What is the period of small oscillations of the block of mass $m$ if the springs are ideal and pulleys are massless ?

Two springs of constant ${k_1}$and ${k_2}$are joined in series. The effective spring constant of the combination is given by

A particle of mass $m$ is attached to one end of a mass-less spring of force constant $k$, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t=0$ with an initial velocity $u_0$. When the speed of the particle is $0.5 u_0$, it collies elastically with a rigid wall. After this collision :

$(A)$ the speed of the particle when it returns to its equilibrium position is $u_0$.

$(B)$ the time at which the particle passes through the equilibrium position for the first time is $t=\pi \sqrt{\frac{ m }{ k }}$.

$(C)$ the time at which the maximum compression of the spring occurs is $t =\frac{4 \pi}{3} \sqrt{\frac{ m }{ k }}$.

$(D)$ the time at which the particle passes througout the equilibrium position for the second time is $t=\frac{5 \pi}{3} \sqrt{\frac{ m }{ k }}$.

Two masses ${m_1}$ and ${m_2}$ are suspended together by a massless spring of constant k. When the masses are in equilibrium, ${m_1}$ is removed without disturbing the system. Then the angular frequency of oscillation of ${m_2}$ is

In arrangement given in figure, if the block of mass m is displaced, the frequency is given by