Properties of Matter
314 Questions
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 25th June Morning Shift
The velocity of upper layer of water in a river is 36 kmh$-$1. Shearing stress between horizontal layers of water is 10$-$3 Nm$-$2. Depth of the river is __________ m. (Co-efficient of viscosity of water is 10$-$2 Pa.s)
Correct Answer: 100
Explanation:
$F = - \eta A{{du} \over {dx}}$
$ \Rightarrow {10^{ - 3}} = {10^{ - 2}} \times {{10} \over h}$
$ \Rightarrow h = {{{{10}^{ - 1}}} \over {{{10}^{ - 3}}}}$ m = 100 m
$\Rightarrow$ (100)
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 24th June Evening Shift
In an experiment to verify Newton's law of cooling, a graph is plotted between the temperature difference ($\Delta$T) of the water and surroundings and time as shown in figure. The initial temperature of water is taken as 80$^\circ$C. The value of t2 as mentioned in the graph will be __________.
Correct Answer: 16
Explanation:
Temperature of surrounding = 20$^\circ$C
For 0 $\to$ 6 minutes, average temp. = 70$^\circ$C
$\to$ Rate of cooling $\propto$ 70$^\circ$C $-$ 20$^\circ$C = 50$^\circ$C
For 6 $\to$ t2 minutes, average temp. = 50$^\circ$C
$\to$ Rate of cooling $\propto$ 30$^\circ$C
$ \Rightarrow {t_2} - 6 = {5 \over 3}$ (6 minutes)
$\Rightarrow$ t2 = 16 minutes
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 31st August Evening Shift
Four identical hollow cylindrical columns of mild steel support a big structure of mass 50 $\times$ 103 kg. The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local distribution, calculate the compression strain of each column. [Use Y = 2.0 $\times$ 1011 Pa, g = 9.8 m/s2]
A.
3.60 $\times$ 10$-$8
B.
2.60 $\times$ 10$-$7
C.
1.87 $\times$ 10$-$3
D.
7.07 $\times$ 10$-$4
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 31st August Morning Shift
A uniform heavy rod of weight 10 kg ms$-$2, cross-sectional area 100 cm2 and length 20 cm is hanging from a fixed support. Young modulus of the material of the rod is 2 $\times$ 1011 Nm$-$2. Neglecting the lateral contraction, find the elongation of rod due to its own weight.
A.
2 $\times$ 10$-$9 m
B.
5 $\times$ 10$-$8 m
C.
4 $\times$ 10$-$8 m
D.
5 $\times$ 10$-$10 m
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th August Morning Shift
In Millikan's oil drop experiment, what is viscous force acting on an uncharged drop of radius 2.0 $\times$ 10$-$5 m and density 1.2 $\times$ 103 kgm$-$3 ? Take viscosity of liquid = 1.8 $\times$ 10$-$5 Nsm$-$2. (Neglect buoyancy due to air).
A.
3.8 $\times$ 10$-$11 N
B.
3.9 $\times$ 10$-$10 N
C.
1.8 $\times$ 10$-$10 N
D.
5.8 $\times$ 10$-$10 N
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th August Evening Shift
Two blocks of masses 3 kg and 5 kg are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is ${{24} \over \pi } \times {10^2}$ Nm-2. What is the minimum radius of the wire ? (Take g = 10 ms-2)
A.
125 cm
B.
1250 cm
C.
12.5 cm
D.
1.25 cm
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th August Morning Shift
Two narrow bores of diameter 5.0 mm and 8.0 mm are joined together to form a U-shaped tube open at both ends. If this U-tube contains water, what is the difference in the level of two limbs of the tube. [Take surface tension of water T = 7.3 $\times$ 10$-$2 Nm$-$1, angle of contact = 0, g = 10 ms2 and density of water = 1.0 $\times$ 103 kg m$-$3]
A.
3.62 mm
B.
2.19 mm
C.
5.34 mm
D.
4.97 mm
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Evening Shift
A raindrop with radius R = 0.2 mm falls from a cloud at a height h = 2000 m above the ground. Assume that the drop is spherical throughout its fall and the force of buoyance may be neglected, then the terminal speed attained by the raindrop is :
[Density of water fw = 1000 kg m$-$3 and Density of air fa = 1.2 kg m$-$3, g = 10 m/s2, Coefficient of viscosity of air = 1.8 $\times$ 10$-$5 Nsm$-$2]
[Density of water fw = 1000 kg m$-$3 and Density of air fa = 1.2 kg m$-$3, g = 10 m/s2, Coefficient of viscosity of air = 1.8 $\times$ 10$-$5 Nsm$-$2]
A.
