What is Mass-spring system: Definition and 48 Discussions

In a real spring–mass system, the spring has a non-negligible mass


{\displaystyle m}
. Since not all of the spring's length moves at the same velocity


{\displaystyle v}
as the suspended mass


{\displaystyle M}
, its kinetic energy is not equal to





{\displaystyle {\tfrac {1}{2}}mv^{2}}
. As such,


{\displaystyle m}
cannot be simply added to


{\displaystyle M}
to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to


{\displaystyle M}
to correctly predict the behavior of the system.

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  1. mattlfang

    Find the velocity and acceleration of a pulley in a mass-spring system

    This looks like a classical setup but I can't find a solution. We can calculate the energy of the system by looking at the work done by the gravity and the spring. But how do we divide the energy between the kinetic energy of the pulley and the rotation of the pulley?
  2. M_Abubakr

    Infinite-Extent 2D Mass-Spring System vibration and dispersion relation

    Hi, I am trying to find equation of motion and its solutions for a 2D infinite lumped mass spring system as depicted in figure. All the masses are identical, All the springs are identical, and even the horizontal and vertical periodicity is the same n=a. I need to try find dispersive relation...
  3. U

    Derivation of the oscillation period for a vertical mass-spring system

    I understand the derivation of T= 2π√m/k is a= -kx/m, in a mass spring system horizonatally on a smooth plane, as this equated to the general equation of acceleration of simple harmonic motion , a= - 4π^2 (1/T^2) x but surely when in a vertical system , taking downwards as -ve, ma = kx - mg...
  4. Moara

    Max Elongation of Mass-Spring System

    For the maximum elongation the velocities of the blocks should be equal. With that i used two conservation of energy for m1 and m2, for m1 the work of the elastic force is integral of kx(dx1) and for m2 kx(dx2), where X1+x2=x. I got three equation and four variables since i don't know the final...
  5. Y

    Solve a system of two linked harmonic oscillators

    $$m_1 \ddot{x} - m_1 g + \frac{k(d-l)}{d}x=0$$ $$m_2 \ddot{y} - m_2 \omega^2 y + \frac{k(d-l)}{d}y=0$$ It is two masses connected by a spring. ##d=\sqrt{x^2 + y^2}## and ##l## is the length of the relaxed spring (a constant). What is the strategy to solve such a system? I tried substituting...
  6. P

    Mass-spring question (frequency and amplitude)

    I am confused by this question and was hoping someone could clear this up for me, I know this is simple. A mass-spring system vibrates at 10 Hz. Ideally, i.e., without friction, it would continue forever at the same amplitude. In practice, its amplitude is found to decay such that it decreases...
  7. C

    Value of Spring Stiffness k in Free Vibration of Mass-Spring System

    A mass-spring system is in free vibration after an initial excitation. There are no outside forces acting on the system. What is the value of the spring stiffness k (units of N/m; round your answer to a single decimal place)? Mass m = 0.6 kg Amplitude A = 0.4 Using this equation: z(t) = A sin...
  8. L

    Amplitude of damped mass-spring system

    Homework Statement Given the IVP \ddot{r}+\dot{r}+20r=0\\ r(0) = 0.8\\ \dot{r}(0) = 0 for the length of an oscillating spring (damped), we find that the general solution is r=e^{-0.5t}[0.8\cos(\sqrt{19.75}t)+\frac{0.4}{\sqrt{19.75}}\sin(\sqrt{19.75}t)] and I wish to find the curve bounding...
  9. S

    Probability density function of simple Mass-Spring system.

