Mechanics – Page 2

Mirror on a spring

January 19th, 2022jeefirst0 Comments

JEE Advanced 2019 Paper 2, Question 12

A perfectly reflecting mirror of mass $M$ mounted on a spring constitutes a spring-mass system of angular frequency $\Omega$ such that $\frac{4 \pi M \Omega}{h} = 10^{24} \, {\rm m}^{-2}$ with $h$ as Planck’s constant. $N$ photons of wavelength $\lambda = 8 \pi \times 10^{-6} \, {\rm m}$ strike the mirror simultaneously at normal incidence such that the mirror gets displaced by $1 \mu{\rm m}$ .
If the value of $N$ is $x \times 10^{12}$ , what is the value of $x$ ?

[Consider the spring as massless]

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Solution

The momentum carried by a single photon is given by the de Broglie relation $p_{\rm ph} = h/\lambda$ . The total momentum carried by $N$ photons is therefore $N \times p_{\rm ph} = Nh/\lambda$ . All the photons hit the mirror simultaneously and are reflected …

Particle in a tube

January 16th, 2022jeefirst0 Comments

JEE Advanced 2019 Paper 2, Question 3

A small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from the closed end is $L=L_{0}$ the particle speed is $v=v_{0} .$ The piston is moved inward at a very low speed $V$ such that $V \ll \frac{{\rm d} L}{L} v_{0}$ , where ${\rm d} L$ is the infinitesimal displacement of the piston. Which of …

Pulleys and masses connected to a spring

January 14th, 2022jeefirst0 Comments

JEE Advanced 2019 Paper 2, Question 2

A block of mass $2 M$ is attached to a massless spring with spring-constant $k$ . This block is connected to two other blocks of masses $M$ and $2 M$ using two massless pulleys and strings. The accelerations of the blocks are $a_{1}, a_{2}$ and $a_{3}$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_{0}$ . Which of the following option(s) is/are correct?

[ $g$ is the acceleration due to gravity. Neglect friction]

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$x_{0}=\frac{4 M g}{k}$
When spring achieves an extension of $\frac{x_{0}}{2}$ for the first time, the speed of the

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A falling rod

January 11th, 2022jeefirst0 Comments

JEE Advanced 2019 Paper 2, Question 1

A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle $60^{\circ}$ with vertical?

[ $g$ is the acceleration due to gravity]

The angular speed of the rod will be $\sqrt{\frac{3 g}{2 L}}$
The angular acceleration of the rod will be $\frac{2 g}{L}$
The radial acceleration of the rod’s center of mass will be $\frac{3 g}{4}$
The normal reaction force from the

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Infinitely bouncing ball

January 10th, 2022jeefirst0 Comments

JEE Advanced 2019 Paper 2, Question 9

A ball is thrown from ground at an angle $\theta$ with horizontal and with an initial speed $u_{0}$ . For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $V_{1}$ . After hitting the ground, the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_{0}/\alpha$ . Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8 V_{1}$ , what is the …