Find a unit vector in the direction of where P and Q
Question: Find a unit vector in the direction of where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively. Solution: Given coordinates are P(5, 0, 8) and Q(3, 3, 2)....
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Question: For the following matrices, verify that $A(B C)=(A B) C$ : $A=\left[\begin{array}{lll}1 2 5 \\ 0 1 3\end{array}\right], B=\left[\begin{array}{ccc}2 3 0 \\ 1 0 4 \\ 1 -1 2\end{array}\right]$ and $C=\left[\begin{array}{l}1 \\ 4 \\ 5\end{array}\right]$ Solution:...
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Question: If find the unit vector in the direction of $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-2 \hat{k}$ (i) $6 \vec{b}$ (ii) 90$2 \vec{a}-\vec{b}$ Solution:...
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Question: If $A=\left[\begin{array}{ccc}0 c -b \\ -c 0 a \\ b -a 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}a^{2} a b a c \\ a b b^{2} b c \\ a c b c c^{2}\end{array}\right]$, show that $A B$ is a zero matrix. Solution: Given : $\mathrm{A}=\left[\begin{array}{ccc}0 \mathrm{c} -\mathrm{b} \\ -\mathrm{c} 0 \mathrm{a} \\ \mathrm{b} -\mathrm{a} 0\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}\mathrm{a}^{2} \mathrm{ab} \mathrm{ac} \\ \mathrm{ab} \mathrm{b}^{2} \mathrm{bc} \\ \ma...
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Question: If $\mathrm{A}=\left[\begin{array}{ccc}2 -3 -5 \\ -1 4 5 \\ 1 -3 -4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}2 -2 -4 \\ -1 3 4 \\ 1 -2 -3\end{array}\right]$, shown that $\mathrm{AB}=\mathrm{A}$ and $\mathrm{BA}=\mathrm{B}$. Solution: Given : $\mathrm{A}=\left[\begin{array}{ccc}2 -3 -5 \\ -1 4 5 \\ 1 -3 -4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}2 -2 -4 \\ -1 3 4 \\ 1 -2 -3\end{array}\right]$, Matrix A is of order $3 \times 3$ and Matrix $B$ is of or...
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Question: Show that $\mathrm{AB}=\mathrm{BA}$ in each of the following cases: $A=\left[\begin{array}{rrr}1 3 -1 \\ 2 2 -1 \\ 3 0 -1\end{array}\right]$ and $B=\left[\begin{array}{rrr}-2 3 -1 \\ -1 2 -1 \\ -6 9 -4\end{array}\right]$ Solution: $\left[\begin{array}{lll}a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33}\end{array}\right] \times\left[\begin{array}{lll}b_{11} b_{12} b_{13} \\ b_{21} b_{22} b_{23} \\ b_{31} b_{32} b_{33}\end{array}\right]$ $=\left[\begin{array}{lll}a_{...
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Question: Find the unit vector in the direction of sum of vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2 \hat{j}+\hat{k}$ Solution: Given vectors are,...
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Question: A uniform chain of mass in and length I overhangs a table with its two third part on the table. Find the work to be done by a person to put the hanging part back on the table. Solution: $W=(m / l) d x g(x)$ Total work done is : $\mathrm{W}=1 / 3 \int_{0}(\mathrm{~m} / \mathrm{l}) \mathrm{dx} \mathrm{g}(\mathrm{x})$ $\mathrm{W}=\mathrm{mgl} / 18$...
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Question: A block weighing $10 \mathrm{~N}$ travels down a smooth curved track $A B$ joined to a rough horizontal surface (figure 8-E5). The rough surface has a friction coefficient of $0.20$ with the block. If the block starts slipping on the track from a point $1.0 \mathrm{~m}$ above the horizontal surface, how far will it move on the rough surface? Solution: Work done, $\mathrm{W}=\mathrm{mgh}=10 \times 1$ $W=10 \mathrm{~J}$ Frictional force, $\mathrm{F}=\mu \mathrm{R}=\mu \mathrm{mg}=0.2 \ti...
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Question: Figure (8-E4) shows a particle sliding on a frictionless track which terminates in a straight horizontal section. If the particle starts slipping from the point $A$, how far away from the track will the particle hit the ground? Solution: By conservation of energy of point $A$ and $B$ $\mathrm{mgH}=\frac{\frac{1}{2}}{2} \mathrm{~m} \mathrm{} \mathrm{v}^{2}+\mathrm{mgh}$ $g \times 1=\frac{1}{2} v^{2}+g \times 0.5$ $v=\sqrt{9.8}=0.3 \mathrm{~m} / \mathrm{s}$ After B, body obeys projectile...