250.6 ms$-$1
B.
43.56 ms$-$1
C.
4.94 ms$-$1
D.
14.4 ms$-$1
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Morning Shift
A light cylindrical vessel is kept on a horizontal surface. Area of base is A. A hole of cross-sectional area 'a' is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is (a < < A) :
A.
${A \over {2a}}$
B.
None of these
C.
${{2a} \over A}$
D.
${{a} \over A}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Morning Shift
A body takes 4 min. to cool from 61$^\circ$ C to 59$^\circ$ C. If the temperature of the surroundings is 30$^\circ$ C, the time taken by the body to cool from 51$^\circ$ C to 49$^\circ$ C is :
A.
4 min.
B.
3 min.
C.
8 min.
D.
6 min.
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th July Evening Shift
Two spherical soap bubbles of radii r1 and r2 in vacuum combine under isothermal conditions. The resulting bubble has a radius equal to :
A.
${{{r_1}{r_2}} \over {{r_1} + {r_2}}}$
B.
$\sqrt {{r_1}{r_2}} $
C.
$\sqrt {r_1^2 + r_2^2} $
D.
${{{r_1} + {r_2}} \over 2}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th July Morning Shift
Two wires of same length and radius are joined end to end and loaded. The Young's modulii of the materials of the two wires are Y1 and Y2. The combination behaves as a single wire then its Young's modulus is :
A.
$Y = {{2{Y_1}{Y_2}} \over {3({Y_1} + {Y_2})}}$
B.
$Y = {{2{Y_1}{Y_2}} \over {{Y_1} + {Y_2}}}$
C.
$Y = {{{Y_1}{Y_2}} \over {2({Y_1} + {Y_2})}}$
D.
$Y = {{{Y_1}{Y_2}} \over {{Y_1} + {Y_2}}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 20th July Evening Shift
The length of a metal wire is l1, when the tension in it is T1 and is l2 when the tension is T2. The natural length of the wire is :
A.
$\sqrt {{l_1}{l_2}} $
B.
${{{l_1}{T_2} - {l_2}{T_1}} \over {{T_2} - {T_1}}}$
C.
${{{l_1}{T_2} + {l_2}{T_1}} \over {{T_2} + {T_1}}}$
D.
${{{l_1} + {l_2}} \over 2}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 20th July Evening Shift
Two small drops of mercury each of radius R coalesce to form a single large drop. The ratio of total surface energy before and after the change is :
A.
${2^{{1 \over 3}}}:1$
B.
$1:{2^{{1 \over 3}}}$
C.
2 : 1
D.
1 : 2
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 20th July Morning Shift
The value of tension in a long thin metal wire has been changed from T1 to T2. The lengths of the metal wire at two different values of tension T1 and T2 are l1 and l2 respectively. The actual length of the metal wire is :
A.
${{{l_1} + {l_2}} \over 2}$
B.
$\sqrt {{T_1}{T_2}{l_1}{l_2}} $
C.
${{{T_1}{l_2} - {T_2}{l_1}} \over {{T_1} - {T_2}}}$
D.
${{{T_1}{l_1} - {T_2}{l_2}} \over {{T_1} - {T_2}}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 17th March Evening Shift
An object is located at 2 km beneath the surface of the water. If the fractional compression ${{\Delta V} \over V}$ is 1.36%, the ratio of hydraulic stress to the corresponding hydraulic strain will be ____________. [Given : density of water is 1000 kgm$-$3 and g = 9.8 ms$-$2]
A.
1.44 $\times$ 107 Nm$-$2
B.
1.44 $\times$ 109 Nm$-$2
C.
1.96 $\times$ 107 Nm$-$2
D.
2.26 $\times$ 109 Nm$-$2
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 17th March Morning Shift
When two soap bubbles of radii a and b (b > a) coalesce, the radius of curvature of common surface is :
A.
${{b - a} \over {ab}}$
B.
${{a + b} \over {ab}}$
C.
${{ab} \over {a + b}}$
D.
${{ab} \over {b - a}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 16th March Evening Shift
What will be the nature of flow of water from a circular tap, when its flow rate increased from 0.18 L/min to 0.48 L/min? The radius of the tap and viscosity of water are 0.5 cm and 10$-$3 Pa s, respectively. (Density of water : 103 kg/m3)
A.
