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A spherical body of mass m and radius r is allowed to fall in a medium of viscosity η. The time in which the velocity of the body increases from zero to 0.63 times the terminal velocity (v) is called time constant (τ). Dimensionally τ can be represented by:
The distance covered by a particle undergoing SHM in one time period is: (amplitude = $A$)
To get output $1$ for the following circuit, the correct choice for the input is:
An excited heavy nucleus ${}_{Z}^{A}\mathrm{X}$ emits radiations in the following sequence: $${}_{Z}^{A}\mathrm{X} \rightarrow {}_{Z-2}^{A-4}\mathrm{D}_1 \rightarrow {}_{Z-1}^{A-4}\mathrm{D}_2 \rightarrow {}_{Z-3}^{A-8}\mathrm{D}_3 \rightarrow {}_{Z-4}^{A-8}\mathrm{D}_4 \rightarrow {}_{Z-4}^{A-8}\mathrm{D}_5$$ where $Z, A$ are the atomic and mass number of element $\mathrm{X}$, respectively. The possible emitted particles or radiations in the sequence, respectively are:
When an object is shot from the bottom of a long, smooth inclined plane kept at an angle of 60$^{\circ}$ with horizontal, it can travel a distance $x_1$ along the plane. But when the inclination is decreased to 30$^{\circ}$ and the same object is shot with the same velocity, it can travel $x_2$ distance. Then $x_1:x_2$ will be:
A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k'$. Then they are connected in parallel and the force constant is $k''$. Then $k':k''$ is:
Two cars moving in opposite directions approach each other with speed of $22 \text{ m/s}$ and $16.5 \text{ m/s}$ respectively. The driver of the first car blows a horn having a frequency $400 \text{ Hz}$. The frequency heard by the driver of the second car is [velocity of sound $340 \text{ m/s}$]
Find the value of the angle of emergence from the prism given below for the incidence ray shown. The refractive index of the glass is $\sqrt{3}$.
The rms value of potential difference V shown in the figure is
An intrinsic semiconductor is converted into an n-type extrinsic semiconductor by doping it with:
Two simple harmonic motions of angular frequencies $100$ and $1000 \text{ rad s}^{-1}$ have the same displacement amplitude. The ratio of their maximum acceleration is:
Two particles of masses $m_1, m_2$ move with initial velocities $u_1$ and $u_2$. On collision, one of the particles gets excited to a higher level, after absorbing energy $\varepsilon$. If the final velocities of the particles are $v_1$ and $v_2$, then we must have:
Two points are located at a distance of $10\text{ m}$ and $15\text{ m}$ from the source of oscillation. The period of oscillation is $0.05\text{ s}$ and the velocity of the wave is $300\text{ m/s}$. What is the phase difference between the oscillations of two points?
The displacement of a particle executing simple harmonic motion is given by $y = A_0 + A \sin \omega t + B \cos \omega t$. Then the amplitude of its oscillation is given by:
A second's pendulum is mounted in a rocket. Its period of oscillation decreases when the rocket:
Two planets orbit a star in circular paths with radii $R$ and $4R$, respectively. At a specific time, the two planets and the star are aligned in a straight line. If the orbital period of the planet closest to the star is $T$, what is the minimum time after which the star and the planets will again be aligned in a straight line (at the initial position)?
A rocket is fired vertically upward with a speed of $v = \frac{v_e}{\sqrt{2}}$ from the Earth's surface, where $v_e$ is escape velocity on the surface of Earth. The distance from the surface of Earth upto which the rocket can go before returning to the Earth is: (given, the radius of Earth $R = 6400$ km)
A particle executes simple harmonic oscillation with an amplitude $A$. The period of oscillation is $T$. The minimum time taken by the particle to travel half of the amplitude from the equilibrium position is:
Out of the following functions, which represents SHM? I. $y = \sin \omega t - \cos \omega t$ II. $y = \sin^3 \omega t$ III. $y = 5 \cos\left(\frac{3\pi}{4} - 3\omega t\right)$ IV. $y = 1 + \omega t + \omega^2 t^2$
A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is $20 \text{ m/s}^2$ at a distance of $5 \text{ m}$ from the mean position. The time period of oscillation is: