Spring Mass System

Springs are very basic objects with a simple function. We all have intuitive ideas of how springs work, whether that be from playing with toys with springs in them when we were young or from jumping on a trampoline. However, springs are probably more important than you realize and can be found everywhere. They play a key role in cars, electrical circuits and clocks, as well as even being useful for many scientific models!  Springs and elasticity have been studied greatly for a long time and the function of systems involving springs can be explained by simple equations. Read on to learn more about them.

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All mass-spring systems consist of one mass connected to one spring. Is this statement true or false?

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What is the relationship between the magnitude of acceleration and distance from equilibrium in simple harmonic motion?

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The period of oscillation for a mass on a spring depends on the amplitude. Is this statement true or false? 

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What two quantities does the angular frequency of an oscillating spring-mass system depend on?

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A spring is attached to a wall on its left side. It is then compressed by someone's hand. In which direction will it exert a force on their hand?

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For an oscillating spring-mass system hanging from a ceiling, why is the equilibrium position lower than the natural length of the spring?

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Springs in ideal spring-mass systems are presumed to be massless. Is this statement true or false? 

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What is a restoring force in simple harmonic motion?

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There are no frictional forces in idealised spring-mass systems. Is this statement true or false? 

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Which of these is not an example of a spring-mass system?

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For an oscillating spring-mass system, what form is the energy in when the mass is at the positions of maximum displacement?

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All mass-spring systems consist of one mass connected to one spring. Is this statement true or false?

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  • Mo

What is the relationship between the magnitude of acceleration and distance from equilibrium in simple harmonic motion?

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  • Mo

The period of oscillation for a mass on a spring depends on the amplitude. Is this statement true or false? 

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  • Immunology
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  • Mo

What two quantities does the angular frequency of an oscillating spring-mass system depend on?

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  • + Add tag
  • Immunology
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  • Mo

A spring is attached to a wall on its left side. It is then compressed by someone's hand. In which direction will it exert a force on their hand?

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  • Immunology
  • Cell Biology
  • Mo

For an oscillating spring-mass system hanging from a ceiling, why is the equilibrium position lower than the natural length of the spring?

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  • Immunology
  • Cell Biology
  • Mo

Springs in ideal spring-mass systems are presumed to be massless. Is this statement true or false? 

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  • Cell Biology
  • Mo

What is a restoring force in simple harmonic motion?

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There are no frictional forces in idealised spring-mass systems. Is this statement true or false? 

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Which of these is not an example of a spring-mass system?

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For an oscillating spring-mass system, what form is the energy in when the mass is at the positions of maximum displacement?

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Table of contents

    Spring-Mass System Meaning

    The simplest case of a spring-mass system consists of a mass attached to the end of a spring. There can be other arrangements with more masses and more springs attached in different ways, but we are just going to consider this simple case. An example of a spring-mass system on a horizontal surface is shown below.

    Mass and Spring Systems horizontal mass and spring StudySmarterA simple horizontal spring-mass system consists of a spring attached to a wall on one side and a mass on the other, ux1.eiu.

    A spring is attached to a fixed point on one side and to a mass m on the other side. In the diagram, k represents the spring constant and is a measure of the stiffness of the sprint. A larger spring constant means a stiffer spring. The more stiff a string is, the more force is exerted by the string when it is displaced by a certain amount. The spring constant is measured in newtons per meter, N/m. The spring-mass system above is shown to be in its relaxed state, meaning that the string is at its natural length - it is not stretched or compressed - and so there is no force on the mass. We say that the mass is at its equilibrium position.

    Spring-Mass System Equation

    Now we can consider the forces acting on the mass when it is moved from its equilibrium position. If we move the mass to a displacement ofxmeters to the right, the spring will be stretched past its natural length, as shown below.

    Mass and Spring Systems stretched spring restoring force StudySmarter

    If the mass is pulled to the right, the string will be stretched and exert a force on the mass directed back towards the left, ux1.eiu.