Steady flow to unsteady flow
B.
Unsteady to steady flow
C.
Remains turbulent flow
D.
Remains steady flow
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 16th March Morning Shift
The pressure acting on a submarine is 3 $\times$ 105 Pa at a certain depth. If the depth is doubled, the percentage increase in the pressure acting on the submarine would be :
(Assume that atmospheric pressure is 1 $\times$ 105 Pa density of water is 103 kg m$-$3, g = 10 ms$-$2)
(Assume that atmospheric pressure is 1 $\times$ 105 Pa density of water is 103 kg m$-$3, g = 10 ms$-$2)
A.
${{200} \over 5}$%
B.
${{200} \over 3}$%
C.
${{3} \over 200}$%
D.
${{5} \over 200}$%
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Evening Shift
The length of metallic wire is l1 when tension in it is T1. It is l2 when the tension is T2. The original length of the wire will be :
A.
${{{T_1}{l_1} - {T_2}{l_2}} \over {{T_2} - {T_1}}}$
B.
${{{l_1} + {l_2}} \over 2}$
C.
${{{T_2}{l_1} + {T_1}{l_2}} \over {{T_1} + {T_2}}}$
D.
${{{T_2}{l_1} - {T_1}{l_2}} \over {{T_2} - {T_1}}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Morning Shift
A large number of water drops, each of radius r, combine to have a drop of radius R. If the surface tension is T and mechanical equivalent of heat is J, the rise in heat energy per unit volume will be :
A.
${{2T} \over J}\left( {{1 \over r} - {1 \over R}} \right)$
B.
${{3T} \over J}\left( {{1 \over r} - {1 \over R}} \right)$
C.
${{3T} \over rJ}$
D.
${{2T} \over rJ}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Morning Shift
The normal density of a material is $\rho$ and its bulk modulus of elasticity is K. The magnitude of increase in density of material, when a pressure P is applied uniformly on all sides, will be :
A.
${{\rho K} \over P}$
B.
${{PK} \over \rho }$
C.
${{\rho P} \over K}$
D.
${K \over {\rho P}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 24th February Morning Shift
If Y, K and $\eta $ are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.
A.
$Y = {{9K\eta } \over {3K - \eta }}N/{m^2}$
B.
$Y = {{9K\eta } \over {2\eta + 3K}}N/{m^2}$
C.
$\eta = {{3YK} \over {9K + Y}}N/{m^2}$
D.
$K = {{Y\eta } \over {9\eta - 3Y}}N/{m^2}$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 1st September Evening Shift
A steel rod with y = 2.0 $\times$ 1011 Nm$-$2 and $\alpha$ = 10$-$5 $^\circ$C$-$1 of length 4 m and area of cross-section 10 cm2 is heated from 0$^\circ$C to 400$^\circ$C without being allowed to extend. The tension produced in the rod is x $\times$ 105 N where the value of x is ____________.
Correct Answer: 8
Explanation:
Given, the Young's modulus of the steel rod, Y = 2 $\times$ 1011 Pa
Thermal coefficient of the steel rod, $\alpha$ = 10$-$5$^\circ$C
The length of the steel rod, l = 4 m
The area of the cross-section, A = 10 cm2
The temperature difference, $\Delta$T = 400$^\circ$C
As we know that,
Thermal strain = $\alpha$ $\Delta$T
Using the Hooke's law
Young's modulus (Y) = ${{Thermal\,stress} \over {Thermal\,strain}} = {{F/A} \over {\alpha \,\Delta \,T}}$
Thermal stress, $F = YA\,\alpha \,\Delta \,T$
Substitute the values in the above equation, we get
$F = 2 \times {10^{11}} \times 10 \times {10^{ - 4}} \times {10^{ - 5}} \times (400)$
$ = 8 \times {10^5}N$
Comparing with, $F = x \times {10^5}N$
The value of the x = 8.
