A cubical metal block of edge 12 cm floats in mercury with
Question: A cubical metal block of edge $12 \mathrm{~cm}$ floats in mercury with one fifth of the height inside the mercury. Water is poured till the surface of the block is just immersed in it. Find the height of the water column to be poured. Specific gravity of mercury = 13.6. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{5}}{\sqrt{1+x^{3}}} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\mathrm{x}}{\left(1+\mathrm{x}^{4}\right)} \mathrm{dx}$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{3}}{\left(1+x^{8}\right)} d x$ Solution:...
Read More →Solve the previous problem if the lead piece is fastened on the top surface
Question: Solve the previous problem if the lead piece is fastened on the top surface of the block and the block is to float with its upper surface just dipping into water. Solution:...
Read More →A cubical block of wood weighing 200 g has a lead piece fastened underneath.
Question: A cubical block of wood weighing $200 \mathrm{~g}$ has a lead piece fastened underneath. Find the mass of the lead piece which will just allow the block to float in water. Specific gravity of wood is $0.8$ and that of lead is $11.3$. Solution:...
Read More →A cubical box is to be constructed with iron sheets
Question: A cubical box is to be constructed with iron sheets $1 \mathrm{~mm}$ in thickness. What can be the minimum value of the external edge so that the cube does not sink ink water? Density of iron $=8000 \mathrm{~kg} / \mathrm{m}^{3}$ and density of water $=1000 \mathrm{~kg} / \mathrm{m}^{3}$ Solution:...
Read More →A cube of ice floats partly in water and partly in K.oil.
Question: A cube of ice floats partly in water and partly in K.oil. Find the ratio of the volume of ice immersed in water to that in K.oil. Specific gravity of K.oil is $0.8$ and that of ice is $0.9$. Solution:...
Read More →A cubical block of ice floating in water has to support a metal piece weighing
Question: A cubical block of ice floating in water has to support a metal piece weighing $0.5 \mathrm{~kg}$. What can be the minimum edge of the block so that it does not sink in water? Specific gravity of ice $=0.9$ Solution:...
Read More →A ferry boat has internal volume 1 m3 and weight 50 kg.
Question: A ferry boat has internal volume $1 \mathrm{~m}^{3}$ and weight $50 \mathrm{~kg}$. (a) Neglecting the thickness of the wood, find the fraction of the volume of the boat immersed in water. (b) If a leak develops in the bottom and water starts coming in, what fraction of the boat's volume will be filled with water before water starts coming in from the sides? Solution:...
Read More →A metal piece of mass 160 g lies in equilibrium inside
Question: A metal piece of mass $160 \mathrm{~g}$ lies in equilibrium inside a glass of water (figure 13-E4). The piece touches the bottom of the glass at a small number of points. If the density of the metal is $8000 \mathrm{~kg} / \mathrm{m}^{3}$ Solution:...
Read More →Refer to the previous problem. Suppose,
Question: Refer to the previous problem. Suppose, the goldsmith argues that he has not mixed copper or any other material with gold, rather some cavities might have been left inside the ornament. Calculate the volume of the cavities left that will allow the weights given that problem. Solution:...
Read More →An ornament weighing 36g in air,
Question: An ornament weighing $36 \mathrm{~g}$ in air, weighs only $34 \mathrm{~g}$ in water. Assuming that some copper is mixed with gold to prepare the ornament, find the amount of copper in it. Specific gravity of gold is $19.3$ and that of copper is $8.9$ Solution:...
Read More →Water is filled in a rectangular tank of size 3m × 2m × 1m.
Question: Water is filled in a rectangular tank of size $3 m \times 2 m \times 1 m$. (a) Find the total force exerted by the water on the bottom surface of the tank. (b) Consider a vertical side of area $2 \mathrm{~m} \times 1 \mathrm{~m}$. Take a horizontal strip of width $\partial \mathrm{x}$ metre in this side, situated at a depth of $x$ metre from the surface of water. Find the force by the water on strip. (c) Find the torque of the force calculated in part (b) about the bottom edge of this ...
