# What is Spring mass system: Definition + 51 Threads

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

F

=

k

x

,

{\displaystyle {\vec {F}}=-k{\vec {x}},}
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:

Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Decay to the equilibrium position, without oscillations (overdamped oscillator).The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

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1. ### Free Body Diagram of Mass-Spring System

Suppose we are given the 2 following masses 1 and 2, where 1 initially moves at velocity ##v_\rm{1}## and 2 is stationary. Note, however, that 2 is not bolted down to any surface - it is free to move around after collision. What would the free-body force diagram of masses 1 and 2 be, after they...
2. ### Potential of particles moving on a circle attracted by elastic force

I use ##l-1## lagrangian coordinates ##\alpha_1,...,\alpha_{l-1}## . ##\alpha_i## is the angle between ##OP_{i-1}## and ##OP_{i}##. As the length of a chord between two rays with angle ##\alpha## is ##d=2Rsin(\alpha/2)##, I write the potential energy of the system as...
3. ### Stretching of a rotating spring

TL;DR Summary: Are the k and the w linked? Yesterday I came across this problem: A mass is attached to a spring and the system rotates (one of the spring end is fixed) in an horizontal plane. Given the mass m, the value of k, the length of the spring l_0 and the angular velocity w, compute...
4. ### Time period of SHM & Energy conservation in pulley-spring-block system

TL;DR Summary: I was solving this problem given in a book. The answer I got was wrong and seems to violate the conservation of mechanical energy. Yet the forces were balanced. Can someone provide an explanation. So here is the problem: In the above arrangement, I had to find the time period...
5. ### I Work-Energy Theorem for Moving Block and Spring System?

In the chapter of center of mass and linear momentum, there are multiple problems involving two block systems connected by a string, with both blocks given certain initial velocities. The goal is to calculate the maximum elongation or compression in the spring. For example, consider this...
6. ### Proving SHM for charged spring mass system in electric field

For part (f) Solution is I don't understand why the bit highlighted in yellow is true. Would anybody be kind enough to help.
7. ### Spring-mass system with a pendulum using Lagrangian dynamics

I'm stuck in a problem of a spring mass system with a pendulum attached to it as showed in the figure below: My goal is to find the movement equation for the mass, using Lagrangian dynamics. If the spring moves, the wire will move the same amount. Therefore, we can write the x and y position...
8. ### Signal transfer properties of a spring-suspended platform?

Ok, the last time I did something with physics, save for simple in-game simulation of gravity, was many moons ago in high school. I certainly did not learn to do this in high school, although two subjects I had there at a basic level, are touched: Waves and Classical mechanics. I have no idea...
9. ### Question about the nonlinear Hooke's law

hi guys i saw this problem online about using the MATLAB ode45 to solve the nonlinear Hoock's law and its specifically stated that the nonlinear hoock's law is given by $$F = k\;u + \epsilon\;u^{3}$$ , but when expanding the potential function in a Taylor series where you obtain the force...
10. ### Lagrangian of a spring mass system

I know that from the given problem, I need to find the expression for Kinetic energy, KE = 1/2 m [r(dot)]^2 and Potential energy, PE = 1/2 k r^2 So L = 1/2 m [r(dot)]^2 - 1/2 k r^2 Hence H = 1/2 m [r(dot)]^2 + 1/2 k r^2 I assume that the fixed length r0 is provided to find the value of end...
11. ### Energy Conservation as an explanation for v=0 for spring mass system

My Solution: For the displacement graph, the gradient is crucial to predict the behaviour of the displacement of the block through time. At 1: System is released - velocity is zero, considering forces acting on block, kx < mg, as block is observed to move downwards, and object is...
12. ### How to find the deceleration of a mass colliding on a spring?

Problem Statement: A known mass at a know velocity collides on a spring of known stiffness. What is the equation that governs the deceleration of the mass, so that the force on the spring could be found? Relevant Equations: 1/2 m*V^2 = 1/2*k*x^2 + 1/2*m*(Vo)^2 Kinetic energy of mass before...
13. ### Mathematical modeling of driveline impact test of an automobile

I am in a process of making a mathematical model of an automobile driveline validation procedure called "Driveline impact". In this test, after the engine in cranked, first gear is selected. While clutch pedal is still pressed, engine rpm is raised unto its max. torque deliverable speed. In this...
14. ### What is the governing equation of a spring with sinusoidal excitation?

