# What is Simple harmonic motion: Definition and 913 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. ### Advantages of a cone-shaped spring?

I'm doing a personal experiment where I take a conical spring (that is, a spring with two different diameters on either end), hang it from the ceiling, and measure the period of oscillation for different masses hanging below the spring. I do this for two different orientations of the spring; one...
2. ### Position and acceleration in simple harmonic motion given velocity

A spring attached to a mass undergoes simple harmonic motion. From Newton's second law we have ##ma=-qx## where ##q## is the spring constant. $$x''+\frac{q}{m}x=0$$ A second order equation with constant coefficients. The characteristic equation is ##r^2+\frac{q}{m}=0##. The roots are...
3. ### What is the max amplitude before the sand particles slip off the vibrating plate?

A horizontal metal plate connected to a vibration generator is oscillating vertically with simple harmonic motion of period 0.080 s and amplitude 1.2 mm. There are dry grains of sand on the plate. The frequency of the vibrating plate is kept constant and its amplitude is slowly increased from...
4. ### What is the definition of a way and how does it relate to travel?

I put 0, but that is incorrect. Why is 0 an incorrect answer? This is confusing, as if the pendulum is in free fall, wouldn't there be no SHM at all?
5. ### I Solutions to Simple Harmonic Motion second order differential equation

All simple harmonic motion must satisfy $$\frac{d^2s}{dt^2}=-k^2s$$ for a positive value k. The most well known solution is the sinusoidal one $$s=Acos/sin(\omega t + \delta)$$ A is amplitude, ##\omega##is related to frequency and ##\delta## is phase displacement. My lecturer said that there...
6. ### 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...
7. ### Solving for Simple Harmonic Motion: A Picture Problem

Here is a picture of the problem It is not clear to me how to really prove that the equation for ##\theta(t)## is simple harmonic motion, and what the period of this motion is.
8. ### 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...
9. ### Bungee jump | simple harmonic motion

As you all know, a bungee jump is where a person is tied to a cord and the person jumps off and bounces up again. The natural length of a cord is 75 metres. Then when a person is attached onto the cord, the length becomes 83 metres when the person is at rest. I am sure that the person is not...
10. ### B Correct SHM Equation: Does € Matter?

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 €?
11. ### I The speed of a waves on a string in Simple harmonic motion

The speed of a wave in simple harmonic motion on a string is $$v= \sqrt{\frac{F}{\mu}}$$ where v= the horizontal velocity of the wave on a string. Is the F the horizontal force or the resultant force (combination of Fy and Fx)?
12. ### B Is simple harmonic motion also a pure translatory motion?

Is simple harmonic motion also a pure translatory motion?"A rigid body moves in pure translation if each particle of the body undergoes the same displacement as every other particle in any given time interval" [Halliday and Resnick, Physics].If not,then how does shm deviate from this definition>
13. ### An object oscillating in simple harmonic motion

Since it passes through the origin every ##3.6s## the period is ##T=3.6s## hence ##\omega=\frac{2\pi}{\omega}=\frac{2\pi}{3.6}\frac{rad}{s}## thus ##A=\frac{v_{max}}{\omega}=\frac{1.2}{\frac{2\pi}{3.6}}m\simeq 0.69m## and ##a_{max}=\omega^2 A=(\frac{2\pi}{T})^2 A=(\frac{2\pi}{3.6})^2 \cdot...
14. ### Superposition of two simple harmonic motion

Hey! I am stuck in this problem, i don't know how to sum this ecuations. I remember that its possible because the direction is the same So, i try to sum like this: cos (t+5325) + 1.5 cos (t+5325) =1.5 cos (t+5325) I don't know if i fine. I thanks your help, please ;)
15. ### Understanding the Forces Behind Part B of the Graph

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...
16. ### 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...
17. ### Is Your Calculation of Simple Harmonic Motion Accurate?

The first ecuation values i am 99% that is correct. But, in the second and three problem i don't know if my results are ok. The problem number 2 i comprobate with the teacher that te aceleration its correct, so, with this i calculate the velocity. I use like example the second problem for try...
18. ### 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...
19. ### Simple Harmonic Motion of a Mass Hanging from a Vertical Spring

Assuming zero spring mass and zero friction, At the greatest value of x, the loss in gravitational potential energy should equal the loss in elastic potential energy. so I did (1/2)kx^2=mgx to isolate x in the formula, x=(2mg)/k then I plugged in my values so: (2*13.6*9.81)/8.8= 30.3218...
20. ### 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...
21. ### 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...
22. ### Why Might a Math Solution Involve Doubling the Percentage Difference?

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%...
23. ### Simple harmonic motion problem

How is the answer to 3 (d) is found?
24. ### 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...
25. ### Simple Harmonic Motion Question

First I use young's modulus to solve for delta y. I get 5.67x10 -5. I am not sure what to do after this, but this is my attempt. Next I do T = 2delta y sqrt(m/k) (I am not sure if I am supposed to put 2 delta y) Solving for f, i get f = 1/(2delta y sqrt(m/k)) F = kx, mg = kx, m = kx/g...
26. ### Energy loss in simple harmonic motion causes the time period to shorten?

https://www.asi.edu.au/wp-content/uploads/2016/10/ASOEsolns2012.pdf Q11 D) Markers comments: Few students reached part (d) and very few of those who did realized that the amplitude does affect the time taken for each of Mordred’s bounces. i.e. the energy losses results in shorter periods...
27. ### Classical Book for simple harmonic motion

sites or books for SHM high school and undergrad level. i want to understand SHM from the ground up and I am finding difficulty with my current sources
28. ### Simple harmonic motion homework