Thermal coefficient of the steel rod, $\alpha$ = 10$-$5$^\circ$C
The length of the steel rod, l = 4 m
The area of the cross-section, A = 10 cm2
The temperature difference, $\Delta$T = 400$^\circ$C
As we know that,
Thermal strain = $\alpha$ $\Delta$T
Using the Hooke's law
Young's modulus (Y) = ${{Thermal\,stress} \over {Thermal\,strain}} = {{F/A} \over {\alpha \,\Delta \,T}}$
Thermal stress, $F = YA\,\alpha \,\Delta \,T$
Substitute the values in the above equation, we get
$F = 2 \times {10^{11}} \times 10 \times {10^{ - 4}} \times {10^{ - 5}} \times (400)$
$ = 8 \times {10^5}N$
Comparing with, $F = x \times {10^5}N$
The value of the x = 8.
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 31st August Morning Shift
When a rubber ball is taken to a depth of __________m in deep sea, its volume decreases by 0.5%.
(The bulk modulus of rubber = 9.8 $\times$ 108 Nm$-$2, Density of sea water = 103 kgm$-$3, g = 9.8 m/s2)
(The bulk modulus of rubber = 9.8 $\times$ 108 Nm$-$2, Density of sea water = 103 kgm$-$3, g = 9.8 m/s2)
Correct Answer: 500
Explanation:
$B = - {{\Delta P} \over {\left( {{{\Delta V} \over V}} \right)}} = - {{\rho gh} \over {\left( {{{\Delta V} \over V}} \right)}}$
$ - {{B{{\Delta V} \over V}} \over {\rho g}} = h$
${{9.8 \times {{10}^8} \times 0.5} \over {100 \times {{10}^3} \times 9.8}} = h$
h = 500
$ - {{B{{\Delta V} \over V}} \over {\rho g}} = h$
${{9.8 \times {{10}^8} \times 0.5} \over {100 \times {{10}^3} \times 9.8}} = h$
h = 500
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 27th August Evening Shift
Wires W1 and W2 are made of same material having the breaking stress of 1.25 $\times$ 109 N/m2. W1 and W2 have cross-sectional area of 8 $\times$ 10$-$7 m2 and 4 $\times$ 10$-$7 m2, respectively. Masses of 20 kg and 10 kg hang from them as shown in the figure. The maximum mass that can be placed in the pan without breaking the wires is ____________ kg. (Use g = 10 m/s2)
Correct Answer: 40
Explanation:
B.S1 = ${{{T_{1\max }}} \over {8 \times {{10}^{ - 7}}}}$ $\Rightarrow$ T1 max = 8 $\times$ 1.25 $\times$ 100 = 1000 N
B.S2 = ${{{T_{2\max }}} \over {4 \times {{10}^{ - 7}}}}$ $\Rightarrow$ T2 max = 4 $\times$ 1.25 $\times$ 100 = 500 N
m = ${{500 - 100} \over {10}} = 40$ kg
B.S2 = ${{{T_{2\max }}} \over {4 \times {{10}^{ - 7}}}}$ $\Rightarrow$ T2 max = 4 $\times$ 1.25 $\times$ 100 = 500 N
m = ${{500 - 100} \over {10}} = 40$ kg
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 26th August Morning Shift
A soap bubble of radius 3 cm is formed inside the another soap bubble of radius 6 cm. The radius of an equivalent soap bubble which has the same excess pressure as inside the smaller bubble with respect to the atmospheric pressure is ................ cm.
Correct Answer: 2
Explanation:
Image
Excess pressure inside the smaller soap bubble
$\Delta P = {{4S} \over {{r_1}}} + {{4S} \over {{r_2}}}$ .... (i)
The excess pressure inside equivalent soap bubble
$\Delta P = {{4S} \over {{R_{eq}}}}$ ....... (ii)
From (i) & (ii)
${{4S} \over {{R_{eq}}}} = {{4S} \over {{r_1}}} + {{4S} \over {{r_2}}}$
${1 \over {{R_{eq}}}} = {1 \over {{r_1}}} + {1 \over {{r_2}}}$
$ = {1 \over 6} + {1 \over 3}$
Req = 2 cm
Excess pressure inside the smaller soap bubble
$\Delta P = {{4S} \over {{r_1}}} + {{4S} \over {{r_2}}}$ .... (i)
The excess pressure inside equivalent soap bubble
$\Delta P = {{4S} \over {{R_{eq}}}}$ ....... (ii)
From (i) & (ii)
${{4S} \over {{R_{eq}}}} = {{4S} \over {{r_1}}} + {{4S} \over {{r_2}}}$
${1 \over {{R_{eq}}}} = {1 \over {{r_1}}} + {1 \over {{r_2}}}$
$ = {1 \over 6} + {1 \over 3}$
Req = 2 cm
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 27th July Evening Shift
The water is filled upto height of 12 m in a tank having vertical sidewalls. A hole is made in one of the walls at a depth 'h' below the water level. The value of 'h' for which the emerging steam of water strikes the ground at the maximum range is ________ m.
