The drawing shows a top view of a frictionless horizontal surface, where there are two indentical springs with particles of mass $m_1$ and $m_2$ attached to them. Each spring has a spring constant of $1200\  N/m.$ The particles are pulled to the right and then released from the  positions shown in the drawing. How much time passes before the particles are again side by side for the first time if $m_1 = 3.0\  kg$ and $m_2 = 27 \,kg \,?$

An object is attached to the bottom of a light vertical spring and set vibrating. The maximum speed of the object is $15\, cm/sec$ and the period is $628$ milli-seconds. The amplitude of the motion in centimeters is

In the given figure, a mass $M$ is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is $k$. The mass oscillates on a frictionless surface with time period $T$ and amplitude $A$. When the mass is in equilibrium position, as shown in the figure, another mass $m$ is gently fixed upon it. The new amplitude of oscillation will be

The total spring constant of the system as shown in the figure will be

A $5\; kg$ collar is attached to a spring of spring constant $500\;N m ^{-1} .$ It slides without friction over a hortzontal rod. The collar is displaced from its equilibrium position by $10.0\; cm$ and released. Calculate

$(a)$ the period of oscillation.

$(b)$ the maximum speed and

$(c)$ maximum acceleration of the collar.

The frequency of oscillations of a mass $m$ connected horizontally by a spring of spring constant $k$ is $4 Hz$. When the spring is replaced by two identical spring as shown in figure. Then the effective frequency is,