Read More →Find the force exerted by the water on a
Question: Find the force exerted by the water on a $2 \mathrm{~m}^{2}$ plane surfaces of a large stone placed at the bottom of a sea $500 \mathrm{~m}$ deep. Does the force depend on the orientation of the surface? Solution:...
Read More →If water be used to construct a barometer,
Question: If water be used to construct a barometer, what would be the height of water column at standard atmospheric pressure (76 cm of mercury)? Solution:...
Read More →Suppose the glass of the previous problem is covered by
Question: Suppose the glass of the previous problem is covered by a jar and the air inside the jar is completely pumped out. (a) What will be the answers to the problem? (b) Show that the answers do not change if a glass of different shape is used provided the height, the bottom area and the volume are unchanged. Solution:...
Read More →A glass of water has a bottom of area
Question: A glass of water has a bottom of area $20 \mathrm{~cm}^{2}$, top of area $20 \mathrm{~cm}^{2}$, height $20 \mathrm{~cm}$ and volume half a litre. (a) Find the force exerted by the water on the bottom. (b) Considering the equilibrium of the water, find the resultant force exerted by the sides of the glass on the water. Atmospheric pressure $=1.01=1000 \mathrm{~kg} / \mathrm{m}^{3}$ and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Take all numbers to be exact. Solution:...
Read More →The area of cross-section of the wider tube shown in figure (13-E2)
Question: The area of cross-section of the wider tube shown in figure (13-E2) is $900 \mathrm{~cm}^{2}$. If the boy standing on the piston weighs $45 \mathrm{~kg}$, find the difference in the levels of water in the two tubes. Solution:...
Read More →The height of mercury surfaces in the two arms of the manometer shown in figure (13-E1)
Question: The height of mercury surfaces in the two arms of the manometer shown in figure (13-E1) are $2 \mathrm{~cm}$ and $8 \mathrm{~cm}$. Atmospheric pressure $=1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}$. Find (a) the pressure of the gas in the cylinder and (b) the pressure of mercury at the bottom of the $U$ tube. Solution:...
Read More →The surface of water in a water tank on the top of a house is
Question: The surface of water in a water tank on the top of a house is $4 \mathrm{~m}$ above the tap level. Find the pressure of water at the tap when the tap is closed? Is it necessary to specify that the tap is closed? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ Solution:...
Read More →A particle is subjected to two simple harmonic motions,
Question: A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of $45^{\circ}$ with the $X$-axis. The two motions are given by $x=x_{0} \sin \omega t$ and $s=s_{0} \sin \omega t$ Find the amplitude of the resultant motion. Solution:...
Read More →A particle is subjected to two simple harmonic motions given by
Question: A particle is subjected to two simple harmonic motions given by $x_{1}=2.0 \sin (100 \pi t)$ and $x_{2}=2.0 \sin (120 \pi t+\pi / 3)$ where $x$ is in centimeter and $t$ in second. Find the displacement of the particle at $(a) t=0.0125,(b) t=0.025$. Solution:...
Read More →Three simple harmonic motions of equal amplitudes A and equal
Question: Three simple harmonic motions of equal amplitudes A and equal time periods in the same direction combine. The phase of the second motion is $60^{\circ}$ ahead of the second. Find the amplitude of the resultant motion. Solution:...
Read More →A particle is subjected to two simple harmonic motions of same time
Question: A particle is subjected to two simple harmonic motions of same time period in the same direction. The amplitude of the first motion is $3.0 \mathrm{~cm}$ and that of the second is $4.0 \mathrm{~cm}$. Find the resultant amplitude if the phase difference between the motions is (a) $0^{\circ}$, (b) $60^{\circ}$, (c) $90^{\circ}$. Solution:...
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