Correct Answer: 6
Explanation:
$R = \sqrt {2gh} \times \sqrt {{{(12 - h) \times 2} \over g}} $
$\sqrt {4h(12 - h)} = R$
For maximum R
${{dR} \over {dh}} = 0$
$\Rightarrow$ h = 6 m
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 27th July Morning Shift
A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section 10$-$6 m2 stretched by an amount 0.04 m. The velocity of the projected stone is ______________ m/s.
(Young's modulus of rubber = 0.5 $\times$ 109 N/m2)
(Young's modulus of rubber = 0.5 $\times$ 109 N/m2)
Correct Answer: 20
Explanation:
By energy conservation
${1 \over 2}.{{YA} \over L}.{x^2} = {1 \over 2}m{v^2}$
${{0.5 \times {{10}^9} \times {{10}^{ - 6}} \times {{(0.04)}^2}} \over {0.1}} = {{20} \over {1000}}{v^2}$
$\therefore$ ${v^2} = 400$
$v = 20$ m/s
${1 \over 2}.{{YA} \over L}.{x^2} = {1 \over 2}m{v^2}$
${{0.5 \times {{10}^9} \times {{10}^{ - 6}} \times {{(0.04)}^2}} \over {0.1}} = {{20} \over {1000}}{v^2}$
$\therefore$ ${v^2} = 400$
$v = 20$ m/s
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 22th July Evening Shift
In 5 minutes, a body cools from 75$^\circ$C to 65$^\circ$C at room temperature of 25$^\circ$C. The temperature of body at the end of next 5 minutes is _________$^\circ$ C.
Correct Answer: 57
Explanation:
${{75 - 65} \over 5} = k\left( {{{75 + 65} \over 2} - 25} \right)$
$ \Rightarrow k = {2 \over {45}}$
${{65 - T} \over 5} = k\left( {{{65 + T} \over 2} - 25} \right)$
$ \Rightarrow T = 57^\circ $ C
$ \Rightarrow k = {2 \over {45}}$
${{65 - T} \over 5} = k\left( {{{65 + T} \over 2} - 25} \right)$
$ \Rightarrow T = 57^\circ $ C
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 18th March Evening Shift
Consider a water tank as shown in the figure. It's cross-sectional area is 0.4 m2. The tank has an opening B near the bottom whose cross-section area is 1 cm2. A load of 24 kg is applied on the water at the top when the height of the water level is 40 cm above the bottom, the velocity of water coming out the opening B is v ms-1.
The value of v, to the nearest integer, is ___________. [Take value of g to be 10 ms-2]
The value of v, to the nearest integer, is ___________. [Take value of g to be 10 ms-2]
Correct Answer: 3
Explanation:
Force at point A,
mg + p0A = pA
$ \Rightarrow $ p = p0 + ${{mg} \over A}$
Given, area of A = 0.4 m2 = 0.4 $\times$ 104 cm2 and area of B = 1 cm2
applying continuity equation
AV1 = av [V1 = velocity at point A]
$ \Rightarrow $ V1 = ${a \over A}v$
As A >>> a so ${a \over A}$ very small
$ \therefore $ V1 <<< v
So we can neglect V1 by assuming V1 = 0
On applying Bernoulli's theorem at points A and B,
${p_0} + {{mg} \over A} + {{\rho V_1^2} \over 2} + \rho gh = {p_0} + {{\rho {v^2}} \over 2} + 0$
$ \Rightarrow {p_0} + {{mg} \over A} + 0 + \rho gh = {p_0} + {{\rho {v^2}} \over 2}$
$ \Rightarrow {{mg} \over A} + \rho gh = {{\rho {v^2}} \over 2}$
$ \Rightarrow {{24 \times 10} \over {0.4}} + 1000 \times 10 \times 0.4 = {{1000 \times {v^2}} \over 2}$
$ \Rightarrow $ v $ \simeq $ 3 m/s
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 18th March Morning Shift
Two separate wires A and B are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire B is 4 times that of the radius of wire A. The length of the wires A and B are in the ratio of a : b. Then a/b can be expressed as 1/x where x is _________.
Correct Answer: 32
Explanation:
${\rho _A} = {\rho _B}$
${r _B} = 4{r _A}$
$\Delta {l_a} = 2$ mm
$\Delta {l_B} = 4$ mm
$y = {{stress} \over {strain}} = {{F/A} \over {\Delta l/l}}$
$ \Rightarrow $ y${{\Delta l} \over l} = {F \over {A}}$
$ \Rightarrow $ $l = {{Ay\Delta l} \over F}$
$ \Rightarrow $ ${{{I_a}} \over {{I_b}}} = {{\pi r_a^2 \times y \times \Delta {I_a} \times F} \over {\pi r_b^2 \times y \times \Delta {I_b} \times F}}$
$ \Rightarrow $ ${{{I_a}} \over {{I_b}}} = {{r_a^2 \times \Delta {I_a}} \over {r_b^2 \times \Delta {I_b}}} = {{r_a^2 \times 2} \over {{{(4{r_a})}^2} \times 4}} = {{r_a^2} \over {16r_a^2 \times 2}}$
$ \Rightarrow $ ${{{I_a}} \over {{I_b}}} = {1 \over {32}}$
$ \therefore $ $x = 32$
${r _B} = 4{r _A}$
$\Delta {l_a} = 2$ mm
$\Delta {l_B} = 4$ mm
$y = {{stress} \over {strain}} = {{F/A} \over {\Delta l/l}}$
$ \Rightarrow $ y${{\Delta l} \over l} = {F \over {A}}$
$ \Rightarrow $ $l = {{Ay\Delta l} \over F}$
$ \Rightarrow $ ${{{I_a}} \over {{I_b}}} = {{\pi r_a^2 \times y \times \Delta {I_a} \times F} \over {\pi r_b^2 \times y \times \Delta {I_b} \times F}}$
$ \Rightarrow $ ${{{I_a}} \over {{I_b}}} = {{r_a^2 \times \Delta {I_a}} \over {r_b^2 \times \Delta {I_b}}} = {{r_a^2 \times 2} \over {{{(4{r_a})}^2} \times 4}} = {{r_a^2} \over {16r_a^2 \times 2}}$
$ \Rightarrow $ ${{{I_a}} \over {{I_b}}} = {1 \over {32}}$
$ \therefore $ $x = 32$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 17th March Evening Shift
Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes 0.01 cm3 of oleic acid per cm3 of the solution. Then you make a thin film of this solution (monomolecular thickness) of area 4 cm2 by considering 100 spherical drops of radius ${\left( {{3 \over {40\pi }}} \right)^{{1 \over 3}}} \times {10^{ - 3}}$ cm. Then the thickness of oleic acid layer will be x $\times$ 10$-$14 m. Where x is ____________.
Correct Answer: 25
Explanation:
4tT = 100 $ \times $ ${4 \over 3}\pi {r^3}$
= $100 \times {4 \over 3}\pi \times {3 \over {40\pi }} \times {10^{ - 9}}$
= 10-8 cm3
$ \Rightarrow $ tT = 25 $ \times $ 10-10 cm
= 25 $ \times $ 10-12 m
t0 = 0.01 tT = 25 $ \times $ 10-14 m
$ \therefore $ x = 25
= $100 \times {4 \over 3}\pi \times {3 \over {40\pi }} \times {10^{ - 9}}$
= 10-8 cm3
$ \Rightarrow $ tT = 25 $ \times $ 10-10 cm
= 25 $ \times $ 10-12 m
t0 = 0.01 tT = 25 $ \times $ 10-14 m
$ \therefore $ x = 25
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 24th February Evening Shift
A uniform metallic wire is elongated by 0.04 m when subjected to a linear force F. The elongation, if its length and diameter is doubled and subjected to the same force will be ________ cm.
Correct Answer: 2
Explanation:
Let initial length and diameter be l1 and d1, whereas final length and diameter be l2 and d2.
