Elongation due to a suspended weight

JEE Advanced 2019 Paper 1, Question 14

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area 0.5 cm^2 and, length \sqrt{3} m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30^{\circ} and 60^{\circ}, respectively.
If elongation in copper wire is \left(\Delta l_{C}\right) and elongation in steel wire is \left(\Delta l_{S}\right), then the ratio \frac{\Delta l_{C}}{\Delta l_{S}} is

[Young’s modulus for copper and steel are 1 \times 10^{11} N/m^2 and 2 \times 10^{11} N/m^2, respectively.]

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Solution

We will label the tension in the steel and copper wires by F_S and …

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Doppler effect and beats

JEE Advanced 2019 Paper 1, Question 15

A train S1, moving with a uniform velocity of 108 km/h, approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2, as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When \mathrm{O} is 600 m away from S2 and distance between \mathrm{S} 1 and \mathrm{S} 2 is 800 m, what is the number of beats heard by O?

[Speed of the sound =330 m/s ]

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Recommended reading: How does Doppler effect work?

Solution

When there is relative motion between the source and observer, the observed frequency f_o is …

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Convex lens with two materials

JEE Advanced 2019 Paper 1, Question 10

A thin convex lens is made of two materials with refractive indices n_1 and n_2 as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. f is the focal length of the lens when n_1 = n_2 = n. The focal length is f + \Delta f when n_1 = n and n_2 = n + \Delta n. Assuming \Delta n \ll (n-1) and 1 < n < 2, the correct statement(s) is/are,

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  1. \bigg| \frac{\Delta f}{f} \bigg| < \bigg| \frac{\Delta n}{n} \bigg|
  2. For n=1.5, \Delta n = 10^{-3} and f = 20 cm, the value of |\Delta f| will be 0.02 cm (round off to 2^{\rm nd} decimal place).
  3. If \frac{\Delta n}{n} < 0 then \frac{\Delta f}{f} > 0
  4. The relation between \frac{\Delta f}{f} and \frac{\Delta n}{n} remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.

Related

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Thermodynamic cycle on a VT diagram

JEE Advanced 2019 Paper 1, Question 9

One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume vs. temperature (VT) diagram. The correct statement(s) is/are:

[R is the gas constant]

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  1. Work done in this thermodynamic cycle (1 \to 2 \to 3 \to 4 \to 1) is |W| = \frac{1}{2} R T_0.
  2. The above thermodynamic cycle exhibits only isochoric and adiabatic processes.
  3. The ratio of heat transfer during processes 1 \to 2 and 2 \to 3 is \left| \frac{Q_{1 \to 2}}{Q_{2 \to 3}} \right| = \frac{5}{3}.
  4. The ratio of heat transfer during processes 1 \to 2 and 3 \to 4 is \left| \frac{Q_{1 \to 2}}{Q_{3 \to 4}} \right| = \frac{1}{2}.

Related article: Thermodynamic processes on an ideal gas

Solution

An ideal gas obeys the relations

(1)   \begin{equation*}   \text{The ideal gas law: } P V = n R T \end{equation*}

(2)   \begin{equation*}   \text{Work done {\it by} the gas: } W = \int_a^b P dV \end{equation*}

(3)   \begin{equation*}   \text{Internal energy of a monatomic gas: } U = \frac{3}{2} n R T \end{equation*}

We …

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Flux from a charged shell

JEE Advanced 2019 Paper 1, Question 8

A charged shell of radius R carries a total charge Q. Given \Phi as the flux of electric field through a closed cylindrical surface of height h, radius r
and with its center same as that of the shell. Here, the center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct?

[\epsilon_0 is the permittivity of free space]

  1. If h > 2R and r > R then \Phi = Q/\epsilon_0
  2. If h < 8R/5 and r = 3R/5 then \Phi = 0
  3. If h > 2R and r = 3R/5 then \Phi = Q/5\epsilon_0
  4. If h > 2R and r = 4R/5 then \Phi = Q/5\epsilon_0

Related problems:
Electric field from a sphere

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