JEE Advanced – Page 6

Elongation due to a suspended weight

January 6th, 2022jeefirst0 Comments

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 …

Doppler effect and beats

January 6th, 2022jeefirst0 Comments

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 …

Convex lens with two materials

January 2nd, 2022jeefirst0 Comments

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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$\bigg| \frac{\Delta f}{f} \bigg| < \bigg| \frac{\Delta n}{n} \bigg|$
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).
If $\frac{\Delta n}{n} < 0$ then $\frac{\Delta f}{f} > 0$
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 …

Flux from a charged shell

December 30th, 2021jeefirst0 Comments

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]

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

Related problems:
Electric field from a sphere …

Conducting wire in a magnetic field

December 30th, 2021jeefirst0 Comments

JEE Advanced 2019 Paper 1, Question 6

A conducting wire of parabolic shape, initially $y = x^2$ , is moving with velocity $\vec v = v_0 \hat i$ in a non-uniform magnetic field $\vec B = B_0 \left( 1 + \left( \frac{y}{L} \right)^\beta \right) \hat k$ , as shown in the figure below. If $v_0, B_0, L$ and $\beta$ are positive constants and $\Delta \phi$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

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$|\Delta \phi| = \frac{1}{2} B_0 v_0 L$ for $\beta = 0$ .
$|\Delta \phi| = \frac{4}{3} B_0 v_0 L$ for $\beta = 2$ .
$|\Delta \phi|$ remains the same if the parabolic wire is replaced by a straight wire, $y=x$ initially, of length $\sqrt{2} L$ .
$|\Delta \phi|$ is proportional to the length of the wire projected on the $y$ axis.