    Homework Statement We know that after long run of simple mass-spring system, there should be a probability of finding the mass at certain points between -A and A.. Obviously in probability of finding the particle near A or -A is higher than finding the particle at 0, because the speed is the...
  10. D

    Mass spring system - unknown spring constant

    Homework Statement Problem: [/B] Given the mass spring system solve for k1 The natural frequency wn = 10 s-1 k1=2k2 m=1kg Homework Equations wn=(k/m).5 The Attempt at a Solution solve for k1 m x w2n =2.5k1 k1 = 40 N/mNote: I calculate 40 N/m and the solution states 250 N/m. I think I...
  11. A

    Frequency of damped mass-spring system

    [Mentor's note: Thread title changed to reflect question content] I really need some help with this one: 1. Homework Statement An unhappy rodent of mass 0.307kg , moving on the end of a spring with force constant 2.48N/m , is acted on by a damping force Fx=−b⋅vx. Part A If the constant b has...
  12. arpon

    Problem on a mass-spring system

    Homework Statement Suppose the surface is completely frictionless. Will the spring experience any length change?[/B]Homework EquationsThe Attempt at a Solution To change the length of the spring, force should be applied from both ends. In this case, there is no force of friction. So, my...
  13. A

    Conservation of energy: mass-spring system

    1. A 1 kg block is on a flat frictionless surface. Attached to a relaxed spring (k=50N/m). A light string is attached to the block and runs over a frictionleas pulley to a .45kg dangling mass. If the dangling mass is released from rest, how far will if fall before stopping? Homework Equations...
  14. N

    Mass-spring system (not exactly homework)

    Case of a spring with a mass,m, that has been stretch beyond the equilibrium in the positive x direction. mg + (-kx0) = 0 k = mg/x0 stretch: F = mg+ [-k(x0+x1)] = mg-kx0-kx1 but since mg-kx0 = 0 F = 0-kx1 = ma ma = -kx1 a = -kx1/m How do I arrive at the corollary from a = -...
  15. M

    Equation of motion of a mass-spring system

    hi, all. I am trying to derive the equation of motion of a mass spring system without using the energy method but I am wrong somewhere and I can't find it, can you help me find where I am wrong. Equation of motion of a simple mass spring system is indeed mx''+kx=0 but here I am thinking that...
  16. Y

    Mass-spring system: Steady state solution

    Homework Statement Find the steady-state motion of the mass–spring system modeled by the ODE: 4y''+12y'+9y=225-75sin(3t) Homework Equations for a diff eq modeled as: my''+cy'+ky=F0cos(ωt), yp=acos(ωt)+bsin(ωt) a=F0*(m(ω02-ω2))/(m2*(ω02-ω2)2+ω2c2) b=F0*(ωc)/(m2*(ω02-ω2)2+ω2c2)...
  17. K

    Solve for position given force in a mass-spring system?

    Picture of the system (it's sideways): http://tinypic.com/r/28qq4pi/5 Consider the mass from the spring in the figure. Let m=1, k=1, and g=9.81. Assume that x and y are zero when the 2 springs are unstretched. Assume the system starts from rest with the first spring extended by an amount...
  18. F

    Mass-spring system in SHM concept question

    Homework Statement Homework Equations The Attempt at a Solution Well since spring is in SHM, only conservative forces are at play here. So using conservation of energy, the kinetic energy would be the change in potential energy. Which I have set up as KE = 1/2k(A^2)2 -...
  19. C

    Solving Mass-Spring System: Find Spring Constant & Max Stretch

    Homework Statement A mass of 122 g is attached to a vertically hung spring. The mass stretches the spring 13.8 cm. (a) What is the spring constant of the spring? (b) If the mass is dropped from rest from a position where the spring is just relaxed, what will be the maximum distance the...
  20. T

    Amplitude of horizontal mass-spring system

    Homework Statement This is a horizontal mass spring system in a simple harmonic motion problem set. Vmax = 20 m/s F = 10 N m = 0.5 kg Find Amplitude (A) and spring constant (k) Homework Equations The Attempt at a Solution I could not figure out a way to solve this problem, and the only...
  21. Z

    Differential equation for motion in mass-spring system with impulse

    I am reviewing for a final and I don't know how an impulse affects the differential equation for motion in this mass-spring system. Can someone please help? A mass m=1 is attached to a spring with constant k=2 and damping constant c=2. x(0)=0 & x'(0)=0. At the instant t=π, the mass is struck...
  22. S

    Vertical mass-spring system - force or energy?