Given, l2 = 2l1, d2 = 2d1, $\Delta$l1 = 0.04 m
By using formula of Young's modulus of elasticity,
$Y = {{F\,.\,l} \over {A\Delta l}}$
$\therefore$ ${Y_1} = {Y_2}$
$ \Rightarrow {{F{l_1}} \over {{A_1} \times \Delta {l_1}}} = {{F{l_2}} \over {{A_2} \times \Delta {l_2}}}$
$ \Rightarrow {{F{l_1}} \over {\pi {{({d_1}/2)}^2} \times 0.04}} = {{F2{l_1}} \over {\pi {{(2{d_1}/2)}^2} \times \Delta {l_2}}}$
$ \Rightarrow {1 \over {1/4 \times 0.04}} = {2 \over {\Delta {l_2}}}$
$ \Rightarrow \Delta {l_2} = 0.02$ m = 2 cm
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 24th February Morning Shift
A hydraulic press can lift 100 kg when a mass 'm' is placed on the smaller piston. It can lift ___________ kg when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times keeping the same mass 'm' on the smaller piston.
Correct Answer: 25600
Explanation:
According to Pascal's law,
${{{F_1}} \over {{A_1}}} = {{{F_2}} \over {{A_2}}}$
Initially, ${{100g} \over {{A_1}}} = {{mg} \over {{A_2}}}$ .... (i)
Finally, ${{Mg} \over {16{A_1}}} = {{mg} \over {\left( {{{{A_2}} \over {16}}} \right)}}$ ..... (ii)
On dividing Eqs. (i) by (ii), we get
${{100 \times 16} \over M} = {1 \over {16}}$
$\therefore$ M = 25600 kg
${{{F_1}} \over {{A_1}}} = {{{F_2}} \over {{A_2}}}$
Initially, ${{100g} \over {{A_1}}} = {{mg} \over {{A_2}}}$ .... (i)
Finally, ${{Mg} \over {16{A_1}}} = {{mg} \over {\left( {{{{A_2}} \over {16}}} \right)}}$ ..... (ii)
On dividing Eqs. (i) by (ii), we get
${{100 \times 16} \over M} = {1 \over {16}}$
$\therefore$ M = 25600 kg
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Evening Slot
A fluid is flowing through a horizontal pipe of varying cross-section, with
speed v ms–1 at a point where the pressure is P pascal. At another point where pressure is ${P \over 2}$ Pascal its speed is V ms–1. If the density of the fluid is $\rho $ kg m–3 and the flow is streamline, then V is equal to :
speed v ms–1 at a point where the pressure is P pascal. At another point where pressure is ${P \over 2}$ Pascal its speed is V ms–1. If the density of the fluid is $\rho $ kg m–3 and the flow is streamline, then V is equal to :
A.
$\sqrt {{P \over {2\rho }} + {v^2}} $
B.
$\sqrt {{P \over \rho } + {v^2}} $
C.
$\sqrt {{{2P} \over \rho } + {v^2}} $
D.
$\sqrt {{P \over \rho } + {v}} $
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Morning Slot
An object of mass m is suspended at the end of a massless wire of length L and area of crosssection A. Young modulus of the material of the wire is Y. If the mass is pulled down slightly its
frequency of oscillation along the vertical direction is :
A.
$f = {1 \over {2\pi }}\sqrt {{{YA} \over {mL}}} $
B.
$f = {1 \over {2\pi }}\sqrt {{{mL} \over {YA}}} $
C.
$f = {1 \over {2\pi }}\sqrt {{{YL} \over {mA}}} $
D.
$f = {1 \over {2\pi }}\sqrt {{{mA} \over {YL}}} $
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 5th September Evening Slot
In an experiment to verify Stokes law, a small
spherical ball of radius r and density $\rho $ falls
under gravity through a distance h in air before
entering a tank of water. If the terminal velocity
of the ball inside water is same as its velocity
just before entering the water surface, then the
value of h is proportional to :
(ignore viscosity of air)
(ignore viscosity of air)
A.
r
B.
r4
C.
r3
D.
r2
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 5th September Morning Slot
A hollow spherical shell at outer radius R floats
just submerged under the water surface. The
inner radius of the shell is r. If the specific
gravity of the shell material is ${{27} \over 8}$ w.r.t water,
the value of r is :
A.
${{2} \over 3}$R
B.
${{4} \over 9}$R
C.
${{1} \over 3}$R
D.
${{8} \over 9}$R
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Evening Slot
A cube of metal is subjected to a hydrostatic pressure of 4 GPa. The percentage change in the
length of the side of the cube is close to :
(Given bulk modulus of metal, B = 8 $ \times $ 1010 Pa)
(Given bulk modulus of metal, B = 8 $ \times $ 1010 Pa)
A.
0.6
B.
20
C.
1.67
D.