Hi, Most of the spring vibration lectures assume spring to be fixed on one end and mass on the other end [Example]. In my case, spring has a sinusoidal excitation on one end and mass on other end. Pl. refer the image below. How to get the governing equation? With that I also want to find the...
15. ### 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...
16. ### Modes of Vibration of 3-DOF Spring Mass System

Homework Statement Homework Equations a11 = a21 = a31 = a12 = a13 a22 = a32 = a23  \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} =ω^2m \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}...
17. ### Spring mass system attached to a disc

Hello to all, first sorry about any mistakes because English is not my native language. I'm trying to solve this problem and I can't seem to find the equation of motion Homework Statement The disc and the point A both have mass m; θ(0) = 0 2. The attempt at a solution I tried to use the...
18. ### Position of Mass (Spring Mass Sytem)

Homework Statement A force of 10 Newtons can stretch a spring by 0.04 m. Suppose a mass of 5 kg is attached to the lower end of the spring. We stretch the mass downward by 0.05 m from its equilibrium position and release it from rest. Determine the position of the mass relative to its...
19. ### I with the free vibration of bar problem

Homework Statement Find the free vibration of the bar supported by a spring and a viscous damper (See attached figure) Homework Equations (a)Derive the equation of motion (b) Obtain the moment of inertia of bar (c) obtain natural frequency when d=0. The Attempt at a Solution
20. ### Finding the total response of an undamped spring mass system

Homework Statement Homework Equations The response of a spring mass system can be simplified to equal: x(t) = (x0 - (F0 / (k - mω2))cos(ωnt) + (x'/ωn)sin(ωnt) + (F0 / (k - mω2))cos(ωt) where x & x' are the initial conditions ω is the exciting frequency ωn is the natural frequency k is the...
21. ### Designing a driven spring mass system for school project

Homework Statement I have a math project due in a month. The topic i chose to present was second order differential equations, so i felt like making a driven spring mass system to demonstrate resonance and nice things like that. I have a few questions about materials and design and such and...
22. ### Kinetic/Potential energy of spring mass system

Homework Statement Question: Does the force F(t) acting on mass m1 affect the equations for kinetic and potential energy of the system? Homework Equations F(t) is known V(q) is potential energy | T(q) is kinetic energy The Attempt at a Solution I think PE is not affected. I think KE...
23. ### Need confirmation for spring mass system question

Homework Statement A mass of 1.20 kg. originally at rest, sits on a frictionless surface. It is attached to one end of an unstretched spring (k = 790 N/m), the other end of which is fixed to a wall. The mass is then pulled with a constant force to stretch the spring. As a result, the system...
24. ### Simple Harmonic Motion problem with spring mass system

Homework Statement A 50g mass is attached to a spring and undergoes simple harmonic motion. It's maximum acceleration is 15m/s and maximum speed is 3.5m/s. Determine a)angular frequency, b) spring constant, c) amplitude. Homework Equations ω = √k/m X(t) = A*cos(ωt + ∅) The Attempt at a...
25. ### Regarding Energy and Work done in a Spring Mass System

Homework Statement I am trying to solve a spring mass system (mass m and constant spring stiffness k) where instead of applied force a displacement is applied as u=sin(ωt) where ω is the frequency and t is the time. I need to find the difference of the total work done and the energy (KE and PE)...
26. ### Oscillation in spring mass system questions

Homework Statement I am having an issue with answering number 4 in the attached image. Homework Equations Relevant equations are given in question 3. The Attempt at a Solution Squaring the equation would only make it easier to solve for any of the other variables, or the period...
27. ### Acceleration of Spring Mass System - oneplusone

I always see those spring mass type questions on AP M.C. questions. Ones with a horizontal spring connected to a mass. I understand that at the maximum amplitude, the velocity is zero since it is changing direction, and the displacement is maximum. However, what is the acceleration like? I...
28. ### Velocity of spring mass system with spring mass

I am having a hard time figuring out how to calculate the velocity of a mass in a spring-mass system with respect to distance, where the spring has a mass to it. The spring would be a long steel cable and a force of 4000 lbs would be applied to the system along the line of action. The cable is...
29. ### Graphical analysis of a spring mass system over a function of time

Homework Statement Homework Equations http://en.wikipedia.org/wiki/Simple_harmonic_motionThe Attempt at a Solution part a) frequency of oscillation can be found on the graph, to find frequency I need to find the period (T) first, the period is when the object is back to it's initial...
30. ### How Is the Amplitude Calculated in a Damped Spring-Mass System?

Homework Statement A mass of 4 kg is stretches a spring by 1 m. An external force of cos (!t) N acts on the mass. Assume that the damping constant is nonzero and gravity is 10 ms^-2. Consider a spring mass system described by the following IVP. u''+yu'+u = cos(wt) u(0) = 0 u'(0) = 0 1) Find...
31. ### Potential Energy in a vertical spring mass system

Homework Statement A mass of 0.5 kg hangs motionless from a vertical spring whose length is 0.90 m and whose unstretched length is 0.45 m. Next the mass is pulled down to where the spring has a length of 1.15 m and given an initial speed upwards of 1.4 m/s. What is the maximum length of the...
32. ### Equation of motion: spring mass system - free undamped vibration

Homework Statement i have uploaded my question. please check out the attached .pdf file. Homework Equations The Attempt at a Solution
33. ### Spring Mass System | Calculate Mass m

Sorry for such a simple question. Unfortunately, this problem was originally written in Japanese and was just translated in English by a coworker. I don't know if there was a problem during the translation but we cannot produce the same answer as was shown in the Japanese answer key. Problem...
34. ### Spring mass system and Simple harmonic motion

