The displacement of a body executing SHM is given by x = A sin (2\$\pi t\$ + \$\dfrac{\pi} {3}\$). The first time from t = 0 when the velocity is maximum is:

The displacement of a body executing SHM is given by: \$x=Asin(2\pi t+\pi/3)\$Velocity, \$v=\dfrac{dx}{dt}=2\pi Acos(2\pi t+\pi/3)\$For maximum velocity, \$\dfrac{dv}{dt}=0\$\$\Rightarrow-4\pi^2 A sin(2\pi t+\pi/3)=0\$\$\Rightarrow sin(2\pi t+\pi/3)=sin(n\pi)\$\$\Rightarrow2\pi t+\pi/3=n\pi\$for \$n=1\$\$\Rightarrow2\pi t=\pi-\pi/3=\dfrac{2\pi}{3}\$\$t=\dfrac{1}{3}=0.33sec\$The correct option is A.

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Oscillations

Q: A small block is connected to one end of a massless spring of unstretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t=0. lt then executes simple harmonic motion with angular frequency \$\omega =\frac { \pi }{ 3 } rad/s\$. Simultaneously at t=0, a small pebble is projected with speed v from point P at an angle of 45 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is (take \$g = 10 m/{ s }^{ 2 }\$)

Time is given by \$\displaystyle t=\frac{2vsin \theta}{g}\$\$\displaystyle 1= \frac{2vsin45^{o}}{10}\$\$\displaystyle v= \sqrt{50}m/s\$

Q: A mass \$m\$ is undergoing SHM in the vertical direction about the mean position \$y_{0}\$ with amplitude A and angular frequency \$\omega\$. At a distance \$y\$ from the mean position, the mass detaches from the spring. Assume that the spring contracts and does not obstruct the motion of \$m\$. Find the distance \$y^{*}\$ (measured from the mean position) such that the height \$h\$ attained by the block is maximum. \$(\mathrm{A}\mathrm{\omega})^{2}>g\$

Let \$v\$ be the upward velocity of the block when it detaches from the spring at a distance \$y\$ above the mean position.\$v = w \sqrt{A^2 - y^2} \implies v^2 = w^2 (A^2 - y^2)\$ ..........(1)At the detaching point, the block will continue to move upwards due to inertia and reach the height \$h\$ above the mean position. At that height the velocity of the block will be zero.Thus \$0 - v^2 = 2(-g)s \implies s = \dfrac{v^2}{2g}\$Now the total height \$h =s + y = \dfrac{v^2}{2g} + y \$\$ \therefore h = \dfrac{w^2 (A^2 - y^2)}{2g} + y \$For \$h\$ to be maximum, \$\dfrac{dh}{dy} = 0\$Thus \$\dfrac{w^2}{2g} (-2 y) + 1 = 0\$\$\implies y = \dfrac{g}{w^2}\$

Q: A particle of mass m moves due to a conservative force with potential energy V(x) = \$\frac{Cx}{x^{2}+a^{2}}\$, where C and a are positive constants. The position(s) of stable equilibrium is/are given as

At equilibrium position\$\displaystyle \mathrm{F}=-\dfrac{dV}{dx}=0\$\$\dfrac{dV}{dx }=\dfrac{C(a^{2}-x ^{2})}{(x ^{2}+a^{2})^{2}}=0\$\$\therefore \$ There are two equilibrium positions\$x_1 = a\$\$x_2 = -a\$Consider \$\dfrac{d^{2}V}{dx ^{2}}=\dfrac{2Cx (x ^{2}-3a^{2})}{(x^{2}+a^{2})^{3}}\$\$\dfrac{d^{2}V}{dx ^{2}}|_{x_1} 0\$\$\because \$There is a maxima at \$x=a\$ and minima at \$x=-a\$\$\Rightarrow x _{1}\$ is a position of unstable equilibrium and \$x _{2}\$ is a position of stable equilibrium.

Q: A block rides on a piston that is moving vertically with simple harmonic motion. The maximum speed of the piston is \$2m/s\$. At what amplitude of motion will the block and piston separate? \$(g=10m/s^{2})\$

Let \$A\$ be amplitudeNow, maximum speed of piston \$ = 2 m/s\$\$\therefore w A = 2 m/s\$If \$w^{2} A = g \$ then they will separate.\$\Rightarrow 2\times w = g\$\$\Rightarrow w = 5 rad/sec\$\$\therefore A = \dfrac{2}{5} m = 40 cm\$

Q: The angular frequency of a spring block system is \$\omega _{0}\$ This system is suspended from the ceiling of anelevator moving downwards with a constant speed v\$_{0}\$. The block is at rest relative to the elevator. Lift issuddenly stopped. Assuming the downwards as a positive direction, choose the wrong statement:

Since \${ V }_{ 0 }\$ is the maximum speed of the ball in its equilibrium position.\$\therefore { V }_{ 0 }={ \omega }_{ 0 }A\\ A=\dfrac { { V }_{ 0 } }{ { \omega }_{ 0 } }\$So equation of motion becomes \$\dfrac { { V }_{ 0 } }{ { \omega }_{ 0 } } =Asin{ \omega }_{ 0 }t\$ as there was no initial phase in the SHM.There was no initial phase in the SHM.Hence B is correct.

