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Question: A small heavy block is attached to the lower end of a light rod of length I which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle? Solution: Minimum velocity at $A=v$ Minimum velocity at $B=0$ By law of conservation of energy ${ }^{\frac{1}{2}} m v^{2}=m g(2 l)$ $v=2 \sqrt{g l}$...
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Question: A small block of mass $100 \mathrm{~g}$ is pressed against a horizontal spring fixed at one end to compress the spring through $5.0 \mathrm{~cm}$ (figure 8-E11). The spring constant is $100 \mathrm{~N} / \mathrm{m}$. When released, the block moves horizontally till it leaves the spring. Where will it hit the ground $2 \mathrm{~m}$ below the spring? Solution: By law of conservation of energy: $\frac{1}{2} m v^{2}=\frac{1}{2} k x^{2}$ $v=1.58 \mathrm{~m} / \mathrm{s}^{2}$ Now, projectile...
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Question: A block of mass $m$ sliding on a smooth horizontal surface with a velocity $v$ meets a long horizontal spring fixed at one end and having spring constant $k$ as shown in figure (8-E10). Find the maximum compression of the spring. Will the velocity of the block be the same as when it comes back to the original position shown? Solution: (a) By law of conservation of energy: $\frac{\frac{1}{2}}{2} \mathrm{~m} \mathrm{v}^{2}=\frac{1}{2} \mathrm{kx} \mathrm{x}^{2}$ $\operatorname{Max} x=v \...
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Question: A block of mass $\mathrm{m}$. is attached to two upstretched springs of spring constants $\mathrm{k}$, and $\mathrm{k} 2$ as shown in figure (8-E9). The block is displaced towards right through a distance $x$ and is released. Find the speed of the block as it passes through the mean position shown. Solution: By law of conservation of energy $\frac{1}{2} \mathrm{~m} \mathrm{v}^{2}=\frac{\overline{1}}{2} \mathrm{k}_{1} \mathrm{x}^{2}+\frac{\overline{1}}{2} \mathrm{k}_{2} \mathrm{x}^{2}$ ...
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Question: Consider the situation shown in figure (8-E8). Initially the spring is unstretched when the system is released from rest. Assuming no friction in the pulley, find the maximum elongation of the spring. Solution: Solution 45 From figure $\operatorname{mgx}=\frac{2}{2} \mathrm{x}^{2}$ $\mathrm{X}=\frac{2 m g}{R}$...
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Question: A block of mass $m$ moving at a speed $v$ compresses a spring through a distance $x$ before its speed is halved. Find the spring constant of the spring. Solution: Energy at $A=$ Energy at $B$ $\frac{1}{2} \mathrm{mv}_{\mathrm{a}}=\frac{1}{2} \mathrm{mv}_{\mathrm{b}}{ }^{2}+\mathrm{k} \mathrm{x}^{2}$ $m v^{2}=\frac{m v^{2}}{4}+k x^{2}$ $m v^{2}=\frac{m v^{2}}{4}+k x^{2}$ $\mathrm{k}=\frac{3 m v^{2}}{2 x^{2}}$...
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Question: Figure (8-E7) shows a spring fixed at the bottom end of an incline of inclination $37^{\circ}$. A small block of mass $2 \mathrm{~kg}$ starts slipping down the incline from a point $4.8 \mathrm{~m}$ away from the spring. The block compresses the spring by $20 \mathrm{~cm}$, stops momentarily and then rebounds through a distance of $1 \mathrm{~m}$ up the incline. Find (a) the friction coefficient between the plane and the block and (b) the spring constant of the spring. Take $g=10 \math...
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Question: A block of mass $250 \mathrm{~g}$ is kept on a vertical spring of spring constant $100 \mathrm{~N} / \mathrm{m}$ fixed from below. The spring is now compressed to have a length $10 \mathrm{~cm}$ shorter than its natural length and the system is released from this position. How high does the block rise? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: By law of conservation of energy: $\frac{1}{2} \mathrm{kx} \mathrm{x}^{2}=\mathrm{mgh}$ $\frac{1}{2} \times 100 \times 0.1=\m...
