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The period of oscillation of a simple pendulum is $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}$. Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. The accuracy in the determination of g is
1. 2 %
2. 3 %
3. 1 %
4. 5 %

A pendulum made of a uniform wire of cross sectional area A has time period T. When an additional mass M is added to its bob, the time period changes to T_M. If the Young's modulus of the material of the wire is Y then $\frac{1}{\mathrm{Y}}$ is equal to (g = gravitational acceleration)
1. $\left[{\left(\frac{{\mathrm{T}}_{\mathrm{M}}}{\mathrm{T}}\right)}^2-1\right]\frac{\mathrm{A}}{\mathrm{Mg}}$
2. $\left[{\left(\frac{{\mathrm{T}}_{\mathrm{M}}}{\mathrm{T}}\right)}^2-1\right]\frac{Mg}{A}$
3. $\left[1-{\left(\frac{{\mathrm{T}}_{\mathrm{M}}}{\mathrm{T}}\right)}^2\right]\frac{\mathrm{A}}{\mathrm{Mg}}$
4. $\left[1-{\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{M}}}\right)}^2\right]\frac{\mathrm{A}}{\mathrm{Mg}}$

Two masses m₁ and m₂ are suspended together by a massless spring of constant K. When the masses are in equilibrium, m₁ is removed without disturbing by the system. The amplitude of oscillations is 

1. $\frac{{\mathrm{m}}_1\mathrm{g}}{\mathrm{K}}$
2. $\frac{{\mathrm{m}}_2\mathrm{g}}{\mathrm{K}}$
3. $\frac{\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}}{\mathrm{K}}$
4. $\frac{\left({\mathrm{m}}_1-{\mathrm{m}}_2\right)\mathrm{g}}{\mathrm{K}}$

Two particles executes S.H.M of same amplitude and frequency along the same straight line. They pass one another when going in opposite directions, and each time their displacement is half of their amplitude. The phase difference between them is 
1. 30$°$
2. 60$°$
3. 90$°$
4. 120$°$

A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constants k. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $\mathrm{\theta }$ in one direction and released. The frequency of oscillation is 

1. $\frac{1}{2\mathrm{\pi }}\sqrt{\frac{2\mathrm{k}}{\mathrm{M}}}$
2. $\frac{1}{2\mathrm{\pi }}\sqrt{\frac{\mathrm{k}}{\mathrm{M}}}$
3. $\frac{1}{2\mathrm{\pi }}\sqrt{\frac{6\mathrm{k}}{\mathrm{M}}}$
4. $\frac{1}{2\mathrm{\pi }}\sqrt{\frac{24\mathrm{k}}{\mathrm{M}}}$

A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is $\mathrm{U}\left(\mathrm{x}\right)=\mathrm{k}{\left(\mathrm{x}\right)}^3$ where k is a positive constant. If the amplitude of oscillation is  a, then its time period T is 
1. Proportional to $\frac{1}{\sqrt{\mathrm{a}}}$
2. Independent of a
3. Proportional to $\sqrt{\mathrm{a}}$
4. Proportional to ${\mathrm{a}}^{3/2}$

A small block is connected to one end of a massless spring of un-stretched 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. It then executes simple harmonic motion with angular frequency $\mathrm{\omega }=\frac{\mathrm{\pi }}{3} \mathrm{rad}/\mathrm{s}$. 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/s²)

1. $\sqrt{50} \mathrm{m}/\mathrm{s}$
2. $\sqrt{51} \mathrm{m}/\mathrm{s}$
3. $\sqrt{52} \mathrm{m}/\mathrm{s}$
4. $\sqrt{53} \mathrm{m}/\mathrm{s}$

The period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination $\mathrm{\alpha }$, is given by
1. $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} \mathrm{cos\alpha }}}$
2. $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} \sin \mathrm{\alpha }}}$
3. $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} }}$
4. $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} \tan \mathrm{\alpha }}}$

One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross section and Young's modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period T equal to 
1. $2\mathrm{\pi }\left(\frac{\mathrm{m}}{\mathrm{K}}\right)$
2. $2\mathrm{\pi }{\left\{\frac{\left(\mathrm{YA}+\mathrm{KL}\right)\mathrm{m}}{\mathrm{YAK}}\right\}}^{1/2}$
3. $2\mathrm{\pi }\left(\frac{\mathrm{mYA}}{\mathrm{KL}}\right)$
4. $2\mathrm{\pi }\left(\frac{\mathrm{mL}}{YA}\right)$