    A lot of problems I see have vertical mass-spring systems, where a mass m hangs from a spring with spring constant k stretching it a distance D, and usually all but one of those quantities is known. But would you equate the forces or the energies, i.e. which is the correct equation to use: mg=kD...
  23. C

    Help Needed: Deriving Expressions for Mass-Spring System

    Could somebody help me in deriving the following expressions? I can judge that the first equation is a common expression for finding Keq for a mass-spring system in series.. Can't do the rest :/ anybody please ? Thanks in advance.
  24. B

    The natural frequency of a mass-spring system.

    Homework Statement A certain spring elongates 9mm when it is suspended vertically and a block of mass M is hung on it. The natural frequency of this mass-spring system is: a)0.014 b) 5.3Hz c) 31.8Hz d) 181.7 e) need to know M x=9mm mass=M Homework Equations I don't know any...
  25. K

    Mass-spring system on an incline

    A spring is mounted at an angle of theta = 39degrees on a frictionless incline as illustrated in the figure below. The spring is compressed to 15 cm where it is allowed to propel a mass of 4.9 kg up the incline. (a) If the spring constant is 580 N/m, how fast is the mass moving when leaves...
  26. H

    Inverse Laplace of a Mass-Spring System

    Homework Statement Given a transfer function in the Laplace Domain Detemine an expression for x(t), given f(t) is a sinusodial input with frequency omega = root(k2/m2) and amplitude of 1 N (initial conditions equal 0)Homework Equations...
  27. B

    Solving the Oscillation of Mass-Spring System

    Homework Statement Mass m attached to spring with spring constant k=Am. It feels a resistive force magnitude Bmv where v is the speed. and A, B are constants such that 4A > B^2 What is the fractional change in amplitude of oscillation in one complete oscillation? Homework Equations...
  28. E

    Mass-spring system on an incline

    Homework Statement A spring is mounted at an angle of = 38 on a frictionless incline as illustrated in the figure below. The spring is compressed to 15 cm where it is allowed to propel a mass of 5.4 kg up the incline. If the spring constant is 565 N/m, how fast is the mass moving when leaves...
  29. N

    2D isotropic mass-spring system, need to find maximum distance from origin

    Homework Statement I have a 2D isotropic mass-spring system. The mass is pulled a distance A=1m, and then given an upwards kick with a velocity v_0. The k=1, m=1kg. I need to find the furthest distance from the origin the mass will travel in its orbit. Homework Equations x(t)=A_x\cos(\omega...
  30. V

    Mass-Spring System Beats and Resonance

    Beats and Resonance In the Beat not have friction force, correct ? m \frac{d^2x}{dt^2} + kx = F_o cos(wt) We can write as \frac{d^2x}{dt^2} + w_o^2 x = \frac{F_o}{m} cos(wt) If w \not= w_o Assuming (Particular solution) x_p = acos(wt) + bsin(wt) Why we have assuming this ...
  31. V

    Mass-spring system damped

    Damped vibration m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0 Characteristic equation is mr^2 + \gamma r + k = 0 r_1 = \frac{- \gamma + \sqrt{( \gamma )^2 - 4mk}}{2m} r_2 = \frac{- \gamma - \sqrt{( \gamma )^2 - 4mk}}{2m} In overdamped ( \gamma )^2 - 4mk > 0 What I...
  32. V

    Solve Mass-Spring System Vibration Free

    Vibration Free Please, are correct? m \frac{d^2x}{dt^2} + kx = 0 Where frequency is w = \sqrt{\frac{k}{m}} \frac{d^2x}{dt^2} + \frac{k}{m}x = 0 The characteristic equation is: r^2 + w^2 = 0 r = +or- iw where i^2 = -1 Then x(t) = C_1e^{iwt} + C_2e^{-iwt} Calculating...
  33. K

    Dual Mass-Spring System help

    Lets see if anyone can help me with this. I have to derive a transfer function for the following: A small satellite with a moment of inertia J1 that has a instrument with a moment of inertia J2. The instrument is at the end of a small strut that has a stiffness constant of k and a damping...
  34. C

    Converting a simple mass-spring system to state-space model, how?

