Practice Questions

Q8. Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, T1 and T2 . The temperature of the hot reservoir of the first engine is T1 and the temperature of the cold reservoir of the second engine is T2 . T is temperature of the sink of first engine which is also the source for the second engine. How is T related to T1 and T2 , if both the engines perform equal amount of work? JEE Main 2020 (07 Jan Shift 2) JEE Main Previous Year Paper (1) T = 2T1T2 (2) T = T1+T22 T1+T2 (3) T = √T1T2 (4) T = 0

Q8. In a dilute gas at pressure P and temperature ' t', the time between successive collision of a molecule varies with T as : (1) T (2) 1 √T (3) 1 (4) √T T

Q8. 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: (1) √2 : 1 (2) 1 : 2 (3) 2 : 1 (4) 1 : √2

Q8. Two isolated conducting spheres S1 and S2 of radius 32 R and 31 R have 12 μC and −3 μC charges, respectively, and are at a large distance from each other, They are now connected by a conducting wire. A long time after this is done the charges on S1 and S2 are respectively: (1) 4. 5 μC of both (2) +4. 5 μC and −4. 5 μC (3) 3 μC and 6 μC (4) 6 μC and 3 μC

Q8. 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 ω rad s−1 . The difference in the height, h (in cm) of liquid at the Centre of vessel and at the sides of the vessel will be : JEE Main 2020 (02 Sep Shift 1) JEE Main Previous Year Paper (1) 2ω2 (2) 5ω2 25g 2g (3) 25ω2 (4) 2ω2 2g 5g

Q8. Consider two ideal diatomic gases A and B at some temperature T . Molecules of the gas A are rigid, and have a mass m . Molecules of the gas B have an additional vibrational mode and have a mass m . The ratio of the 4 specific heats (C V )Aand ( CV )B of gas A and B, respectively is: JEE Main 2020 (09 Jan Shift 1) JEE Main Previous Year Paper (1) 7 : 9 (2) 5 : 9 (3) 3 : 5 (4) 5 : 7

Q8. Match the Cp ratio for ideal gases with different type of molecules: Cv Molecule Type Cp/Cv (A) Monoatomic (I) 7/5 (B) Diatomic rigid molecules (II) 9/7 (C) Diatomic non-rigid molecules (III) 4/3 (D) Triatomic rigid molecules (IV) 5/3 (1) (A) – (IV), (B) – (II), (C) – (I), (D) – (III) (2) (A) – (III), (B) – (IV), (C) – (II), (D) – (I) (3) (A) – (IV), (B) – (I), (C) – (II), (D) – (III) (4) (A) – (II), (B) – (III), (C) – (I), (D) – (IV)

Q8. Number of molecules in a volume of 4 cm3 of a perfect monoatomic gas at some temperature T and at a pressure of 2 cm of mercury is close to? (Given, mean kinetic energy of a molecule (at T) is 4 × 10−14 erg, g = 980 cm s−2 density of mercury = 13. 6 g cm−3 ) (1) 4 .8 ×1018 (2) 4. 0 × 1016 (3) 5. 8 × 1016 (4) 5. 8 × 1018

Q8. The displacement time graph of a particle executing SHM is given in figure: (sketch is schematic and not to scale) Which of the following statements is/are true for this motion? (A) The force is zero at t = 3T4 (B) The magnitude of acceleration is maximum at t = T (C) The speed is maximum at t = T4 (D) The P. E. is equal to K. E. of the oscillation at t = T2 (1) (A), (B) and (C) (2) (B), (C) and (D) (3) (A), (B) and (D) (4) (A) and (D)

Q8. A cube of metal is subjected to a hydrostatic pressure 4 GPa . The percentage change in the length of the side of the cube is close to : (Given bulk modulus of metal, B = 8 × 1010 Pa ) (1) 5 (2) 0. 6 (3) 20 (4) 1. 67

Q8. A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period T1 and (ii) back and forth in a direction perpendicular to its plane, with a period T2. The ratio T1 will be : T2 (1) 2 (2) 2 √3 3 (3) 3 (4) √2 √2 3

Q8. An object of mass m is suspended at the end of a massless wire of length L and area of cross-section, 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 : (1) 1 (2) 1 A f = 2π √mLY A f = 2π √YmL (3) 1 (4) 1 L f = 2π √mAV L f = 2π √YmA