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Question: Figure (8-E7) shows a spring fixed at the bottom end of an incline of inclination $37^{\circ}$. A small block of mass $2 \mathrm{~kg}$ starts slipping down the incline from a point $4.8 \mathrm{~m}$ away from the spring. The block compresses the spring by $20 \mathrm{~cm}$, stops momentarily and then rebounds through a distance of $1 \mathrm{~m}$ up the incline. Find (a) the friction coefficient between the plane and the block and (b) the spring constant of the spring. Take $\mathrm{g}...
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Question: A block of mass $250 \mathrm{~g}$ is kept on a vertical spring of spring constant $100 \mathrm{~N} / \mathrm{m}$ fixed from below. The spring is now compressed to have a length $10 \mathrm{~cm}$ shorter than its natural length and the system is released from this position. How high does the block rise? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: By law of conservation of energy: $\frac{1}{2} \mathrm{kx}^{2}=\mathrm{mgh}$ $\frac{1}{2} \times 100 \times 0.1=\mathrm{mgh}=...
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Question: A block of mass $5.0 \mathrm{~kg}$ is suspended from the end of a vertical spring which is stretched by $10 \mathrm{~cm}$ under the load of the block. The block is given a sharp impulse from below so that it acquires an upward speed of $2.0 \mathrm{~m} / \mathrm{s}$. How high will it rise? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: $\mathrm{k}=(\mathrm{mg}) / \mathrm{x}=50 / 0.1=500 \mathrm{~N} / \mathrm{m}$ Total energy T.E. $=\frac{1}{2} \mathrm{~m} \mathrm{v}^{2}+\...
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Question: A block of mass $1 \mathrm{~kg}$ is placed at the point $A$ of a rough track shown in figure (8-E6). If slightly pushed towards right, it stops at the point B of the track. Calculate the work done by the frictional force on the block during its transit from $A$ to $B$. Solution: Work done = change in P.E. $\mathrm{W}=\mathrm{mgh}-\mathrm{mgH}$ $W=1 \times 10 \times(0.8-1)$ $W=-2 \mathrm{~J}$...
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Question: A uniform chain of length $L$ and mass $M$ overhangs a horizontal table with its two third part on the table. The friction coefficient between the table and the chain is 11 . Find the work done by the friction during the period the chain slips off the table. Solution: Work done for small element is: $d \mathrm{~W}=\mu \mathrm{Rx}$ $d w=\mu(M / L d x) g(x)$ Total work done is : $W={ }_{2 L / 3} \int^{0} \mu \mathrm{M} / \mathrm{L} \mathrm{g}(x) d x$ $\mathrm{W}=(2 \mu \mathrm{MgL}) / 9$...
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Question: Without expanding the determinant, prove that: $\left|\begin{array}{ccc}1 \mathrm{a} \mathrm{a}^{2} \\ 1 \mathrm{~b} \mathrm{~b}^{2} \\ 1 \mathrm{c} \mathrm{c}^{2}\end{array}\right|=\left|\begin{array}{ccc}1 \mathrm{bc} \mathrm{b}+\mathrm{c} \\ 1 \mathrm{ca} \mathrm{c}+\mathrm{a} \\ 1 \mathrm{ab} \mathrm{a}+\mathrm{b}\end{array}\right|$ Solution:...
Read More →Without expanding the determinant, prove that:
Question: Without expanding the determinant, prove that: $\left|\begin{array}{ccc}1 \mathrm{a} \mathrm{bc} \\ 1 \mathrm{~b} \mathrm{ca} \\ 1 \mathrm{c} \mathrm{ab}\end{array}\right|=\left|\begin{array}{ccc}1 \mathrm{a} \mathrm{a}^{2} \\ 1 \mathrm{~b} \mathrm{~b}^{2} \\ 1 \mathrm{c} \mathrm{c}^{2}\end{array}\right|$ Solution:...
Read More →The length of the string of a simple pendulum is measured
Question: The length of the string of a simple pendulum is measured with a meter scale to be $90.0 \mathrm{~cm}$. The radius of the bob plus the length of the hook is calculated to be $2.13 \mathrm{~cm}$ using measurements with a side calipers. What is the effective length of the pendulum? (The effective length is defined as the distance between the point of suspension and the center of the bob.) Solution: Effective Length $=(90.0+2.13) \mathrm{cm}$ In the measurement of the $90.0 \mathrm{~cm}$,...