    This causes the spring to exert a force on the mass as the spring is trying to return to its natural length, hence the mass accelerates back towards the equilibrium position when released.

    On the other hand, if the mass is moved to the left, the spring will be compressed.

    Mass and Spring Systems compressed string restoring force StudySmarter

    If the mass is pushed to the left, the string will be stretched and exert a force on the mass directed back towards the right, ux1.eiu.

    The spring will attempt to return to its natural length and will exert a force on the mass to the right which will cause it to move back towards its equilibrium position.

    From the two situations above, you can see that when the mass is displaced, the spring will always exert a force to bring the mass back to its equilibrium position - this is called a restoring force. This force is given by a simple relation:

    F=-kx.

    F is the force acting on the mass in Newtons,N

    k is the spring constant in newtons-per-meter,N/m

    x is the displacement of the mass from its equilibrium position in meters,m.

    This equation is Hooke's law. It was named after the English scientist Robert Hooke, who did work on a variety of subjects in the 17th century including on elasticity.

    We must be careful to say 'displacement' and not 'distance' as the displacement can be both positive and negative, which is important to find the right direction of the force.

    Hooke's law clearly shows that the force acting on the mass increases as the distance of the mass from the equilibrium position increases. However, the minus sign in the equation shows that the force is always directed in the opposite direction to the displacement - when the mass is on the right of the equilibrium, the force will be directed towards the left and vice versa. The force is always directed towards the equilibrium position.

    Another example of a spring-mass system that you will often find in problems is a vertically oriented spring attached to a ceiling at its top end and a mass at its bottom end.

    Mass and Spring Systems vertical spring-mass system StudySmarterA mass hanging from a vertical spring can be treated in the same way as a horizontal spring-mass system, khanacademy.

    Hooke's law can be applied to this situation in exactly the same way as for the horizontal system, but this time the equilibrium position has shifted due to gravity. The point where there is no force acting on the mass occurs when the upwards force due to the stretched spring is equal to the downwards force due to the mass. The mass of the spring is assumed to be zero for an ideal mass-spring system. The weight of the mass is equal to:

    W=mg,

    where g is the gravitational constant on the surface of the Earth in meters-per-second-squared,m/s2. This can be equated to the spring force to find the equilibrium position x0.

    -kx0=mgx0=-mgk

    The minus sign shows that the equilibrium position is below the position of the mass when the spring is at its natural length, which is as expected.

    Spring-Mass System Simple Harmonic Motion

    If the mass on a spring-mass system is displaced from its equilibrium position and released, it will exhibit simple harmonic motion. An object performing simple harmonic motion (SHM) moves back and forth between points of maximum displacement from the equilibrium position on either side. The mass will be accelerated towards its equilibrium position when it is displaced. When it reaches this position, it will have a velocity and so carry on to the other side, at which point it will experience a restoring force in the opposite direction and the process will be repeated. This motion will continue indefinitely under ideal conditions (ignoring any frictional forces present).

    Simple Harmonic Motion (SHM) is defined as an oscillation in which the acceleration of an object is inversely proportional to its displacement from the equilibrium position and this acceleration must always be directed towards the equilibrium.

    From the definition of simple harmonic motion, the acceleration of an object undergoing SHM is proportional to its displacement and is given by:

    a=-ω2x,where a is the acceleration inm/s2,ωis the angular frequency of the motion israd/sandxis the displacement of the object from its equilibrium position inm. The angular frequencyωis defined to be equal to:

    ω=2πT.

    T is the time period of the motion, which is equal to the time for the mass to move from one position of maximum displacement and back again. For the spring-mass system, we can find the acceleration of the mass from the equation for the restoring force acting on the mass:

    a=Fm=-kmx.

    We can compare this to the equation for SHM stated above, from which we can see that the angular frequency for a mass-spring system is:

    ω=km.

    We can then use the equation for angular frequency to find the time period insof the simple harmonic motion of a spring-mass system.

