrm{~m} / \mathrm{s}$ And $F=m a$ $\mathrm{a}=\frac{F}{m}=2.5 / 0.015=166.66 \mathrm{~m} / \mathrm{s}...
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Question: A block of mass $5.0 \mathrm{~kg}$ slides down an incline of inclination $30^{\circ}$ and length $10 \mathrm{~m}$. Find the work done by the force of gravity. Solution: Work done, $\mathrm{W}=\mathrm{mgh}$ $=5 \times 9.8 \times\left(\sin 30^{\circ} \times 10\right)$ $W=245 \mathrm{~J}$...
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Question: A box is pushed through $4.0 \mathrm{~m}$ across a floor offering $100 \mathrm{~N}$ resistance. How much work is done by the resisting force? Solution: Resisting force $F_{R}=F . d$ $F_{R}=F d \cos \theta=100 \times 4$ $F_{R}=400 \mathrm{~J}$...
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Question: A block of mass $2.00 \mathrm{~kg}$ moving at a speed of $10.0 \mathrm{~m} / \mathrm{s}$ accelerates at $3.00 \mathrm{~m} / \mathrm{s}^{2}$ for $5.00 \mathrm{~s}$. Compute its final kinetic energy. Solution: Final speed $v=u+$ at $=10+3 \times 5$ $v=25 \mathrm{~m} / \mathrm{s}$ Now, $K . E .=\frac{1}{2} m v^{2}$ $=\frac{1}{2} \times 2 \times 25^{2}$ K.E. $=625 \mathrm{~J}$...
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Question: The mass of cyclist together with the bike is $90 \mathrm{~kg}$. Calculate the increase in kinetic energy if the speed increases from 6 ' $0 \mathrm{~km} / \mathrm{h}$ to $12 \mathrm{~km} / \mathrm{h}$. Solution: $K . E_{x}=\frac{1}{2} m v^{2}$ Change in K.E. $=\frac{1}{2} \mathrm{~m}\left[\mathrm{v}^{2}-\mathrm{u}^{2}\right]$ $=\frac{1}{2} \times 90 \times\left[12^{2}-6^{2}\right]$ $=375 \mathrm{~J}$...
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Question: A person stands on a spring balance at the equator. (a) By what fraction is the balance reading less than his true weight? (b) If the speed of earth's rotation is increased by such an amount that the balance reading is half the true weight, what will be the length of the day in this case? Solution: (a) At equator, normal contact force $\mathrm{N}+\mathrm{mR} \omega^{2}=\mathrm{mg}$ Reading $N=m g-m R \omega^{2}$ $t=\frac{m g-\left(m g-m R w^{2}\right)}{m g}$ $\begin{aligned} \frac{R w^...
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Question: A table with smooth horizontal surface is placed in a cabin which moves in a circle of a large radius $\mathrm{R}$ (figure 7-E5). A smooth pulley of small radius is fastened to the table. Two masses $m$ and $2 m$ placed on the table are connected through a string going over the pulley. Initially the masses are held by a person with the strings along the outward radius and then the system is released from rest (with respect to the cabin). Find the magnitude of the initial acceleration o...
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Question: A car moving at a speed of $36 \mathrm{~km} / \mathrm{hr}$ is taking a turn on a circular road of radius $50 \mathrm{~m}$. A small wooden plate is kept on the seat with its plane perpendicular to the radius of the circular road (figure 7-E4). A small block of mass $100 \mathrm{~g}$ is kept on the seat which rests against the plate. The friction coefficient between the block and the plate is $1.1=0.58$. (a) Find the normal contact force exerted by the plate on the block. (b) The plate i...
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Question: A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity $c o$ in a circular path of radius $R$ (figure 7-E3). A smooth groove $A B$ of length $L( R)$ is made on the surface of the table. The groove makes an angle 0 with the radius $\mathrm{OA}$ of the circle in which the cabin rotates. $A$ small particle is kept at the point $A$ in the groove and is released to move along $A B$. Find the time taken by the particle to reach the point $B$. ...
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Question: A block of mass $m$ moves on a horizontal circle against the wall of a cylindrical room of radius $R$. The floor of the room on which the block moves is smooth but the friction coefficient between the wall and the block is $g$. The block is given an initial speed $v_{0}$. As a function of the speed $v$ write (a) the normal force by the wall on the block, (b) the frictional force by the wall and (c) the tangential acceleration of the block. (d) Integrate the tangential acceleration ( $\...
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Question: What is the radius of curvature of the parabola traced out by the projectile in the previous problem at a point where the particle velocity makes an angle $0 / 2$ with the horizontal? Solution: Since acceleration in horizontal direction is zero $u \cos \theta=\operatorname{vcos}\left(\frac{\theta}{2}\right)$ $v=\frac{u \cos \theta}{\cos _{\frac{\theta}{2}}^{\theta}}$ $\mathrm{a}_{\text {radial }}=\frac{V^{2}}{R}$ $R=\frac{\frac{w^{2} \cos ^{2} \theta}{\cos ^{2} \theta / 2}}{g \cos \the...
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Question: A particle is projected with a speed $u$ at an angle 0 with the horizontal. Consider a small part of its path near the highest position and take it approximately to be a circular arc. What is the radius of this circle? This radius is called the radius of curvature of the curve at the point. Solution: At highest point $\mathrm{A}_{\text {radial }}=\frac{v 2}{R}$ $g=\frac{(u \cos \theta)^{2}}{R}$ $R=\frac{u^{2} \cos ^{2} \theta}{g}$...
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