The velocity vector v and displacement vector x of a particle executing SHM are related as frac{v d v}{d x}=-omega^2 x with the initial condition

Q: The velocity vector $v$ and displacement vector $x$ of a particle executing SHM are related as $\frac{v d v}{d x}=-\omega^2 x$ with the initial condition $v=v_0$ at $x=0$. The velocity $v$, when displacement is $x$, is

b (b) As it is $SHM$ so the equation of motion will be $F=- kx$ or $\frac{ vdv }{ dx }=-\omega^2 x$ Now integrating the expression with boundary condition, $\int \limits_{ v _0}^v v d v=-\omega^2 \int \limits_0^\pi xdx$ or $\frac{1}{2}\left[v^2-v_0^2\right]=-\frac{\omega^2 x^2}{2}$ or $v =\sqrt{ v _0^2-\omega^2 x ^2}$

SECTION - A [PHYSICS - MCQ]

GSEB - English Medium

The velocity vector $v$ and displacement vector $x$ of a particle executing SHM are related as $\frac{v d v}{d x}=-\omega^2 x$ with the initial condition $v=v_0$ at $x=0$. The velocity $v$, when displacement is $x$, is

A$v=\sqrt{v_0^2+\omega^2 x^2}$
B$v=\sqrt{v_0^2-\omega^2 x^2}$
C$v=\sqrt[3]{v_0^3+\omega^3 x^3}$
D$v=v_0-\left(\omega^3 x^3 e^{x^3}\right)^{1 / 3}$

AIIMS 2015, Diffcult

STD 11 Science
NEET
STD 11 - 13. oscillations
Physics

Download our app
and get started for free

Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*

Download App

Similar Questions

1
A body is executing simple harmonic motion with frequency $'n',$ the frequency of its potential energy is :
View Solution
2
In the figure, ${S_1}$ and ${S_2}$ are identical springs. The oscillation frequency of the mass $m$ is $f$. If one spring is removed, the frequency will become
View Solution
3
A simple pendulum of length $l$ is made to oscillate with an amplitude of $45$ degrees. The acceleration due to gravity is $g$. Let $T_0=2 \pi \sqrt{l / g}$. The time period of oscillation of this pendulum will be
View Solution
4
In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $k,$ the frequency of oscillation of given body is :
View Solution
5
A simple pendulum is suspended from the roof of a trolley which moves in a horizontal direction with an acceleration $a$, then the time period is given by $T = 2\pi \sqrt {\frac{l}{{g'}}} $, where $g'$ is equal to
View Solution
6
To make the frequency double of a spring oscillator, we have to
View Solution
7
The angular velocities of three bodies in simple harmonic motion are ${\omega _1},\,{\omega _2},\,{\omega _3}$ with their respective amplitudes as ${A_1},\,{A_2},\,{A_3}$. If all the three bodies have same mass and velocity, then
View Solution
8
Which of the diagrams shown in figure represents variation of total mechanical energy of a pendulum oscillating in water as function of time ?
View Solution
9
A small body of mass $0.10\, kg$ is executing $S.H.M.$ of amplitude $1.0 \,m$ and period $0.20\, sec$. The maximum force acting on it is.... $N$
View Solution
10
A person of mass $M$ is, sitting on a swing of length $L$ and swinging with an angular amplitude $\theta_0$. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance $l\, ( l < < L)$, is close to
View Solution

Download our appand get started for free

Similar Questions

Download our app
and get started for free