    Homework Statement Hi guys/girls Professor gave this very simple homework where I need to convert the system below to a state-space model. The system itself is represented by the equation m*x" + k*x = f(t) Where m = 5 and k = 1. Note that, " (doublequote) is a second-order derivative...
  35. D

    Solving a Mass-Spring System: Max. Displacement of Mass

    Homework Statement A block of mass 1.2kg, resting on a horizontal surface of coefficient of kinetic friction of .15, is attached to a linear spring of force constant 12.0 N/m. When the spring is unstretched the mass is at a position x=0. THe mass is pulled to the right streching the spring a...
  36. T

    Simple Mass-Spring System Problem

    Simple Mass-Spring System Problem! A mass M slides across a frictionless horizontal table with constant speed V_0_. It collides with a spring of spring constant k, compressing it. The mass-spring system then rebounds. Take the position of the mass when it first hits the spring to be x=0. a)...
  37. D

    Mass-spring system; Hill equation

    When trying to solve a problem I arrive at the following equation of motion / Hill equation: \frac{d^{2}y}{dx^2} + \frac{4 k_0}{m w^2} cos(2x)y = 0 There exists a value x_0 such that for all x>x_0 the motion is stable. I actually don't know what is meant by this 'stability'. Can...
  38. A

    Does a vertical mass-spring system adhere to the law of conservation of energy?

    does anybody know if the action of a real mas-spring system suspended vertically, support or refute the law of conversation of energy
  39. G

    Mass-spring system with friction

    A mass M slides across a horizontal table. It collides with a spring, compresses the spring, and then the mass-spring system rebounds. This system can be used to find the spring constant k. When the mass first hits the spring at x = 0, it has speed v0. a.) Let the coefficient of kinetic...
  40. D

    DE - mass-spring system

    (I had made a thread with a problem similar to this one, but it turned a bit messy after finding out the professor made some mistakes and the wording of his problem was awkward ... however, this is it) Problem: Differential equation governing a forced, mass-spring system...
  41. D

    Forced Mass-Spring System - Diff Eq

    Homework Statement You have a forced, mass-spring system, without damping. Spring constant = 4 N/m weight of mass = 9.8 N mass = 1 Kg Find the motion X(t) of the mass if ω = 1.5 (Hz) and deduce the maximum elongation of the spring. Sketch the vibrations X(t). Do same for ω = 1.9, 3. Find...
  42. T

    SHM - mass-spring system on frictionless horizontal surface

    Homework Statement A 250 g block is resting on a frictionless horizontal surface. The block is attached to a spring. The mass-spring system is compressed by a 2.5 N force and then released from rest. (a) The resulting oscillation has a 1.0 s period. What is the spring constant? (b) What is...
  43. D

    Need help with mass-spring system

    I need help in finding the equations that describe this system. http://img530.imageshack.us/img530/2584/44567537be2.jpg Any help will be appreciated. Thank you, Kfir
  44. Y

    Solving Oscillating Mass-Spring System w/ Non-Resonant Force

    1. Homework Statement and 2. Homework Equations Find position/velocity of a mass m attached to a spring of constant k when subjected to an oscillatinf roce F(t) = F sin(Bt) With B\not = \sqrt{k/m} The Attempt at a Solution Model; mx'' + kx = F \sin(Bt) I have no...
  45. T

    A mass-spring system with recoil and friction

    An object of mass m is traveling on a horizontal surface. There is a coefficient of kinetic friction, mu , between the object and the surface. The object has speed v when it reaches x=0 and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite...
  46. H

    Mechanical energy of a mass-spring system

    Hey, Stuck on finding the mechanical energy of a mass-spring system, my question is as follows > A mass-spring system oscillates with an amplitude of .026 m. The spring constant is 290 N/m and the mass is 0.50 kg, it asks for the mechanical energy in (J). and the maximum acceleration of the...
  47. H

    A Mass-Spring System with Recoil and Friction

    Here's the question: "An object of mass m is traveling on a horizontal surface. There is a coefficient of kinetic friction mu between the object and the surface. The object has speed v when it reaches x = 0 and encounters a spring. The object compresses the spring, stops, and then recoils and...