    2πT=km

    T=2πkm=2πkm

    This does not depend on the initial displacement of the system - known as the amplitude of the oscillation. A spring-mass system will always oscillate with the same time period no matter how far the mass is originally displaced (as long as frictional forces are ignored).

    Spring-Mass System Examples

    The equations stated above can be utilized in many different practice problems - you just need to identify which equation is the correct one to use.

    A2 kgmass is hung from a ceiling from a spring of natural length2 mand spring constantk=100 N/m. After initially being supported so that the spring is at its natural length, the mass is released and undergoes simple harmonic motion about an equilibrium position.

    What is the distance of this equilibrium position from the ceiling? What is the amplitude of the oscillation? What is the angular frequency of the oscillation? The spring can be assumed to be massless.

    Mass and Spring Systems vertical spring-mass system StudySmarterA mass is hung from a ceiling before being pulled, which courses it to undergo simple harmonic motion, sites.google

    At the equilibrium position, the weight of the mass has to match the upwards force due to the extension of the spring. We considered this situation earlier on in the article and found an equation for the equilibrium position, x0. We can plug the given values in to that equation to find the answer:

    x0=-mgk=-0.196 m.

    This position is below the unstretched length of the string, so we must add its magnitude to the unstretched length of the string to find the distance of the equilibrium position from the ceiling, x:

    x=(2+0.196) m=2.20 m

    We have already found the amplitudeAof the oscillation; it is equal to the extra distance between the unstretched string and the equilibrium point:

    A=0.196 m

    To find the angular frequencyωof the simple harmonic motion of this system, we use the equation in terms of the mass and spring constant that was stated above:

    ω=km=7.07 rad/s

    ~

    The same spring is used as in the question above but this time it is attached to a5 kgmass at one end and a wall at the other. The mass rests on the floor. The mass is pulled to a distance of2.5 mfrom the wall and released, causing it to undergo simple harmonic motion.

    What is the time period of this motion? What is the closest distance to the wall that the mass will reach during the motion? Assume that the spring is massless and the floor is frictionless.

    Mass and Spring Systems horizontal spring-mass system StudySmarterThe spring-mass system is now positioned horizontally, Wikimedia commons.

    The time period of the oscillation of the mass can be found from the equation stated above as we are given values for the mass and the spring constant in the question.

    T=2πmk=1.4 s

    The mass will be closest to the wall when its displacement is equal to the negative of the amplitude of the oscillation. The amplitude of the oscillation is given by the initial position minus the length of the spring, as in a horizontal position the equilibrium point will be the unstretched spring.

    A=(2.5-2) m=0.5 m

    The shortest distance from the mass to the wall S will be the amplitude of the oscillation subtracted from the natural length of the spring.

    S=(2-0.5) m=1.5 m

    Spring Mass System Energy

    The work done by a force moving an object is given by:

    W=Fx

    whereWis the work done measured in Joules,J,

    Fis the applied force measured in Newtons,N,

    xis the distance moved in the direction of the force measured in metres,m.

    The equation for the force applied by a spring is:

    F=-kx.

    We want to use this in the equation to find the work done by a spring to move a mass from its equilibrium position to its position of maximum displacement. However, the force changes as the length of the spring changes. It can be seen from the above equation that the force is directly proportional to the distance that the spring is stretched. As the relationship between these two quantities is linear, we can simply suppose that a constant force acts which is equal to the average of the initial and final forces.

    Faverage=Ffinal-Finitial2=kA-02=kA2

    In the above expression,Arepresents the amplitude of a spring mass oscillation. The magnitude of the force has been taken as we are only considering distances and not displacements. We can now use this expression for the average force to find the work done in moving a mass from its equilibrium position to its initial position.

    W=Faveragex=kA2×A=12kA2

    The work done in stretching a spring is converted into the elastic potential energy stored in the spring, so this expression above gives the maximum potential energy of a spring mass system.