Q9. To raise the temperature of a certain mass of gas by 50 °C at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by 100°C at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal)? (1) 5 (2) 6 (3) 3 (4) 7

Q9. 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 : (1) gdS(x22 + x21) (2) gdS(x2 + x1)2 (3) 4 3 gdS(x2 −x1)2 (4) 41 gdS(x2 −x1)2

Q9. A small spherical droplet of density d is floating exactly half immersed in a liquid of density ρ and surface tension T. The radius of the droplet is (take note that the surface tension applies an upward force on the droplet): r = (1) r = √ 3(d+ρ)g2T (2) √ (d−ρ)gT (3) T (4) 3T r = r = √ (d+ρ)g √ (2d−ρ)g

Q9. A charge Q is distributed over two concentric conducting thin spherical shells radii r and R (R > r) . If the surface charge densities on the two shells are equal, the electric potential at the common centre is : (R+r) (2R+r) 1 1 (1) Q (2) Q 2(R2+r2) (R2+r2) 4πϵ0 4πϵ0 (3) 1 (R+2r)Q (4) 1 (R+r) Q 4πϵ0 2(R2+r2) 4πϵ0 (R2+r2)

Q9. Assuming the nitrogen molecule is moving with r. m. s. velocity at 400 K, the de-Broglie wave length of nitrogen molecule is close to : (Given : nitrogen molecule weight : 4 .64 ×10−26 kg, Boltzman constant : 1 .38 ×10−23 J K−1 , Planck constant : 6 .63 ×10−34 J s) ∘ ∘ (1) (2) 0. 24 A 0. 20 A ∘ ∘ (3) (4) 0. 34 A 0. 44 A

Q9. The mass density of a spherical galaxy varies as K over a large distance r from its center. In that region, a small r star is in a circular orbit of radius R. Then the period of revolution, T depends on R as: (1) T 2 ∝ R (2) T 2 ∝ R3 (3) T 2 ∝ 1 (4) T ∝R R3

Q9. A bullet of mass 5 gram, travelling with a speed of 210 m s−1 strikes a fixed wooden target. One half of its kinetic energy is converted into heat in the wood. The rise of temperature of the bullet if the specific heat of its material is 0. 030 (gram °C)−1 (1 calorie = 4 .2 ×107 ergs) close to : (1) 87. 5°C (2) 83. 3°C (3) 119. 2°C (4) 38. 4°C

Q9. In the circuit shown in the figure, the total charge is 750 μC and the voltage across capacitor C2 is 20 V. Then the charge on capacitor C2 is : (1) 450 μC (2) 590 μC (3) 160 μC (4) 650 μC

Q9. A driver in a car, approaching a vertical wall notices that the frequency of his car horn has changed from 440 Hz to 480 Hz, when it gets reflected from the wall. If the speed of sound in air is 345 m s−1, then the speed of the car is: (1) 54 km/hr (2) 36 km/hr (3) 18 km/hr (4) 24 km/hr

Q9. Three harmonic waves having equal frequency v and same intensity I0 , have phase angles 0, π4 and −π4 respectively. When they are superimposed the intensity of the resultant wave is close to: (1) 5.8I0 (2) 0.2I0 (3) 3I0 (4) I0

Q9. For a transvers wave travelling, along a straight line, the distance between two peaks (crests) is 5 m , while the distance between one crest and one trough is 1. 5 m. The possible wavelengths (in m ) of the waves are: (1) 1, 3, 5 (2) 11 , 13 , 51 , . . . . . (3) 1, 2, 3, . . . . (4) 12 , 14 , 61 , . . . . .

Q9. Consider a mixture of n moles of helium gas and 2n moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its CP value will be: CV (1) 19 (2) 67 13 45 (3) 40 (4) 23 27 15

Q9. Two infinite planes each with uniform surface charge density +σ are kept in such a way that the angle between them is 30o . The electric field in the region shown between them is given by: (1) + 2∈0 σ [(1 √3)ˆy −ˆx2 ] (2) ∈0σ [(1 + √32 )ˆy + ˆx2 ] (3) + 2∈0 σ [(1 √3)ˆy + ˆx2 ] (4) 2∈0σ [(1 −√32 )ˆy −ˆx2 ]