What is the difference between a Periodic, Oscillatory and a Simple Harmonic Motion ?
Derivation and Understand of Equation of SHM
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 is
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 = Aeiwt
b) x = Ae-wt
A particle moves on the x-axis according to the equation x = A sin2 wt.
The motion is simple harmonic
a) With amplitude A
b) With amplitude A/2
c) With time period 2p/w
d) With time period p/w
Velocity and Acceleration of a particle in SHM
For an SHM with amplitude A and angular speed w, starting from mean position.
a) the time taken to reach 1/2 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 10 cm/s and 50 cm/s2 . Locate the position of the particle where its velocity is 8 cm/s.
SHM as a projection of Uniform Circular Motion. Real meaning and understanding of Phase.
Write equation of SHM of angular frequency w and amplitude A if the particle is situated at A / 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.
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?
Analysis pf a Spring-Block system doing SHM. Does a higher amplitude mean a higher Time period ?
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 massless 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:
Find the period of oscillation and equivalent spring constant in each case.
Block has mass m. Friction is absent.
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.
Variation of Potential energy, Kinetic energy and Total Mechanical energy in an SHM.
In a simple harmonic motion
(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 cos wt. 1st graph represents its PE against time and 2nd graph represents its PE against position. Two options are shown in each graph, marked as 1 and 2 in the 1st and 3 and 4 in the 2nd. Which of these are correct?
(a) 1 and 3
(b) 2 and 4
(c) 2 and 3
(d) 1 and 4
Analysis of a Simple Pendulum as SHM. Is a pendulum with high angular amplitude an SHM ?
Find the time period of oscillation of pendulum in the given situation?
In answering this question, you may assume that the angle between the moving string and the vertical stays small throughout the motion.
Analysis of the SHM of simple pendulum placed in an accelerating frame of reference.
A simple pendulum is initially oscillating in a stationery 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 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.
Analysis of SHM of a Rigid body acting as a pendulum.
A circular ring hanging from a nail in a wall undergoes oscillations of amplitude q = 5o and period T = 4s. Find
(a) the radius of the ring
(b) speed and acceleration of CoM 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 CoM and point in part
(e) when the ring is at one of the extreme positions.
Given: g = p2 m/s2
Analysis of angular SHM of a Torsional pendulum.
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.
Resultant motion of a particle whose motion is a superposition of two Collinear SHMs
Resultant motion of a particle whose motion is a superposition of two Perpendicular SHMs
What is the motion of particle for each of the following equation
(1) x = A sin wt + B cos wt
(2) r = A ( i cos wt + j sin wt)
(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 m and the horizontal surface on which the bigger block rests is smooth. If the blocks always 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.
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,
a) collision between the blocks is elastic
b) collision between the blocks is 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 r is n times less than the density r 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 a with the vertical (as shown in the figure). The thread with the ball is now deviated through a small angle b ( b > a ) 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 f. 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 mid point, the threads are deviated by an angle f.
Find the time period of oscillation of the rod.
A simple pendulum having a bob of mass m undergoes small oscillations with amplitude qo. 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?
In figure a stick of length l oscillates as a physical pendulum.
a) What value of distance x between the sticks 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 tubes 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 that the total mass of liquid in tube is m and the area of cross section of tube is A. 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, r is the density of water, and g is gravitational acceleration.
(a)The mass of the block is m. Find the value of d for which the block is in equilibrium.
(b)Show that is 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 massless spring such that it is half submerged in a liquid of density 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 r, area of cross section ?A? floats in two immiscible liquids of densities r1 and r2. 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 CoM and each roller. The rollers, whose direction of rotation are shown in figure, slip against the bar with coefficient of kinetic friction mk. 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 wo , 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 wo , 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 w.r.t their Center 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 massless spring with spring const 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 period of its vertical oscillations. The pulley is light and smooth and the spring and string are light.
Find the time period for small vertical oscillations of the mass m, if
a) pulley is massless
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 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. It makes small oscillations about the lowest point. Find the period.
Two simple harmonic motions are represented by the following equations:
y1 = 10 sin pie/4 ( 12t + 1 )
y2 = 5(sin 3 pie t + root 3 cos 3 pie 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 an angle of 45o with the x axis. The two motions are given by x = xo sin wt and s = so sin wt.
Find the amplitude of the resultant SHM.
Two linear simple harmonic motions of equal amplitudes and frequencies w and 2w are impressed on a particle along the axes of X and Y respectively. If the
initial phase difference between them is , find the resultant path followed by the particle.
A point executes two harmonic oscillations simultaneously along the same direction: x1 = A cos wt and x2 = A cos 2 wt. 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.