    There are presumed to be no frictional forces in the spring-mass systems that we are considering, this means that no energy is lost in the motion. The energy is shared between the potential and the kinetic energies of the system.

    Etotal=PE+KE

    When the mass is at maximum displacement, the spring is the most stretched so the potential energy is maximum. At this point, the speed is zero so its kinetic energy is zero. When the mass passes through its equilibrium position, its speed is a maximum and hence so is its kinetic energy. The spring is not stretched at this point so the potential energy is zero. Along with the equation above, this leads us to conclude that the maximum kinetic energy of the mass of a mass-spring system is equal to the maximum potential energy stored in the spring, which are both also equal to the total energy.

    The formula for kinetic energy is

    KE=12mv2.

    We can equate the maximum kinetic energy - when the mass is moving at its maximum speedv0through the equilibrium position - to the maximum potential energy to find a value for the maximum speed.

    12mv2=12kA2mv2=kA2v2=kA2mv=km×A=ωA

    Spring-Mass System Application

    Spring-mass systems can be seen very often in everyday life and they are also used in scientific models.

    Car Shock Absorbers

    A spring-mass system is used to great effect in cars in the form of shock absorbers, which are placed above the wheel. They are designed for preventing the car from being damaged when it passes over bumps and other obstacles. As a car goes over a bump, the spring above the shock absorber will compress and these springs must be designed to oscillate at the right amplitude and frequency to make the ride of the car comfortable.

    Mass and Spring Systems car shock absorber StudySmarterShock absorbers are placed just above each wheel on a car to prevent damage from bumps, cardealpage.

    Diatomic Molecules

    Diatomic molecules consist of two atoms that are chemically bonded together. These molecules can oscillate and (for small vibrations) the bond can be approximated as a spring with the two atoms as masses on the ends of the spring. The oscillations obey Hooke's law for small amplitudes. This approximation is used to simplify models for scientific research.

    Mass and Spring Systems diatomic molecule StudySmarterOxygen exists as a diatomic molecule and can be approximated as a spring-mass system, bartlby.

    Spring Mass System - Key takeaways

    • The simplest case of a spring-mass system consists of a mass attached to the end of a spring.
    • Hooke's law states that the force generated by a spring is proportional to its extension or compression from the natural length.
    • A spring will always exert a force to bring itself back to its natural length.
    • Hooke's law applies in the same way to horizontal and vertical spring-mass systems.
    • For a vertical spring-mass system, the equilibrium position is displaced downwards due to the gravitational force on the mass.
    • Springs are presumed to be massless in ideal spring-mass systems.
    • There are presumed to be no frictional forces in ideal spring-mass systems.
    • The oscillations of mass on springs can be described by the equations of simple harmonic motion.
    • Simple harmonic motion is defined as an oscillation in which the acceleration of an object is directly proportional to its displacement from its equilibrium position and this acceleration must always be directed towards the equilibrium.
    • The period of oscillation of a spring-mass system does not depend on the amplitude of oscillation.
    Frequently Asked Questions about Spring Mass System

    What is the formula for spring mass system?

    The formula to find the time period T of a spring-mass system is:

    T=2π sqrt(m/k)

    What is an ideal mass spring system?

    In an ideal mass system, the spring is massless and there are no frictional forces present.

    Where are mass spring systems used? 

    Spring-mass systems have many applications such as:

    • Shock absorbers in cars.
    • Scientific models.
    • Trampolines.
    • Pogo sticks.

    What is an example of a spring mass system?

    An example of a spring mass system is a mass hung vertically from a spring that is attached to a fixed point on a ceiling.

    How do you find the mass of a spring?

    The mass of a spring can be found by placing it on a weight measuring scale, just like any other object.

    Test your knowledge with multiple choice flashcards

    All mass-spring systems consist of one mass connected to one spring. Is this statement true or false?

    What is the relationship between the magnitude of acceleration and distance from equilibrium in simple harmonic motion?

    The period of oscillation for a mass on a spring depends on the amplitude. Is this statement true or false? 

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