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Question: Prove that $\left|\begin{array}{lll}1 a^{2}+b c a^{3} \\ 1 b^{2}+c a b^{3} \\ 1 c^{2}+a b c^{3}\end{array}\right|=-(a-b)(b-c)(c-a)\left(a^{2}+b^{2}+c^{2}\right)$ Solution:...
Read More →The thickness of a plate is measured to
Question: The thickness of a plate is measured to be $2.17 \mathrm{~mm}, 2.17 \mathrm{~mm}$ and $2.18 \mathrm{~mm}$ at three different places. Find the average thickness of the plate from this data. Solution: Average thickness $=\frac{2.17+2.17+2.18}{3}=2.1733 \mathrm{~mm}$ Now we have to round it off to three significant figures. So, after 7 , the digit is 3 . Hence, Average thickness $=2.17 \mathrm{~mm}$....
Read More →The length and the radius of a cylinder measured
Question: The length and the radius of a cylinder measured with a slide calipers are found to be $4.54 \mathrm{~cm}$ and $1.75 \mathrm{~cm}$ respectively. Calculate the volume of the cylinder. Solution: Length $L=4.54 \mathrm{~cm}$, radius $r=1.75 \mathrm{~cm}$ Volume $=\pi r^{2} L=\pi \times(4.54) \times(1.75)^{2}=43.6577 \mathrm{~cm}^{3}$. Now, it has to be rounded off to 3 significant digits, so we see that after 6 , the value is 5 . So 6 becomes 7 , and thus the rounded off value is given by...
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Question: If $x \neq y \neq z$ and $\left|\begin{array}{lll}x x^{3} x^{4}-1 \\ y y^{3} y^{4}-1 \\ z z^{3} z^{4}-1\end{array}\right|=0$, prove that $x y z(x y+y z+z x)=(x+y+z)$ Solution:...
Read More →Round the following numbers to 2 significant digits.
Question: Round the following numbers to 2 significant digits. (a) 3472 , (b) $84.16$, (c) $2.55$ and (d) $28.5$. Solution: (a) Here after 4 , the number is 7 , which is greater than 5 . So, the next two digits are neglected and the 4 becomes 5 . Hence, rounded value $=3500$ (b) Here after 4 , the number is 1 , which is less than 5 . So, the next two digits are neglected and the 4 doesn't change. Hence, rounded value $=84$ (c) Here after 5 , the number is 5 . So, the next two digits are neglecte...
Read More →A meter scale is graduated at every millimeter.
Question: A meter scale is graduated at every millimeter. How many significant digits will be there in a length measurement with this scale? Solution: It is stated that the meter scale is graduated at every millimeter. Now, we know $1 \mathrm{~m}=1000 \mathrm{~mm}$. So minimum measurement can be of 1 digit value(say, $3 \mathrm{~mm}, 5 \mathrm{~mm})$, then there can be up to two digits and three digits(say, $76 \mathrm{~mm}, 305 \mathrm{~mm}$ ) and the maximum value can be $1000 \mathrm{~mm}$. S...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{lll}(a+1)(a+2) a+2 1 \\ (a+2)(a+3) a+3 1 \\ (a+3)(a+4) a+4 1\end{array}\right|=-2$ Solution:...
Read More →Write the number significant digits in (a) 1001 ,
Question: Write the number significant digits in (a) 1001 , (b) $100.1$, (c) $100.10$, (d) $0.001001 .$ Solution: The number of significant digits are as follows: (a) 4 (b) 4 (c) 5 (d) 4...
Read More →The changes in a function
Question: The changes in a function $\mathrm{y}$ and the independent variable $\mathrm{x}$ are related as $\frac{d y}{d x}=x^{2}$. Find $y$ as a function of $x$. Solution: It is given that $\frac{d y}{d t}=x^{2}$ We can write $d y=x^{2} d x$ or, $y=\int^{x^{2} d x}=\frac{x^{3}}{3}+c$...
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