Force and Energy Method in SHM

The potential energy of a particle of mass $m$ free to move along $x$ axis has the form $U\left(x\right)=\frac{a}{x^2}-\frac{b}{x}$, where $a$ and $b$ are positive constant and $x$ is $x$-coordinate of the particle. The time period of SHM (along $x$-axis) of small amplitude around mean position is $\frac{P\pi a}{b^2}\sqrt{2ma}$. Find the value of $P$.

A cone with half the density of water is floating in water as shown in figure. It is depressed down by a small distance $\delta \left(\ll \mathrm{H}\right)$ and released. The frequency of simple harmonic oscillations of the cone is

In the arrangement shown in figure, pulleys are small, light and frictionless and springs are ideal. $k_1,k_2,k_3$ and $k_4$ are the spring constants. The period of small vertical oscillations of block of mass $m$ is

The bob of a pendulum is attached to a horizontal spring of spring constant $k$. The pendulum will undergo simple harmonic motion with period$\left(T\right)$

Cotyledons are also called-

A cylinder of mass $\mathrm{M}$ and radius $\mathrm{R}$ lies on a plank of mass $\mathrm{M}$ as shown. The surface between plank and ground is smooth, and between cylinder and plank is rough. Assuming no slipping between cylinder and plank, the time period of oscillations (when displaced from equilibrium) of the system is

Find the period of low amplitude vertical vibrations of the system shown The mass of the block is m. The pulley hangs from the ceiling on a spring with a force constant k. The block hangs from an ideal spring.

A particle of mass 5 x 10^-5 kg is placed at the lowest point of a smooth parabola having the equation x² = 40y (x, y in cm). If it is displaced slightly and it moves such that it is constrained to move along the parabola, the angular frequency of oscillation will be, approximately

A block of mass ${\mathrm{M}}_1$ is constrained to move along with a moveable pulley of mass ${\mathrm{M}}_2$ which is connected to a spring of force constant $\mathrm{k}$, as shown in the figure. If the mass of the fixed pulley is negligible and friction is absent everywhere, then the period of small oscillations of the system is