Oscillations Test

Result

Oscillations Test - 53

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Weekly Quiz Competition

Question 1
1 / -0

A string fixed at both ends, oscillate in $$4^{th}$$ harmonic. The displacement of a particle of string is given as:
$$Y = 2A \sin (5\pi x)\cos (100\pi t)$$. Then find the length of the string?

A
$$80cm$$

B
$$100cm$$

C
$$60cm$$

D
$$120cm$$

Solution

$$\dfrac{2\pi}{\lambda}=5\pi$$
$$4\left(\dfrac{\lambda}{2}\right)= \ell \Rightarrow 2\lambda = \ell \Rightarrow 2\times \dfrac{2}{5} = \ell$$
$$\Rightarrow \ell = 80cm$$
Question 2
1 / -0

A particle is executing the motion $$x = a\cos (\omega t - \theta)$$. The maximum velocity of the particle is

A
$$a\omega \cos \theta$$

B
$$a\omega$$

C
$$a\omega \sin \theta$$

D
None of these

Solution
Question 3
1 / -0

A spring has a spring constant of $$6.0$$ N $$cm^{-1}$$. It is joined to another spring whose spring constant is $$4.0$$ N $$cm^{-1}$$. A load of $$80$$N is suspended from this composite spring.
What is the extension of this composite spring?

A
$$8.0$$cm

B
$$16$$cm

C
$$17$$cm

D
$$33$$cm

Solution

For series combination: $$x_{eff}=x_1+x_2$$
$$k_1x_1+k_2x_2=F$$
$$x_1=\dfrac{F}{k_1} \ , x_2=\dfrac{F}{k_2}$$
$$x_{eff}=\dfrac{F}{k_1}+\dfrac{F}{k_2}$$
$$x_{eff}=20+13.33=33.33cm$$
$$approx\ x_{eff}=33cm$$
Question 4
1 / -0

A spring of length $$'l'$$ has spring constant $$'k'$$ is cut into two parts of length $$l_{1}$$ and $$l_{2}$$. If their respective spring constants are $$k_{1}$$ and $$k_{2}$$, then $$\dfrac {k_{1}}{k_{2}}$$ is

A
$$\dfrac {l_{2}}{l_{1}}$$

B
$$\dfrac {2l_{2}}{l_{1}}$$

C
$$\dfrac {l_{1}}{l_{2}}$$

D
None

Solution

$$k_{1}l_{1} = k_{2}l_{2} = kl$$
$$\dfrac {k_{1}}{k_{2}} = \dfrac {l_{2}}{l_{1}}$$
Question 5
1 / -0

The displacement of a damped harmonic oscillator is given by $$x(t)=e^{-01.1t}\, cos (10\pi t+\Phi ).$$ Here $$t$$ is in seconds. The time taken for its amplitude of vibration to drop to half for its initial value is close to :

A
$$13s$$

B
$$7s$$

C
$$27s$$

D
$$4s$$

Solution

$$A=A_0e^{-0.1t}=\dfrac{A_0}{2}$$
$$ln2=0.1t$$
$$t=10ln2=6.93 \approx 7 sec$$
Question 6
1 / -0

A particle of mass $$m$$ is moving along the X-axis under the potential $$U(x)=\dfrac{kx^2}{2}+{\lambda}{}$$ where $$k$$ and $$\lambda$$ are positive constants of appropriate dimensions. The particle is slightly displaced from its equilibrium position. The particle oscillates with the angular frequency $$(\omega )$$ given by

A
$$3\dfrac{k}{m}$$

B
$$3\dfrac{m}{k}$$

C
$$\sqrt{\dfrac{k}{m}}$$

D
$$\sqrt{3\dfrac{m}{k}}$$

E
$$\sqrt{3\dfrac{k}{m}}$$

Solution

Given, $$U(x)=\dfrac{Kx^2}{2}+\lambda$$

We know,

$$F=-\dfrac{\partial u}{\partial x}=\dfrac{-\partial(\dfrac{Kx^2}{2}+\lambda)}{\partial 2}$$

$$=2\dfrac{Kx}{2}+0=-Kx$$

$$a=\dfrac{F}{m}=-\dfrac{K}{m}x$$............(1)

and we also know

$$a=-w^2x$$

compare equation (1) with (ii)

$$w^2=\dfrac{K}{m}$$.........(2)

$$\boxed{w=\sqrt{\dfrac{K}{m}}}$$ Ans ..........(c)
Question 7
1 / -0

If $$\vec{s}=a\sin \omega t\ \hat{i}+b\cos \omega t\ \hat{j}$$, the equation of path of particle is:

A
$$x^2+y^2=\sqrt{a^2+b^2}$$

B
$$\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1$$

C
$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$

D
None of these

Solution
Question 8
1 / -0

Two S.H.M.'s $$x=a\sin\omega t$$ and $$y=b\cos \omega t$$ directed along y-axis respectively are acted on particle. The path of the particle is:

A
Circle

B
Straight line

C
Ellipse

D
Parabola

Solution
Question 9
1 / -0

A particle moves along y-axis according to equation $$y=3+4\cos \omega t$$. The motion of particle is:

A
Not S.H.M.

B
Oscillatory but not S.H.M.

C
S.H.M.

D
None of the above

Solution
Question 10
1 / -0

A particle of mass $$10kg$$ is executing S.H.M. of time period $$2$$ second and amplitude $$0.25m$$. The magnitude of maximum force on the particle is:

A
$$5N$$

B
$$24.65N$$

C
Zero

D
$$40.6N$$

Solution