I don't know how to start doing this homework. I would like help to orient myself.
29. ### Simple harmonic motion equations as a function of time

I conducted a mass-sprig experiment to see how stiffness of a spring and mass affect the frequency of oscillation. In addition to this to this i have to plot a graph to show displacement,velocity and acceleration of the mass as a function of time.From my research online For the displacement as...
30. ### Simple Harmonic motion calculation for a mass on a spring

Im not sure how to find k if I'm not given a force or period.
31. ### Zero Amplitude Damped Simple Harmonic Motion with k=0.7s^-1 and f=3Hz

Hi guys sorry if this is the wrong thread, I have a damped simple harmonic motion pictured below, i have to find the inerval t=0 and t=1 for which the amplitude of x(t) is considered to be zero. The behaviour of the graph below can be described as e^-kt cos(2πft) k=0.7s^-1 and f= 3Hz
32. ### Hello Reality Anyone familiar with the Davisson-Germer Experiment?

Greetings, I'm happy to find such an enthusiastic community with an encyclopedic knowledge and mathematical rigor. I'm a Biomedical Engineering Researcher that's had to breach into the world of condensed matter physics to better understand the physical principles of the piezoelectric crystal...
33. ### 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.
34. ### How to know whether motion is simple harmonic motion or not?

I am reading "Coulomb and the evolution of physics and engineering in eighteenth-century France". There it is said in page 152 para 1 that "Coulomb found that within a very wide range, the torsion device oscillated in SHM". My questions are: (1) By just looking at the time period of the...
35. ### 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...
36. ### Simple Harmonic Motion Amplitude

Using A = x0, B = v0/ω I get ω = 4π, A = 1, B = 1/4π then converting to phase/magnitude form \sqrt{A^{2} + B^{^{2}}} = \alpha \sqrt{1^{2} + \left ( \frac{1}{4\pi }\right )^{^{2}}} = \alpha = \frac{1}{4\pi }\sqrt{16\pi^{2} +1} However the answer in the back of the book has α = 1 Is...
37. ### 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...
38. ### Initial conditions in simple harmonic motion

I have the formula for amplitude ##A=\sqrt (x_0^2 + \frac{\dot x_0 ^2}{\omega^2})##. But ##x_0## and ##\dot x_0## refers to the initial conditions, and the information that I'm given is not related to the initial conditions, or at least I'm not told so.
39. ### Finding equations for simple harmonic motion given a graph

Well, this is a problem which makes you think more about concepts than numbers, so I want to see if I've done it correctly. 1) I draw a simple pendulum in an elevator, where you have weight, tension and a pseudo-force. In this situation the effective gravity may be changing due to different...
40. ### Prove that a mass has simple harmonic motion

If I write Newton's equations, seen inside the room and with non tilted axis we have: ##x) N.sin(\alpha)-Fe.cos(\alpha)=m.a_x## ##y) N.cos(\alpha)+Fe.sin(\alpha)-m.g-f*=m.a_y## Where ##f*=ma##, ##Fe## is the elastic force. Then, how can I realize about simple harmonic motion? I also can think...
41. ### Simple harmonic motion -- manipulating SHM equations

My workings: ##D(t) = Asin\omega t## ##v(t) = \frac{\text{dD}}{\text{dt}}=Acos(\omega t)\omega## ##v(t) =Acos(\omega t)\omega## When displacement half of amplitude, ## Asin\omega t## = 0.5##A## ## sin\omega t## = 0.5 ##v(t) =Acos(\omega t)\omega## ##v(t) =\omega (0.5Asin\omega t)cos \omega t ##...
42. ### 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...
43. ### I Find Frequency of Block Oscillation Due to Shear Force

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...
44. ### 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)...
45. ### How to find the velocity of a wave in simple harmonic motion given time

The graph provided is below. The problem asks for the speed of the wave at 0.12s. I used the formula v=w*xmax*cos(wt), provided in our textbook where xmax is the amplitude of 2 cm, w (omega) is 2pi divided by the period of 0.2. However, for some reason this formula doesn't give me the correct...
46. ### Simple Harmonic Motion Experiment Problem

I think you could try to solve for the forces based on when the spring falls from an incline at various angles theta, but I am not sure. Or spring potential energy? I'm really confused. Is there any other method? Could it involve using water and wave harmonics? (We learned waves and sound...
47. ### Simple Harmonic motion of a Pendulum

First, I decided to solve for the coefficient in front of the cosine simple harmonic function for velocity. I know there is max velocity of 30cm/s at time = 0 , so I plug it into velocity function. xmax * w = A v(t) = Acos(wt) 0.3 = Acos(w*0) A = 0.3 Then I have my velocity function...
48. ### Is this Simple Harmonic Motion?

I've been going to the theme park almost every year-and this year in my Physics class we are learning mechanics, more specifically Simple Harmonic Motion. My teacher told us that for an object to have 'Simple Harmonic Motion' it must have oscillatory motion (like a pendulum going back and...
49. ### Simple harmonic motion of a bar pivoted at one end

Hi, I am unsure how to proceed with this problem. I believe that I can correctly calculate the frequency of the oscillations for a bar that is not suspended from a spring but I do not know how to take the effect of the spring into account. The answer given by my professor is ...
50. ### 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...