$A$ mass of $2.0\, kg$ is placed on a flat pan attached to a vertical spring fixed on the ground as shown in the figure. The mass of the spring and the pan is negligible. When pressed slightly and released,the mass executes simple harmonic motion. The spring constant is $200\, N/m$. What should be the minimum amplitude of the motion so that the mass gets detached from the pan? (Take $g = 10\, m/s^2$)

In a spring-block system as shown in the figure,if the spring constant $K = 9 \pi^2 \ Nm^{-1}$,then the time period of oscillation is (in $s$)

Two blocks of masses $m$ and $M$ $(M > m)$ are placed on a frictionless table as shown in the figure. $A$ massless spring with spring constant $k$ is attached to the lower block. If the system is slightly displaced and released,then ($\mu =$ coefficient of friction between the two blocks):
$(A)$ The time period of small oscillation of the two blocks is $T = 2\pi \sqrt{\frac{M + m}{k}}$
$(B)$ The acceleration of the blocks is $a = \frac{kx}{M + m}$ ($x =$ displacement of the blocks from the mean position)
$(C)$ The magnitude of the frictional force on the upper block is $f = \frac{mkx}{M + m}$
$(D)$ The maximum amplitude of the upper block,if it does not slip,is $A = \frac{\mu mg(M + m)}{mk} = \frac{\mu g(M + m)}{k}$ (Wait,let's re-evaluate: $f_{max} = \mu mg$. Since $f = ma = m \cdot \frac{kx}{M+m}$,at max amplitude $A$,$m \cdot \frac{kA}{M+m} = \mu mg \implies A = \frac{\mu g(M+m)}{k}$)
$(E)$ Maximum frictional force can be $\mu mg$.
Choose the correct answer from the options given below.

Figure $(a)$ shows a spring of force constant $k$ clamped rigidly at one end and a mass $m$ attached to its free end. $A$ force $F$ applied at the free end stretches the spring. Figure $(b)$ shows the same spring with both ends free and attached to a mass $m$ at either end. Each end of the spring in Figure $(b)$ is stretched by the same force $F$.
$(a)$ What is the maximum extension of the spring in the two cases?
$(b)$ If the mass in Figure $(a)$ and the two masses in Figure $(b)$ are released,what is the period of oscillation in each case?

$A$ block $P$ of mass $m$ is placed on a frictionless horizontal surface. Another block $Q$ of the same mass $m$ is kept on $P$ and connected to a wall with the help of a spring of spring constant $k$ as shown in the figure. $\mu_s$ is the coefficient of static friction between $P$ and $Q$. The blocks move together performing simple harmonic motion $(SHM)$ of amplitude $A$. The maximum value of the friction force between $P$ and $Q$ is

Two springs of force constants $K$ and $2K$ are connected to a mass $m$ as shown in the figure. The frequency of oscillation of the mass is: