Linear and Angular Simple Harmonic Motions

Electrons moving with different speeds enter a uniform magnetic field in a direction perpendicular field. They will move along circular paths, time periods of rotation will be :

Two point-like objects of masses $20 \mathrm{gm}$ and $30 \mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10 \mathrm{cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times {10}^{-8} \mathrm{N} \mathrm{m} {\mathrm{rad}}^{-1}$. The angular frequency of the oscillations in $n\times {10}^{-3} \mathrm{rad} {\mathrm{s}}^{-1}$. The value of $n$ is _____.

Given system is in equilibrium. All surfaces are smooth. Spring is ideal and blocks are sticked at the ends of spring. Now $F_{\mathit{0}}$ is removed. Average normal contact force between wall and mass $m_{\mathit{1}}$ upto the time spring attains its natural length for the first time in Newton is: (Given that $F_{\mathit{0}}=18\pi$ Newton)

Find the angular frequency of small oscillation of block $m$ in the arrangement shown.
Rod is massless. [Assume gravity to be absent]
Given : ${\mathrm{K}}_1=1{\mathrm{Nm}}^{-1},{ \mathrm{K}}_2=4{\mathrm{Nm}}^{-1},{ \mathrm{K}}_3=\frac{1}{4}{\mathrm{Nm}}^{-1}, \mathrm{m}=5\mathrm{kg}$.

A block of mass $m$ is attached to a massless spring of force constant $k$, the other end of which is fixed to the wall of a truck as shown. The block is placed over a smooth surface of the truck and initially the spring is unstretched. Suddenly the truck starts moving towards right with a constant acceleration $a$ as seen from the truck.

Density of a liquid varies with depth as $\rho =\alpha h$. A small ball of density ${\rho }_0$ is released from the free surface of the liquid. Then:

In the given arrangement, the balls are identical each of mass $m$ placed in gravity free space with initially springs are at their natural length. When the balls are slightly displaced in the vertical direction, then the balls are in vertical SHM. Then the possible angular frequencies for that SHM (the springs are ideal springs. Each of spring constant is $k$) (Only consider both balls oscillates with same angular frequency but they crosses the mean position their velocities direction are in the same direction (or) opposite direction)

A simple pendulum of length $L= 30 \mathrm{cm}$ and mass $m=2\mathrm{kg}$ is attached to a cart of mass $3\mathrm{kg}$. The cart can move frictionlessly on level ground. The bob is released from rest when string makes an angle of $1.8°$ with the horizontal

A simple pendulum and a homogeneous rod pivoted at its end are free to oscillate with small amplitude. What is the ratio of their periods of swing if their lengths are identical?

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

A particle of mass $m$ moves with its potential energy $U$ varying with its position as shown. If $E$ is total energy of the particle, its motion is periodic with time period equal to

The displacement of two particles executing S.H.M are represented by $y_1=a\sin \omega t$ and $y_2=a\cos \left(\omega t+\varphi \right)$. The phase difference between the velocities of these particles is

A uniform plank of mass $m$, free to move in horizontal direction, is placed on the top of a solid cylinder of mass $2 \mathrm{m}$ and radius $R$. The plank is attached with one end of a mass less spring of force constant $k$ and the other end of spring is fixed with a wall as shown in fig. The plank is shiftily displaced towards spring and released. There is no slipping any where then

A block of mass $m$ is kept on a horizontal platform of mass $M$. The platform is doing SHM in horizontal plane with angular frequency $\omega$. There is no slipping between the block and the platform due to friction. Then

Two blocks $A\left(5 \mathrm{kg}\right)$ and $B\left(2 \mathrm{kg}\right)$ attached to the ends of a spring of stiffness constant $1120 \mathrm{N} {\mathrm{m}}^{-1}$ are placed on a smooth horizontal plane with the spring being initially un-deformed. At time $t=0$, velocities of $3 \mathrm{m} {\mathrm{s}}^{-1}$ and $10 \mathrm{m} {\mathrm{s}}^{-1}$ along the line of the spring in the same direction are simultaneously imparted to $A$ and $B$ respectively as shown. Then,

Figure shows a block $P$ of mass $m$ resting on a horizontal smooth floor at a distance $l$ from a rigid wall. Block is pushed toward right by a distant $\frac{31}{2}$ and released. When block passes from its mean position another block of mass $m_1$ is placed on it which sticks to it due to friction so that the combined block just collides with the left wall.

A plank of area of cross-section $A$ and mass $m$ is half immersed in liquid $1$ of density $\mathrm{\rho }$ and half in liquid $2$ of density $2\mathrm{\rho }.$ What is period of oscillation of the plank if it is slightly depressed downwards.

A particle executes $SHM$ about a point other than $x=0$ as shown in the graph.

The energy of a particle executing simple harmonic motion is given by $E=A{x}^2+B{v}^2$ where $x$ is the displacement from mean position $x=0$ and $v$ is the velocity of the particle at $x$ then choose the correct statement(s)

In the figure shown a block $A$ of mass $m$ is rigidly attached with two light springs each of stiffness $k$ and suspended from a fixed support. Another block $B$ of same mass is just placed on it and blocks are in equilibrium. Suddenly the block $B$ is removed. Choose the CORRECT option(s) afterward.