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Question: In a children's park, there is a slide which has a total length of $10 \mathrm{~m}$ and a height of $8.0 \mathrm{~m}$ (figure 8E3). Vertical ladder are provided to reach the top. A boy weighing $200 \mathrm{~N}$ climbs up the ladder to the top of the slide and slides down to the ground. The average friction offered by the slide is three tenth of his weight. Find (a) the work done by the ladder on the boy as he goes up, (b) the work done by the slide on the boy as he comes down. Neglect...
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Question: A small block of mass $200 \mathrm{~g}$ is kept at the top of a frictionless incline which is $10 \mathrm{~m}$ long and $3.2 \mathrm{~m}$ high. How much work was required (a) to lift the block from the ground and put it at the top, (b) to slide the block up the incline? What will be the speed of the block when it reaches the ground, if (c) it falls off the incline and drops vertically on the ground (d) it slides down the incline? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solut...
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Question: A car weighing $1400 \mathrm{~kg}$ is moving at a speed of $54 \mathrm{~km} / \mathrm{h}$ up a hill when the motor stops. If it is just able to reach the destination which is at a height of $10 \mathrm{~m}$ above the point, calculate the work done against friction (negative of the work done by the friction). Solution: Work done, $\mathrm{W}=\left(0+\frac{1}{2} \mathrm{mv}^{2}\right)-\mathrm{mgh}$ $W=\left[^{\frac{1}{2}} \times 1400 \times 15^{2}\right]-[1400 \times 9.8 \times 10]$ $W=2...
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Question: A block of mass $100 \mathrm{~g}$ is moved with a speed of $5.0 \mathrm{~m} / \mathrm{s}$ at the highest point in a closed circular tube of radius $10 \mathrm{~cm}$ kept in a vertical plane. The cross-section of the tube is such that the block just fits in it. The block makes several oscillations inside the tube and finally stops at the lowest point. Find the work done by the tube on the block during the process. Solution: Work done by block $=$ T.E at $A-$ T.E. at $B$ $=\frac{1}{2} \m...
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Question: Consider the situation shown in figure (8-E2). The system is released from rest and the block of mass $1.0 \mathrm{~kg}$ is found to have a speed $0.3 \mathrm{~m} / \mathrm{s}$ after it has descended through a distance of $1 \mathrm{~m}$. Find the coefficient of kinetic friction between the block and the table. Solution: Work done = change in K.E. $-\mu R S+m g=\left[^{\frac{1}{2}} m_{1} v_{1}^{22} m_{2} v_{2}^{2}\right]-0$$-\mu \times 40 \times 2+1 \times 10=\frac{1}{2}\left(4 \times ...
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Question: The two blocks in an Atwood machine have masses $2.0 \mathrm{~kg}$ and $3.0 \mathrm{~kg}$. Find the work done by gravity during the fourth second after the svstem is released from rest. Solution: $T-3 g+3 a=01$ $T-2 g-2 a=02$ Solving equation 1 and 2 $\mathrm{a}=\frac{\stackrel{\boldsymbol{E}}{5}}{5 \mathrm{~m} / \mathrm{s}^{2}}$ Now, $S=\frac{a}{2}(2 m-1)$ $S=\frac{g / 5}{2}[2 \times 4-1]$ $\mathrm{h}=\mathrm{S}=6.86 \mathrm{~m}$ Mass $\mathrm{m}=\mathrm{m}_{1}-\mathrm{m}_{2}$ $\mathr...
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Question: The heavier block in an Atwood machine has a mass twice that of the lighter one. The tension in the string is $16.0 \mathrm{~N}$ when the system is set into motion. Find the decrease in the gravitational potential energy during the first second after the system is released from rest. Solution: Net force is $\mathrm{T}-2 \mathrm{mg}+2 \mathrm{ma}=0 \ldots \ldots 1$ $\mathrm{T}-\mathrm{mg}-\mathrm{ma}=0 \ldots \ldots 2$ from equation 1 and 2 $a=\frac{T}{4 m}=\frac{16}{4 m}$ $a=\bar{m} \m...
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Question: A block of mass $30.0 \mathrm{~kg}$ is being brought down by a chain. If the block acquires a speed of $40.0 \mathrm{~cm} / \mathrm{s}$ in dropping down $2.00 \mathrm{~m}$, find the work done by the chain during the process. Solution: $v^{2}-u^{2}=2 a s$ $a=\left(0.4^{2}-0\right) / 2 \times 2$ $a=0.04 \mathrm{~m} / \mathrm{s}^{2}$. Force $\mathrm{F}=\mathrm{ma}-\mathrm{mg}$ and Work done $W=F s \cos \theta=m(a-g) s \cos \theta$ $W=30(0.04-9.8) \times 2 \times 1$ $W=-585.5 \mathrm{~J} \...
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Question: A scooter company gives the following specifications about its product. Weight of the scooter $-95 \mathrm{~kg}$ Maximum speed - $60 \mathrm{~km} / \mathrm{h}$ Maximum engine power - $3.5 \mathrm{hp}$ Pick up time to get the maximum speed $-5 \mathrm{~s}$ Check the validity of these specifications. Solution: Max. acceleration, $a=\frac{v-u}{t}$ $a=(50 / 3-0) / 5=3.33 \mathrm{~m} / \mathrm{s}^{2}$ Force $\mathrm{F}=\mathrm{ma}=95 \times 3.33$ $F=316.6 \mathrm{~N}$ Velocity $\mathrm{v}=\...
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Question: In a factory it is desired to lift $2000 \mathrm{~kg}$ of metal through a distance of $12 \mathrm{~m}$ in 1 minute. Find the minimum horsepower of the engine to be used. Solution: $W=F d \cos \theta$ $W=m g \cos \theta=2000 \times 10 \times 12 \times 1$ $W=24 \times 10^{4} J$ $P=\stackrel{\frac{w}{t}}{P}=\left(24 \times 10^{4}\right) / 60=4000 \mathrm{~W}$ $P=\frac{W}{t}=\left(24 \times 10^{4}\right) / 60=4000 W$ Horse power $=\mathrm{P} / 746=4000 / 746=5.3 \mathrm{hp}$...
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Question: An unruly demonstrator lifts a stone of mass $200 \mathrm{~g}$ from the ground and throws it at his opponent. At the time of projection, the stone is $150 \mathrm{~cm}$ above the ground and has a speed of $3.00 \mathrm{~m} / \mathrm{s}$. Calculate the work done by the demonstrator during the process. If it takes one second for the demonstrator to lift the stone and throw, what horsepower does he use? Solution: Work done $\mathrm{W}=\mathrm{mgh}+^{\frac{1}{2}} \mathrm{mv}^{2}$ $W=0.2 \t...
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Question: The US athlete Florence Griffith-Joyner won the $100 \mathrm{~m}$ sprint gold medal at Seol Olympic 1988 setting a new Olympic record of $10.54 \mathrm{~s}$. Assume that she achieved her maximum speed in a very short-time and then ran the race with that speed till she crossed the line. Take her mass to be $50 \mathrm{~kg}$. (a) Calculate the kinetic energy of Griffith-Joyner at her full speed. (b) Assuming that the track, the wind etc. offered an average resistance of one tenth of her ...
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Question: The $200 \mathrm{~m}$ free style women's swimming gold medal at Seol Olympic 1988 went to Heike Friendrich of East Germany when she set a new Olympic record of 1 minute and $57.56$ seconds. Assume that she covered most of the distance with a uniform speed and had to exert $460 \mathrm{~W}$ to maintain her speed. Calculate the average force of resistance offered by the water during the swim. Solution: Power, $\mathrm{P}=\frac{\frac{1}{2}}{2} \mathrm{~V} / \mathrm{t}$ $W=P \times t=460 \...
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Question: A projectile is fired from the top of a 40 m high cliff with an initial speed of 50 m/s at an unknown angle. Find its speed when it hits the ground. Solution: By law of conservation of energy $\mathrm{mgh}+\frac{1}{2} \mathrm{~m} \mathrm{v}_{\mathrm{i}}^{2}=\frac{1}{2} \mathrm{mv}_{\mathrm{f}}$ $10 \times 40+\frac{1}{2} \times 50^{2}=\frac{1}{2} \times V_{f}^{2}$ $v=v_{f}=57.4 \mathrm{~m} / \mathrm{s}$ $v \approx 58 \mathrm{~m} / \mathrm{s}$...
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Question: A person is painting his house walls. He stands on a ladder with a bucket containing paint in one hand and a brush in other. Suddenly the bucket slips from his hand and falls down on the floor. If the bucket with the paint had a mass of $6.0 \mathrm{~kg}$ and was at a height of $2.0 \mathrm{~m}$ at the time it slipped, how much gravitational potential energy is lost together with the paint? Solution: Initial P.E. at heiaht $h$. $P . E_{i}=m g h=6 \times 9.8 \times 2=117.6 \mathrm{~J}$ ...
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