A particle executes linear simple harmonic motion with an amplitude of 2 cm. When the particle is at 1 cm from the mean position the magnitude of its velocity is equal to that of its acceleration . Then its time period in second is

• $\frac{1}{2 \mathrm{\pi }\sqrt{3}}$

• $2 \mathrm{\pi }\sqrt{3}$

• $\frac{2 \mathrm{\pi }}{\sqrt{3}}$

• $\frac{\sqrt{3}}{2\mathrm{\pi }}$

Two springs are joined and attached to a mass of 16 kg. The system is then suspended vertically from a rigid support. The spring constant of the two springs are k₁  and k₂ respectively. The period of vertical oscillations of the system will be

• $\frac{1}{8\mathrm{\pi }}\sqrt{{\mathrm{k}}_1 + {\mathrm{k}}_2}$

• $8\mathrm{\pi } \sqrt{\frac{{\mathrm{k}}_1 + {\mathrm{k}}_2}{{\mathrm{k}}_1{\mathrm{k}}_2}}$

• $\frac{\mathrm{\pi }}{2}\sqrt{{\mathrm{k}}_1 - {\mathrm{k}}_2}$

• $\frac{\mathrm{\pi }}{2}\sqrt{\frac{{\mathrm{k}}_1}{{\mathrm{k}}_2}}$

Two massless springs of force constants k₁ and k₂ are joined end to end. The resultant force constant k of the system is

• $\mathrm{k} = \frac{{\mathrm{k}}_1 + {\mathrm{k}}_2}{{\mathrm{k}}_1{\mathrm{k}}_2}$

• $\mathrm{k} = \frac{{\mathrm{k}}_1 - {\mathrm{k}}_2}{{\mathrm{k}}_1{\mathrm{k}}_2}$

• $\mathrm{k} = \frac{{\mathrm{k}}_1{\mathrm{k}}_2}{{\mathrm{k}}_1 + {\mathrm{k}}_2}$

• $\mathrm{k} = \frac{{\mathrm{k}}_1{\mathrm{k}}_2}{{\mathrm{k}}_1 - {\mathrm{k}}_2}$

A spring of force constant k is cut into two equal halves.  The force constant of each half is

• $\frac{\mathrm{k}}{\sqrt{2}}$

• k

• $\frac{\mathrm{k}}{2}$

• 2k

A particle of mass m is attached to three identical massless springs of spring constant k as shown in the figure. The time period of vertical oscillation of the particle is

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}}$

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{3\mathrm{k}}}$

• $\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$

246.

A spring of force constant k is cut into three equal parts. The force constant of each part would be

• $\frac{\mathrm{k}}{3}$

• 3k

• k

• 2k

A particle is executing linear simple harmonic motion of amplitude A. At what displacement is the energy of the particle half potential and half kinetic ?

• $\frac{\mathrm{A}}{4}$

• $\frac{\mathrm{A}}{2}$

• $\frac{\mathrm{A}}{\sqrt{2}}$

• $\frac{\mathrm{A}}{\sqrt{3}}$

Two identical springs are connected to mass m as shown (k = spring constant). If the period of the configuration in (a) is 2s, the period of the configuration in (b) is

• $\sqrt{2} \mathrm{s}$

• 1 s

• $\frac{1}{\sqrt{2}} \mathrm{s}$ 

• $2\sqrt{2} \mathrm{s}$

A particle of mass m is located in a one dimensional potential field where potential energy is given by V(x) = A(l − cos px), where A and p are constants. The period of small oscillations of the particle is 

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\left(\mathrm{Ap}\right)}}$

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\left({\mathrm{Ap}}^2\right)}}$

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{A}}}$

• $\frac{1}{2\mathrm{\pi }} \sqrt{\frac{\mathrm{Ap}}{\mathrm{m}}}$

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 α is given by

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g} \cos \mathrm{\alpha }}}$

• $2\mathrm{\pi } \sqrt{\frac{\mathrm{l}}{\mathrm{g} \sin \mathrm{\alpha }}}$

• $2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}$

• $2\mathrm{\pi } \sqrt{\frac{\mathrm{l}}{\mathrm{g} \tan \mathrm{\alpha }}}$