Physics
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NEET AIIMS BITSAT Easy Qs Med Qs Hard Qs

A small block is connected to one end of a massless spring of unstretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t=0. lt then executes simple harmonic motion with angular frequency $ω = \frac{π}{3} r a d / s$ .
Simultaneously at t=0, a small pebble is projected with speed v from point P at an angle of 45 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is
(take $g = 10 m / s^{2}$ )

50 m / s

51 m / s

52 m / s

53 m / s

Hard

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In the above question, the velocity of the rear 2 kg block after it separates from the spring will be :

0 m/s

5 m/s

10 m/s

7.5 m/s

Hard

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A mass $m$ is undergoing SHM in the vertical direction about the mean position $y_{0}$ with amplitude A and angular frequency $ω$ . At a distance $y$ from the mean position, the mass detaches from the spring. Assume that the spring contracts and does not obstruct the motion of $m$ . Find the distance $y^{*}$ (measured from the mean position) such that the height $h$ attained by the block is maximum. $(A ω)^{2} > g$

\frac{g}{ω ^{2}}

\frac{2 g}{ω ^{2}}

\frac{g}{2 ω ^{2}}

None of these

Hard

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A particle of mass m moves due to a conservative force with potential energy V(x) = $\frac{C x}{x ^{2} + a ^{2}}$ , where C and a are positive constants. The position(s) of stable equilibrium is/are given as

x = + a

only

x = - a

only

x = - \frac{a}{2}

and

+ \frac{a}{2}

x = - a

and

+ a

Hard

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A particle of mass m moves under a conservative force with potential energy. $V (x) = \frac{C x}{a ^{2} + x ^{2} ,}$ where $C$ and $a$ are positive constants.
If the practicle starts from a point with velocity $v$ , the range of values of $v$ for which it escapes to - $\infty$ are given by

< \frac{C}{m a}

> \frac{C}{m a}

> \frac{2 C}{m a}

< \frac{2 C}{m a}

Hard

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A mass of 0.2kg is attached to the lower end of a massless spring of force-constant 200 N/m, the upper end of which is fixed to a rigid support. Which of the following statements is/are true?

In equilibrium, the spring will be stretched by 1cm.

If the mass is raised till the spring is unstretched state and then released, it will go down by 2cm

before moving upwards.

The frequency of oscillation will be nearly 5 Hz

If the system is taken to the moon, the frequency of oscillation will be the same as on the earth.

Hard

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A block rides on a piston that is moving vertically with simple harmonic motion. The maximum speed of the piston is $2 m / s$ . At what amplitude of motion will the block and piston separate? $(g = 10 m / s^{2})$

20 c m

30 c m

40 c m

50 c m

Hard

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The displacement of a body executing SHM is given by x = A sin (2 $π t$ + $\frac{π}{3}$ ). The first time from t = 0 when the velocity is maximum is:

0.33 sec

0.16 sec

0.25 sec

0.5 sec

Hard

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A particle of mass 200 g executes linear simple harmonic motion with an amplitude 10 cm. When the particles at a point midway between the mean and the extreme position, its kinetic energy is $3 π^{2} \times 1 0^{- 3} J$ . Assuming the initial phase to be $\frac{2 π}{3}$ , the equation of motion of the particle will be :

y=10 sin

(2 π t + \frac{2 π}{3})

y=10 sin

(4 π t + \frac{2 π}{3})

y=10 cos

(2 π t + \frac{π}{6})

y=10 cos

(2 π t + \frac{π}{3})

Hard

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The angular frequency of a spring block system is $ω_{0}$ This system is suspended from the ceiling of anelevator moving downwards with a constant speed v $_{0}$ . The block is at rest relative to the elevator. Lift issuddenly stopped. Assuming the downwards as a positive direction, choose the wrong statement:

The amplitude of the block is

\frac{v _{0}}{ω _{0}}

The initial phase of the block is

π

The equation of motion for the block is

\frac{v _{0}}{ω _{0}} sin ω_{0}

The maximum speed of the block is v

_{0}

Hard

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Oscillations

Physics 4861 Views

In the above question, the velocity of the rear 2 kg block after it separates from the spring will be :

A particle of mass m moves due to a conservative force with potential energy V(x) = x2+a2Cx​, where C and a are positive constants. The position(s) of stable equilibrium is/are given as

A particle of mass m moves under a conservative force with potential energy. V(x)=a2+x2​,Cx​ where C and aare positive constants.If the practicle starts from a point with velocity v, the range of values of v for which it escapes to - ∞ are given by

A mass of 0.2kg is attached to the lower end of a massless spring of force-constant 200 N/m, the upper end of which is fixed to a rigid support. Which of the following statements is/are true?

A block rides on a piston that is moving vertically with simple harmonic motion. The maximum speed of the piston is 2m/s. At what amplitude of motion will the block and piston separate? (g=10m/s2)

The displacement of a body executing SHM is given by x = A sin (2πt + 3π​). The first time from t = 0 when the velocity is maximum is:

A particle of mass 200 g executes linear simple harmonic motion with an amplitude 10 cm. When the particles at a point midway between the mean and the extreme position, its kinetic energy is 3π2×10−3J. Assuming the initial phase to be 32π​, the equation of motion of the particle will be :

Physics
4861 Views

A particle of mass m moves due to a conservative force with potential energy V(x) = $\frac{C x}{x ^{2} + a ^{2}}$ , where C and a are positive constants. The position(s) of stable equilibrium is/are given as

A particle of mass m moves under a conservative force with potential energy. $V (x) = \frac{C x}{a ^{2} + x ^{2} ,}$ where $C$ and $a$ are positive constants.
If the practicle starts from a point with velocity $v$ , the range of values of $v$ for which it escapes to - $\infty$ are given by

A block rides on a piston that is moving vertically with simple harmonic motion. The maximum speed of the piston is $2 m / s$ . At what amplitude of motion will the block and piston separate? $(g = 10 m / s^{2})$

The displacement of a body executing SHM is given by x = A sin (2 $π t$ + $\frac{π}{3}$ ). The first time from t = 0 when the velocity is maximum is: