# Simple harmonic motion Definition and 109 Discussions

In mechanics and physics, simple harmonic motion (sometimes abbreviated SHM) is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.
Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see small-angle approximation). Simple harmonic motion can also be used to model molecular vibration as well.
Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques of Fourier analysis.

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1. ### How to prove that motion is periodic but not simple harmonic?

TL;DR Summary: Prove that a sum of trigonometric ratios is periodic but not not simple harmonic. We need to prove that ##x = sin{\omega t} + sin{2\omega t} + sin{4\omega t}## where ##x## is the displacement from the equilibrium position at time ##t##. I can see that each term is a SHM, but...
2. ### Pan suspended by a spring (Energy + SHM)

I have successfully completed parts A, and B, however, I am confused on Part C. Here was my attempt and the answer key's attempt: My attempt: Since I correctly knew the speed after the collision, and the gravitational potential energy after the collision if I set h=0 at when it was at rest...
3. ### B Correct SHM equation?

A textbook I am using gives the basic eqn of motion of shm as follows : X = Asin(wt + €) V =Awcos(wt+€) But other textbooks and online sources are interchanging sin and cos in above equations, so which is the correct one? Or does it depend on the phase constant €?
4. ### Oscillations of a block on a rough surface while being pulled by a long spring

I am only asking about the answer to part B, but reading through part A may give some some context/familiarity. Below is the answer to part B: I largely understand the graph except for 1 part. My understanding is as such: At first, ##x = \frac {\mu_k m g } {k}##. Force exerted by the...
5. ### Equations of motion of damped oscillations due to kinetic friction

Take rightwards as positive. There are 2 equations of motion, depending on whether ##\frac {dx} {dt} ## is positive or not. The 2 equations are: ##m\ddot x = -kx \pm \mu mg## My questions about this system: Is this SHM? Possible method to solve for equation of motion: - Solve the 2nd ODE...
6. ### Getting the equations of motion for this SHM problem

Summary:: I have come across a situation where I seem to get different equations of motion for an oscillating system. Please do help me find out where I went wrong. *I am not asking how to solve the problem* I am going to consider 4 parts of the cylinder's motion, as listed below. (There is...
7. ### Mass-spring oscillator problem

I understand the masses will accelerate toward each other with the same varying speed before they reach the natural length of the spring. Then they continue to approach each other while compress the spring, that'll slow their speeds down definitely. So my question is, how could we calculate how...
8. ### How to solve spring mass damper system manually?

The other day when I solved a spring mass damper system in Matlab, I was curious how in olden days would have people solved the equation. We all know the 2nd order differential equation of the system: However if I know the time, damping coefficient, stiffness and mass, will I be able to find...
9. ### Percentage difference problem

See the question : https://www.thestudentroom.co.uk/att...hmentid=978958 The mark scheme/answer : https://www.thestudentroom.co.uk/att...hmentid=978956 I have got the answer to the vertical height gained = 1.355 m. No problem. But not the value of the percentage difference. Their value : 23%...
10. ### Simple harmonic motion problem

How is the answer to 3 (d) is found?
11. ### Equation Demonstration -- Comparing a pendulum's motion to an LC circuit

I've just learned about simple harmonic motion and I've been given the following examples: The physical pendulum (for small oscillations sin(theta)~theta), with the formula (1st pic), and the LC circuit, with the formula (2nd pic). If possible, I need the demonstration for these 2 formulas...
12. ### A critically damped simple harmonic oscillator - Find Friction

c = Critically Damped factor c = 2√(km) c = 2 × √(150 × .58) = 18.65 Friction force = -cv Velocity v = disp/time = .05/3.5 Friction force = - 18.65 * .05/3.5 = -.27 N I am not sure if above is correct. Please check and let me know how to do it.
13. ### Maximum compression of a spring?

I found the amplitude of the simple harmonic motion that results (0.367, and I know this is correct because I entered it and it was marked as a correct answer), and assumed it would be the same value for the maximum compression since x(t) = Acos(wt). And, since the maximum value of cosine is 1...
14. ### Find the point of separation in SHM

In the given problem, i can understand that after placing the two blocks in equilibrium it oscillates with an amplitude of The answer for (b) is given as To my knowledge, m2 separate from m1 when the acceleration is greater than gsinø and so they should be separating only at max displacement...
15. ### B Shifting of a Cosine Curve with negative phase angle values

Continuing on from the summary, the chapter has given a graphed example. We are shown a regular cosine wave with phase angle 0 and another with phase angle (-Pi/4) in order to illustrate that the second curve is shifted rightward to the regular cosine curve because of the negative value. Now, my...
16. ### I Shear Oscillation

Hi. I'm trying to determine the frequency of an block (roughly a rectangular prism) when the oscillation is due to a shear restoring force. Here is a diagram: In the derivation, ##\rho## is the density of the block,##G## is the shear modulus of the block, ##y## is the elevation of the element...
17. ### Finding the amplitude of a vertical spring

The question asks for a bunch of stuff, but I have everything except part d down. a) Setting the mass of lemons as m1, I used m1*gh = 1/2mv^2, solving for v of the lemons as v = √2gh, where h is the height at which it is dropped. Then, I used COM and had this equation (not 100% sure if right)...
18. ### High school Physics - Simple Harmonic Motion

I started off by finding when Fg=Fx: (72)(x)=(31)(9.8) x=4.2193m After this I'm stuck and have a few things I'm confused about: When the penguin's jumping, is there elastic energy? (because the rope's getting compressed? Or maybe not). Also, I know you can use energy conservation, but...
19. ### Shm , calculation of amplitude of spring mass system

Homework Statement In A spring mass system , the spring stretches 2 cm from its 's frelength when a force of 10 N is applied . This spring is stretched 10 cm from it's free length , when a body of mass m = 2 kg is attached to it and released from rest at time t = 0 . Find the A) force constant...
20. ### Equation of shm for different positions

Homework Statement Write the equation for a particle in simple harmonic motion with amplitude a and angular frequency w considering all distances from one extreme position and time when it is at other extreme end. Homework Equations X = A sin (wt + ∆) ∆ = phase difference The Attempt at a...
21. ### Simple harmonic motion interpretation problem

I'm in trouble trying to understand the expression ##t= \frac{1}{\omega} cos^{-1}(x/A)## that comes from ##x = Acos(\omega t)##, in which ##A## is the amplitude, ##t## is time and ##x## is displacement. When ##x = 0##, ##t = \frac{\pi}{2\omega} ##, shouldn't it be 0 since there was no movement?
22. ### Does amplitude depend on mass in SHM?

Homework Statement Does amplitude of an oscillating spring with an attached block depend on the block's mass? Assuming the spring has spring constant 'k' and obeys Hooke's law. How would the amplitude of the oscillating spring system be affected if the mass of the block were...
23. ### Derive the formula for the frequency of a spring

Homework Statement Two masses m1 and m2 are joined by a spring of spring constant k. Show that the frequency of vibration of these masses along the line connecting them is ω = √[ k(m1 + m2) / (m1*m2) ] (Hint: Center of mass remains at rest.) Homework Equations f = w/2π w = √(k/m) F = -kx a = -...
24. ### Confused about a simple harmonic motion problem....

Homework Statement A vertical block-spring system on Earth has a period of 6.0 s. What is the period of this same system on the moon where the acceleration due to gravity is roughly 1/6 that of earth? Homework Equations w = √(k/m) w = (2Pi)/T T = 2Pi*√(m/k)[/B] The Attempt at a Solution So...
25. ### Which of the options describes the phase constant for SHM?

The only thing I know is that phase constant tells how much a signal is shifted along the x-axis. The answer of the question is both option a and b. I am not getting it!
26. ### Phase angle and Phase in Simple harmonic motion

I'm a teacher at a Senior High School in Indonesia. I have two Senior High School physics books (Indonesian book) written about simple harmonic motion formula: y = A sin θ = A sin (ωt + θ0) = A sin 2πφ = A sin 2π (t/T + θ0/2π) phase angle = θ = ωt + θ0 phase of wave = φ = t/T + θ0/2π But I...
27. ### SHM - Planet Problem

Homework Statement You are exploring a newly discovered planet. The radius of the planet is 7.20 * 107 m. You suspend a lead weight from the lower end of a light string that is 4.00 m long and has mass 0.0280 kg. You measure that it takes 0.0685 s for a transverse pulse to travel from the...
28. ### Simple Harmonic Oscillator with Boundary Conditions

How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions: (x1,t1)=(0.01, 0) (x2,t2)=(0.04, 5) f=13Hz x values are given in relation to the equilibrium point. Equation of Motion for a spring undergoing...
29. ### Simple Harmonic Motion: why sin(wt) instead of sin(t)?

Hello, I have recently been introduced to the topic of simple harmonic motion for the first time (I'm currently an A-level physics student). I feel that I have understood the fundamental ideas behind SHM very well. However, I have one question which has been bugging me and I can't seem to find a...
30. ### Spring attached to a pendulum

Homework Statement A pendulum of mass ##m## and length ##L## is connected to a spring as shown in figure. If the bob is displaced slightly from its mean position and released, it performs simple harmonic motion. What is the angular frequency of the bob? Homework Equations Angular frequency for...
31. ### Simple Harmonic Motion

Homework Statement x=Acos(wt+phi) Homework Equations can somebody explain to me please when phi=0. I saw many different questions with many solutions and I can't understand when we have just x=Acos(wt) and when x=Acos(wt+phi) The Attempt at a Solution
32. ### Simple Pendulum undergoing harmonic oscillation

Homework Statement Is the time average of the tension in the string of the pendulum larger or smaller than mg? By how much? Homework Equations $$F = -mgsin\theta$$ $$T = mgcos\theta$$ The Attempt at a Solution I'm mostly confused by what it means by time average. However from my...
33. ### A Damped Harmonic Oscillator - Gravity not constant.

Hello, I have a question regarding Damped Harmonic Motion and I was wondering if anyone out there could help me out? Under normal conditions, gravity will not have an affect on a damped spring oscillator that goes up and down. Gravity will just change the offset, and the normal force equation...
34. ### Spring, SHM

Homework Statement A point mass m= 20 kg, is suspended by a massless spring of constant 2000 N/m. The point mass is released when elongation in the spring is 15 cm. The equation of displacement of particle as function of time is : (Take g = 10 m/s2) Homework Equations A is amplitude w is...
35. ### Finding the time period of this System

Homework Statement To find the time period of this simple harmonic motion Homework Equations F= -kx The Attempt at a Solution To check Simple harmonic motion first ,I have to displace the mass by some distance which I take to be x in this case. Therefore the spring will be displaced by a...
36. ### How Is Simple Harmonic motion possible here?

One thing I don't understand is that How Amplitude is conserved on both sides if the mass is subjected to different forces on either side of this shm...
37. ### Simple Harmonic Motion in x direction

Homework Statement A simple harmonic oscillator, with oscillations in the x direction, has velocity given by: $$v_{x} = (2.2 \frac {\mathrm{m}} {\mathrm{s}}) \sin [(6.9 \frac {\mathrm{rad}} {\mathrm{s}}) t]$$. Find the values of ##\omega , A, f , T ,## and ##\phi## Homework Equations v_{x} =...
38. ### Swing and angular displacement

hi, we are a few non-native English speaker physics teacher and we wrote some questions for an assessment book but we can't be sure about this two similar question. a) are they accurate for rules of English, are we use correct terms is there a necessary change? b) are they accurate for rules of...
39. A

### Elastic potential energy problem

Homework Statement A 1.00kg mass and 2.00kg mass are set gently on a platform mounted on an ideal spring of force constant 40.0 N/m. The 2.00 kg mass is suddenly removed. How high above its starting position does the 1.00 kg mass reach? Related to it... An 87 g box is attached to a spring with...
40. E

### SHM Energy Conservation in a Spring

1. The problem statement, all variables, and given/known data Describe the energy conversions in a spring undergoing simple harmonic motion as it moves from the point of maximum compression to maximum stretch in a frictionless environment. Focus on points at which there will be maximum speed...
41. A

### I Solving the differential equations involving SHM

What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?
42. T

### How do i get the frequency of undamped motion?

Homework Statement The single wheel of an aircraft can undergo a max of 7500N at a vertical velocity of 8 m/s on landing. The vertical spring moves in SHM and has a stiffness of 600N/mm. The systems vertical damper has a damping coefficient of 38 x 10^3 Ns.m-1 Homework Equations F=Kx...
43. ### Simple Harmonic Motion

Homework Statement Consider a Simple Harmonic Motion (SHM) for which, at time t = 1 s, the displacement is s=1 cm, the velocity is 2 cm s−1, and the acceleration is −3 cm s−2. Find the angular frequency, 4. amplitude, and phase constant for this motion. Homework Equations f=1/T...
44. A

### Simple harmonic motion -- The spring and mass are immersed in a fluid....

1. A mass, M = 1.61 kg, is attached to a wall by a spring with k = 559 N/m. The mass slides on a frictionless floor. The spring and mass are immersed in a fluid with a damping constant of 6.33 kg/s. A horizontal force, F(t) = Fd cos (ωdt), where Fd = 52.5 N, is applied to the mass through a...
45. ### Magnitude of Frictional Force in Simple Harmonic Motion

Homework Statement I do not fully grasp the concept behind all of these sub questions (i)-(iv). Homework Equations v=wAcos(wt) (SMH)? Friction Force = Coefficient of Friction * Normal Force The Attempt at a Solution (i) Varying as simple harmonic motion sees varying acceleration as it...
46. ### Simple Harmonic Motion and equilibrium of springs

The figure below shows a system in equillibrium. The pulley and the springs (both with constants "k") are ideal. The period of oscillation of the mass A is given by: Relevant equations: F = -kx (SHM) I tried to do a "force diagram" and set up some geometric relations but it´s not working.
47. ### Simple Harmonic Motion/Period of a Physical Pendulum

I'm studying the motion of a physical pendulum, could someone help me make the final step in figuring out how to find the period so I can make predictions before carrying out a practical? Basically I have a meter rule with holes drilled along the length and will be pivoting it at various points...
48. ### Trouble finding the amplitude in a SHM problem

Homework Statement a horizontal spring-mass is composed of a spring with constant 10.0 N/m and an 80.0 gram mass on the end of the spring. the surface supporting the mass is friction less. when the system is first observed, the spring is extended 1.30 cm and the velocity of the mass is 54.1...
49. ### Calculating Amplitude - SHM

I have encountered two separate review problems that have to do with finding a value for amplitude and I am really struggling with it. 1. Homework Statement Question 1- A mass of 3kg is free to move on a horizontal frictionless surface and attached to a spring of k=15 N/m. It is displaced...
50. ### Why are there modes in cantilever beam oscillation equations

I'm doing an experiment measuring the relationship between length of a cantilever beam and period of oscillation when I twang it on one end, but I can't seem to understand the equation. The equation for measuring frequency is given here:https://www.hindawi.com/journals/amse/2013/329530/ but I...