Homework Statement A block of mass 300 gm is attached to a spring with spring constant 1000 N/m. A smaller block of mass 100 gm is pushed against the first block, compressing the spring 5 cm. The coefficient of static friction between the two blocks is u = 0.2. The blocks are then released...
35. ### Stability of a spring mass system in two dimensions

Homework Statement Attached Homework Equations 5.104 mentioned in the problem is Also attached The Attempt at a Solution Suppose the mass is moved to (x,y), the distance from the mass to the left spring is denoted by A, the distance from the mass to the right spring is denoted...
36. ### Conservation of Energy with Spring Mass System on Inclined Plane

Homework Statement (Figure attached to post) A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in Fig. 7.5 and eventually stops. If we...
37. ### Spring Mass System: Calculating Frequency, Acceleration, and Velocity

Homework Statement a mass attached to the lower end of a vertical spring causes the spring to extend by 25mm to its equilibrium position. The mass is then displaced a further 20mm and released. a trace of the vibration and time mesurements are taken. From these mesurements it can be seen...
38. ### Spring Mass system where springs also have mass

hello all, this will be my third time posting this question, i thought perhaps posting in the homework section might have it be answered. Recently I have been attempting to model a spring system (see below) where the springs themselves will have mass. I should know how to do this but i accept...
39. ### How Do You Calculate Mass and Acceleration in a Spring Mass System?

Homework Statement An object suspended from a spring with a spring constant of 2.56 N/m vibrates with a frequency of 0.148 Hz a. What is the mass of the object b. What is the acceleration of the object at a displacement of -0.120m from the equilibriam position Homework Equations a...
40. ### Effect of m, A, k, & Phase Lag on w in Spring Mass System

1. For a given spring mass system, what would be effect on w if: a) incease size of mss, B) inceasing amplitude of vibration, C) increasing spring stiffness, D) increase phase lag 2. w= sqrt (k/m) 3. Therefore using above w = sqrt (k/m) a. Reduces b. no change c. Increases d...
41. ### Solving Spring Mass System Homework: Equations of Motion & Linearization

Homework Statement The system depicted in the attachment consists of a slender rigid link of mass m1 hinged at point O, and supported by a spring of stiffness k2 (200N/m) at point B. The link in turn supports a spring of stiffness k1 (400N/m) at point A connected to a point mass m (2kg)...
42. ### Conservation of energy in a spring mass system help

hi i hope some one can help me with this I'm going out of my mind with it lol. I'm new to this forum so please forgive any errors in where this thread is or if it contains something it shouldn't. a force of 160N is applied to a sliding mass of 13kg fixed on top of a spring. the spring force...
43. ### Spring mass system with applied force kinetic energy

I'm trying to work out a model for a spring mass system with a force acting at the centre of mass of the mass using Lagrangian mechanics. I can't work out the kinetic energy. I know the kinetic energy \text{KE}=\dfrac{1}{2}mv^2. I also have W=\int_a^b F \, dt. Should I use \Delta \text{KE} +...
44. ### What Is the Energy of a Viscously Damped Spring-Mass System?

Homework Statement Use your program for the viscously damped spring to solve this problem. Make sure you change the parameters as specified below. A mass m=7.5 kg is attached to the end of a spring with a spring constant of k=18.3 N/m. The mass moves through a viscous damping medium with a...
45. ### Energy in a vertical spring mass system

Homework Statement I am doing a lab in which we are to show that the energy in a spring mass system is constant throughout the oscillations. Homework Equations I set y initial = 0 to be the point where the spring was in equilibrium when the mass was attached to it. At any given point...
46. ### Double Spring Mass System - Maximizing Velocity

Looking for some pointers on how to approach this problem: I'm considering a system like the following - two springs and two masses connected in sequence: http://www.myphysicslab.com/dbl_spring1.html I would like to find the following: Given m1, m2, k1, k2. The springs are...
47. ### SHM horizontal spring mass system

This problem states that a mass m is attatched to a spring with spring constant k and is pulled aside a distance A from its equilibrium position x=0. a) what is the speed as it passes the position x=0? I know that at A and -A v=0 and at x=0 v=max, but I can't remember how to get an equation...
48. ### What Steps Are Needed to Determine the Mass in a Spring Oscillation Problem?

[b]1. A block of unknown mass is attached to a spring with a spring constant of 6.5 N/m and undergoes simple harmonic motion with an amplitude of 10 cm. when the block is halfway between its equilibrium position and the end point, its speed is measured to be 30 cm/s. Find the mass of the block...
49. ### How can the force of impact on a falling spring mass system be calculated?

i have a spring with stiffness k. The spring is resting on the ground. Now a mass is kept over the spring. Now the spring deforms by an amount x=mg/k where g is the gravity's acceleration.The force acting on the spring is mg Newtons and the reaction force from the ground is also equal to mg...
50. ### Understanding spring mass system

I am just learining basic dynamics from spring mass system. there is a horizontal spring fixed to the wall and a mass is at the end. When the mass is displaced by a force F to the right the spring stretches to a distance X. Now i draw the STATIC FBD. The eqn is f-kx=net force on the mass...