**Simple Harmonic Motion**

- This video introduces the concept of periodic motion and its forms like oscillatory and simple harmonic motion along with the terms used to define their characteristics like time period, frequency and amplitude.
- This video explains in detail the simple harmonic motion with its different equation forms and their graphs. It also discusses the different terms used in the equation forms like angular frequency, phase angle with their respective representation in graphs.
- The time period of a particle in simple harmonic motion is equal to the time between consecutive appearances of the particle at a particular point in its motion. The point in question can be (a) the mean position (b) the extreme position (c) between the mean position and the positive extreme (d) between the mean position and the negative extreme.
- Identify which of the following functions represent simple harmonic motion. (a) x = Ae^(iωt) (b) x = Ae^(- ωt)
- A particle moves on the x-axis according to the equation x = A sin2ωt. Find the amplitude and time period if particle moves in simple harmonic motion.
- This video explains the variation in velocity and acceleration of a particle in simple harmonic motion (SHM) and their derivation from the equation of the motion along with graphs.
- For a simple harmonic motion (SHM) with given amplitude and angular speed, starting from mean position. Find (a) the time taken to reach half amplitude. (b) speed and acceleration of particle at this point (c) position at T/8 (d) position where the speed is half the maximum value (e) position where acceleration is half the maximum value.
- The maximum velocity and acceleration of a particle in simple harmonic motion are known. Locate the position of the particle for a given velocity.
- The maximum velocity and acceleration of a particle in simple harmonic motion are known. Locate the position of the particle for a given velocity.
- Write equation of simple harmonic motion (SHM) of angular frequency ω and amplitude A if the particle is situated at A / √2 at t = 0 and is going toward mean position.
- The figure show the displacement time graph of a particle executing SHM with a time period T. Four points 1, 2, 3 and 4 are marked on the graph where the displacement is half that of the amplitude. (a) Identify the points with same displacement but with opposite direction of motion. Also find the time difference between them. (b) Identify the points where particles are moving in the same direction. Also find the time difference between them.
- Two particles execute simple harmonic motion of same amplitude and frequency along the same straight line. They pass one another, when going in opposite directions, each time their displacement is half of their amplitude. What is the phase difference between them?
- This video discusses the simple harmonic motion of spring-block system along with its function and different parameters.
- A block of mass m is hanging from a spring with spring constant k. (a) Find the elongation in the spring in the equilibrium position. (b) If the block is displaced slightly from the equilibrium position, find the angular frequency of oscillation. (c) What is the mean position of oscillation?
- Two bodies of equal mass are suspended from two separate mass-less springs with force constants k1 and k2, respectively. If the bodies oscillate vertically such that their maximum velocities are equal, the ratio of their amplitudes is.....?
- Two bodies of equal mass are suspended from two separate mass-less springs with force constants k1 and k2, respectively. If the bodies oscillate vertically such that their maximum velocities are equal, the ratio of their amplitudes is.....?
- Two blocks of masses m1 and m2 are connected with a spring of natural length l and spring constant k. The system is lying on a frictionless horizontal surface. Initially the blocks are pulled aside such that the spring is in a stretched state. What will be the frequency of oscillation when the blocks are released?
- Consider the situation as shown in the figure. Mass of lower block is M, that of the upper block is m, and the force constant of the spring is k. Initially, the entire system is stretched and released. Find the period of SHM. All the surfaces are frictionless.
- It discusses the energy considerations like potential, kinetic and total mechanical energy of the spring-block system in simple harmonic motion.
- For a simple harmonic motion (SHM), which of the following is correct? (a) the potential energy is always equal to the kinetic energy. (b) the potential energy is never equal to the kinetic energy. (c) the average potential energy in any time interval is equal to the average kinetic energy in that time period (d) the average potential energy in one time period is equal to the average kinetic energy in one time period.
- For a particle undergoing SHM, the displacement x is related to time t as x = A cosωt. First graph represents its Potential Energy against time and second graph represents its Potential Energy against position. Two options are shown in each graph, marked as 1 and 2 in the first and 3 and 4 in the second. Which of these are correct? (a) 1 and 3 (b) 2 and 4 (c) 2 and 3 (d) 1 and 4.
- This video discusses the motion of a simple pendulum as a special case of simple harmonic motion assuming the angular displacement of the pendulum to be very low.
- Find the time period of oscillation of pendulum in the situation shown in figure, assuming that the angle between the moving string and the vertical stays small throughout the motion.
- This video explains in detail the situation in which the pendulum system is itself accelerating in vertical and horizontal direction and the relative changes in the different parameters.
- A simple pendulum is initially oscillating in a stationary elevator. When the bob is at its lowest point, the elevator starts falling freely under gravity. As seen from the elevator, the bob will (a) continue its oscillation as before (b) stop (c) will move circular path (d) move on straight line.
- A simple pendulum of length l is oscillating at a place where its separation from the earth’s surface is equal to the radius of the earth. By what factor is its time period more than that of a simple pendulum of same length oscillating on the surface of Earth.
- This video discusses the case of a physical pendulum in which a rigid body oscillates about a point with very low angular displacement in simple harmonic motion (SHM).
- A circular ring hanging from a nail in a wall undergoes oscillations with given amplitude and time period. Find (a) the radius of the ring (b) speed and acceleration of Centre of Mass when passing through the mean position (c) speed and acceleration of the point opposite to the point of suspension as it passes through the mean position (d) acceleration of Centre of Mass and point in part (c) when the ring is at one of the extreme positions.
- This video discusses the case of a torsional pendulum and its different parameters in term of restoring torque of the wire.
- A disc with moment of inertia I1 is used in a torsional pendulum. It oscillates with a period of T1. Another disc is placed over the first one and the time period of the system becomes T2. Find the moment of inertia of the second disc about the wire.
- This video explain in detail the superposition of two simple harmonic motion having same angular frequency but different amplitudes along the same line, known as collinear SHMs.
- This video discusses the situation in which two simple harmonic motions having same angular frequency with direction perpendicular to each other are superimposed.
- What is the motion of particle for each of the following equation (1) x = A sin ωt + B cos ωt (2) r = A ( i cosωt + j sinωt) (a) Simple harmonic motion (b) Elliptical motion (c) Circular motion (d) None of these
- The coefficient of friction between the two blocks shown in Figure is known and the horizontal surface on which the bigger block rests is smooth. If the blocks always move together, show that the system executes Simple Harmonic Motion and find the period. Also find the frictional force between the blocks and maximum amplitude for which they can move together.
- Consider the situation shown in figure. The lower block of mass m2 is attached to the spring of spring constant k while the upper block of mass m1 rest on the lower block. The system performs vertical oscillations. Find the following. (a) time period of oscillation (b) normal force on upper block (c) maximum amplitude at which blocks stay together and maximum speed at this amplitude.
- The left block in figure moves at a speed v towards the right block placed in equilibrium. Collision of left block (if any) with the wall is elastic and the surfaces are frictionless. Find the time period of motion if collision between the blocks is (a) elastic (b) completely in-elastic. Neglect the widths of the blocks.
- Find the time period of small oscillations of a ball suspended by a thread of length l if it is placed in a liquid whose density is n times less than the density of the ball. The resistance of the liquid is to be neglected.
- A ball is hung by a thread of length l. There is a wall such that when the pendulum touches the wall the thread makes an angle α with the vertical as shown in the figure. The thread with the ball is now deviated through a small angle β (β > α) and set free. Assuming the collision of the ball with the wall to be perfectly elastic, find the period of such a pendulum.
- A pendulum hangs from the roof of a trolley sliding on a smooth inclined plane of angle φ. If mass of the bob is m and length of the string is l, find the angular frequency for small oscillation of pendulum.
- A uniform rod of mass m is suspended by two identical threads of equal length l, as shown in figure. When turned through a small angle about a vertical axis passing through its midpoint, the threads are deviated by an angle φ. Find the time period of oscillation of the rod.
- A simple pendulum having a bob of mass m undergoes small oscillations with given amplitude. Find the tension in the string as a function of the angle made by the string with the vertical. When is this tension maximum, and when is it minimum?
- As shown in figure, a stick of length l oscillates as a physical pendulum. (a) What value of distance x between the stick's centre of mass and its pivot point O gives the least period? (b) What is that least period?
- A U tube, as shown in figure, contains mass m of a liquid. The tube's area of cross- section is A. When the liquid is displaced by a small amount in the vertical direction it oscillates freely up and down about its position of equilibrium. Compute(a) the effective spring constant for the oscillation, and (b) the period of oscillation. Ignore frictional and surface tension effects.
- Consider the two cases shown below where a liquid is filled in bent tubes. If the liquid is depressed by a small amount in one of the arms of tube, find the time period of oscillation; Given the total mass of liquid in tube and the area of cross section of tube. Neglect viscosity.
- A rectangular block of wood is floating in a large pool of water. A is the area of the face, d is its depth beneath the surface of the water, ρ is the density of water, and g is gravitational acceleration.(a) Find the value of depth d for which the block is in equilibrium.(b)Show that if the block is depressed below its equilibrium depth (but not beneath surface of the water) and then released, it will execute harmonic oscillations. (c)Determine the frequency of the oscillations.
- A uniform cylinder of length l and mass m having cross-sectional area A is suspended with its length vertical from a fixed point by a mass-less spring such that it is half submerged in a liquid of density ρ at equilibrium position. When the cylinder is given a small downward push and released, it starts oscillating vertically with small amplitudes. If the force constant of the spring is k, calculate the frequency of oscillation of the cylinder.
- A vertical pole of length of l, density ρ, area of cross section ‘A’ floats in two immiscible liquids of densities ρ1 and ρ2. In equilibrium position the bottom end is at the interface of the liquids. When the cylinder is displaced vertically, find the time period of oscillation.
- A thin uniform bar of mass m lies symmetrically across two rapidly rotating, fixed rollers with distance L between the bar’s Centre of Mass and each roller. The rollers, whose direction of rotation are shown in figure, slip against the bar with known coefficient of kinetic friction. If the bar is displaced horizontally by distance x and released, what is the angular frequency of the resulting horizontal simple harmonic motion?
- A sleeve of mass m is fixed between two identical springs, each having a force constant equal to k. The sleeve is free to slide without friction over a horizontal bar. The entire setup is made to rotate with a constant angular velocity, about a vertical axis passing through the middle of the bar. Find the time period of small oscillations of the sleeve. For what value(s) of angular velocity, the sleeve will not oscillate at all.
- A horizontal spring-block system of mass M executes simple harmonic motion. When the block is passing through its equilibrium position, an object of mass m is put on it and the two move together. Find the new amplitude and frequency of oscillation.
- A particle of mass m is attached to three identical springs each of force constant k as shown in figure. If the particle is pushed slightly downwards and released, find the time period of oscillation.
- In the given situations, the objects are pivoted with respect to their Centre of Mass and a spring with force constant k is attached to their end as shown. Find the angular frequency of oscillation for small angular displacements.
- In the two situations shown in the figure, a uniform rod of length l and mass m is hinged about one of its ends. Its other end is connected to a spring of spring constant k. Find the frequency of small oscillations.
- A solid disc of mass m is attached to a mass-less spring with spring constant k, so that it can roll without slipping on a rough horizontal surface. Calculate the period of oscillation for small displacements.
- A uniform disc of mass m and radius R is connected with light springs in the situations as shown in the figure. Assuming a perfect rolling of the disc on the horizontal surface, find the angular frequency of oscillation for all the cases.
- Consider the situation shown in the figure. If the block is displaced slightly below its equilibrium position and released, find the time period of its vertical oscillations. The pulley is light and smooth and the spring and string are light.
- For the system shown in figure, find the time period for small vertical oscillations of the mass m, if (a) pulley is mass-less (b) pulley has moment of inertia I. String and spring are light, and the string does not slip over the pulley.
- In the arrangement shown in figure, pulleys are small and light and springs are ideal and have force constants k1, k2, k3, and k4 respectively. Calculate the time period of small vertical oscillation of block of mass m.
- A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R as shown in figure. It makes small oscillations about the lowest point. Find the time period.
- Two simple harmonic motions are represented by the following equations: y1 = 10 sin(π/4) ( 12t + 1 ) y2 = 5(sin 3π t + √3 cos 3π t) Find out the ratio of their amplitudes. What are the time periods of the two motions?
- A particle is subjected to two simple harmonic motions, one along the x axis and the other on a line making a given angle with the x axis. The two motions are given by x = x0 sinωt and s = s0 sinωt. Find the amplitude of the resultant SHM.
- Two linear simple harmonic motions of equal amplitudes and frequencies ω and 2ω are impressed on a particle along the axes of X and Y respectively. If the initial phase difference between them is π/2, find the resultant path followed by the particle.
- A point executes two harmonic oscillations simultaneously along the same direction: x1 = A cosωt and x2 = A cos 2ωt. Determine the maximum velocity of the point.
- The potential energy of a particle of mass M acted upon only by conservative forces varies as U(x) = Uo (1 - cos ax) where Uo and a are constants. Find the time period of small oscillations of the particle about the mean position.