5
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Evening Slot
Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density
d. The area of the base of both vessels is S but the height of liquid in one vessel is x1
and in the
other, x2
. When both cylinders are connected through a pipe of negligible volume very close to the
bottom, the liquid flows from one vessel to the other until it comes to equilibrium at a new height.
The change in energy of the system in the process is:
A.
gdS(x2 + x1)2
B.
gdS$\left( {x_2^2 + x_1^2} \right)$
C.
${1 \over 4}gdS{\left( {{x_2} - {x_1}} \right)^2}$
D.
${3 \over 4}gdS{\left( {{x_2} - {x_1}} \right)^2}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Morning Slot
A air bubble of radius 1 cm in water has an upward acceleration 9.8 cm s–2. The density of water is
1 gm cm–3 and water offers negligible drag force on the bubble. The mass of the bubble is (g = 980
cm/s2).
A.
1.52 gm
B.
4.51 gm
C.
3.15 gm
D.
4.15 gm
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 3rd September Evening Slot
A metallic sphere cools from 50oC to 40o in 300 s. If atmospheric temperature around is 20oC,
then the sphere’s temperature after the next 5 minutes will be close to :
A.
35oC
B.
31oC
C.
33oC
D.
28oC
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
Pressure inside two soap bubbles are 1.01 and 1.02 atmosphere, respectively. The ratio of their
volumes is :
A.
4 : 1
B.
8 : 1
C.
2 : 1
D.
0.8 : 1
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
A capillary tube made of glass of radius 0.15
mm is dipped vertically in a beaker filled with
methylene iodide (surface tension = 0.05 Nm–1,
density = 667 kg m–3) which rises to height h in
the tube. It is observed that the two tangents
drawn from liquid-glass interfaces (from opp.
sides of the capillary) make an angle of 60o
with one another. Then h is close to (g = 10 ms–2)
A.
0.049 m
B.
0.087 m
C.
0.137 m
D.
0.172 m
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
A cylindrical vessel containing a liquid is
rotated about its axis so that the liquid rises at
its sides as shown in the figure. The radius of
vessel is 5 cm and the angular speed of
rotation is $\omega $ rad s–1. The difference in the
height, h (in cm) of liquid at the centre of
vessel and at the side will be :
A.
${{2{\omega ^2}} \over {25g}}$
B.
${{5{\omega ^2}} \over {2g}}$
C.
${{25{\omega ^2}} \over {2g}}$
D.
${{2{\omega ^2}} \over {5g}}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 9th January Evening Slot
A small spherical droplet of density d is floating
exactly half immersed in a liquid of density $\rho $
and surface tension T. The radius of the droplet
is (take note that the surface tension applies an
upward force on the droplet) :
A.
$r = \sqrt {{T \over {\left( {d - \rho } \right)g}}} $
B.
$r = \sqrt {{{2T} \over {3\left( {d + \rho } \right)g}}} $
C.
$r = \sqrt {{T \over {\left( {d + \rho } \right)g}}} $
D.
$r = \sqrt {{{3T} \over {\left( {2d - \rho } \right)g}}} $
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 9th January Evening Slot
Two steel wires having same length are
suspended from a ceiling under the same load.
If the ratio of their energy stored per unit
volume is 1 : 4, the ratio of their diameters is:
A.
1 : 2
B.
2 : 1
C.
$1:\sqrt 2 $
D.
$\sqrt 2 :1$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 9th January Morning Slot
Water flows in a horizontal tube (see figure).
The pressure of water changes by 700 Nm–2
between A and B where the area of cross section
are 40 cm2 and 20 cm2, respectively. Find the
rate of flow of water through the tube.
(density of water = 1000 kgm–3)
The pressure of water changes by 700 Nm–2
between A and B where the area of cross section
are 40 cm2 and 20 cm2, respectively. Find the
rate of flow of water through the tube.
(density of water = 1000 kgm–3)
A.
1810 cm3/s
B.
2420 cm3/s
C.
3020 cm3/s
D.
2720 cm3/s
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 8th January Evening Slot
Two liquids of densities ${\rho _1}$ an ${\rho _2}$ (${\rho _2}$ = 2${\rho _1}$) are
filled up behind a square wall of side 10 m as
shown in figure. Each liquid has a height of
5 m. The ratio of the forces due to these liquids
exerted on upper part MN to that at the lower part
NO is (Assume that the liquids are not mixing)
A.
1/3
B.
1/2
C.
1/4
D.
2/3