The function ${\sin }^2\left(\mathrm{\omega t}\right)$ represents
1. A simple harmonic motion with a period $2\mathrm{\pi }/\mathrm{\omega }$
2. A simple harmonic motion with a period $\mathrm{\pi }/\mathrm{\omega }$
3. A periodic but not simple harmonic motion with period $2\mathrm{\pi }/\mathrm{\omega }$
4. A periodic but not simple harmonic motion with a period $\mathrm{\pi }/\mathrm{\omega }$

A simple pendulum has time period T₁. The point of suspension is now moved upward moved upward according to equation y=kt² and where k  = 1 m/s². If new time period is T₂ then ratio $\frac{{\mathrm{T}}_1^2}{{\mathrm{T}}_2^2}$ will be
1. 2/3
2. 5/6
3. 6/5
4. 3/2 

Graph between velocity and displacement of a particle , executing SHM is 
1. A straight line 
2. A parabola
3. A hyperbola
4. An ellipse

A block of mass m is kept on smooth horizontal surface and connected with two springs as shown in figure. Initially springs are in their natural length. Time period of small horizontal oscillation of the block is :

1. $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$
2. $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{5\mathrm{k}}}$
3. $\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$
4. $\left(\mathrm{\pi }+\frac{2}{\sqrt{3}}\right)\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$

A uniform cyclinder of length L and mass M having cross-sectional area A is suspended with its vertical length, from a fixed point by a massless spring, such that it is half submerged in a liquid of density d at equilibrium position. When the cylinder is given a small downward push and released, it starts oscillating vertically with a small amplitude. If the force constant of the spring is K, the frequency of oscillation of the cylinder is 
1. $\frac{1}{2}{\left(\frac{\mathrm{K}-\mathrm{Adg}}{\mathrm{M}}\right)}^{1/2}$
2. $\frac{1}{2}{\left(\frac{\mathrm{K}+\mathrm{dgL}}{\mathrm{M}}\right)}^{1/2}$
3. $\frac{1}{2\mathrm{\pi }}{\left(\frac{\mathrm{K}+\mathrm{Adg}}{\mathrm{M}}\right)}^{1/2}$
4. $\frac{1}{2\mathrm{\pi }}{\left(\frac{\mathrm{K}-\mathrm{Adg}}{Adg}\right)}^{1/2}$

A particle at the end of a spring executes simple harmonic motion with a period t₁, while the corresponding period for another spring is t₂. If the period of oscillation with the two springs in series is T, then
1. $\mathrm{T}={\mathrm{t}}_1+{\mathrm{t}}_2$
2. ${\mathrm{T}}^2={\mathrm{t}}_1^2+{\mathrm{t}}_2^2$
3. ${\mathrm{T}}^{-1}={\mathrm{t}}_1^{-1}+{\mathrm{t}}_2^{-1}$
4. ${\mathrm{T}}^{-2}={\mathrm{t}}_1^{-2}+{\mathrm{t}}_2^{-2}$

The displacement y of a particle executing a certain periodic motion is given by $\mathrm{y}=4{\cos }^2\left(\frac{1}{2}\mathrm{t}\right)\sin \left(1000\mathrm{t}\right)$. This expression may be considered to be the superposition of n independent harmonic motions. Then, n is equal to 
1. 2
2. 3
3. 4
4. 5

A ring of mass M and radius R is hanged from a point on the rim. The ring displaced slightly along its plane, and free to oscillate. The length of equivalent simple pendulum l is 

1. l = R
2. l = 2R
3. l = $\sqrt{2} \mathrm{R}$
4. None of these

Two pendulums having lengths 1 m and 16 m are both provided small displacements in the same direction at the same instant. They will again be in phase at the mean position after the shorter pendulum completes
1. $\frac{1}{4}\mathrm{th}$ oscillation
2. 4 oscillations
3. 5 oscillations
4. 16 oscillations

A mass m is suspended from a massless pulley which itself is suspended with the help of a  massless extensible spring as shown below. What will be the time period of oscillation of the mass ? The force constant of the spring is k

1. $\mathrm{\pi }\sqrt{\mathrm{m}/\mathrm{k}}$
2. $2\mathrm{\pi }\sqrt{\mathrm{m}/\mathrm{k}}$
3. $4\mathrm{\pi }\sqrt{\mathrm{m}/\mathrm{k}}$
4. $2\mathrm{\pi }\sqrt{\mathrm{m}/2\mathrm{k}}$