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Exercise:
The potential energy of a mass oscillating on a vertical spring consists of both gravitational and elastic potential energy. Show that it can be expressed as an em effective potential energy sscEpot frac k y^t where the extension/compression yt of the spring is calculated with respect to the equlibrium position. vspacemm Remark: Consider that the gravitational potential energy is only defined up to an arbitrary constant which can be chosen appropriately.

Solution:
For a displacement yt from the equlibrium position see figure of the pulum mass the gravitational potential energy is sscEpot G m g yt+E_ with an arbitrary constant E_ which corresponds to the gravitational potential energy at the equilibrium position. vspacemm The elastic potential energy for the same displacement is sscEpot el frac klefty_-ytright^ center includegraphicswidthtextwidth#image_path:potential-energy-mass-on-spring-# center The extension y_ of the spring at the equilibrium position is given by F_k F_G Longrightarrow k y_ m g Longrightarrow y_ fracm gk It follows for the total potential energy sscEpot sscEpot G+sscEpot el m g yt+E_ + frac klefty_-ytright^ m g yt+E_ + frac kleftfracm gk-ytright^ m g yt+E_ + fracm^ g^k+frack y^t-m g yt E_+fracm^ g^k+frack y^t By choosing the reference level for the gravitational potential energy such that E_ -fracm^ g^k we can rewrite the potential energy as sscEpot frack y^t Graphing the gravitational and the elastic potential energy and the kinetic energy for the mass on a spring over one oscillation period reveals the conservation of energy. center includegraphicswidthtextwidth#image_path:potential-energy-mass-on-spring-#center
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Exercise:
The potential energy of a mass oscillating on a vertical spring consists of both gravitational and elastic potential energy. Show that it can be expressed as an em effective potential energy sscEpot frac k y^t where the extension/compression yt of the spring is calculated with respect to the equlibrium position. vspacemm Remark: Consider that the gravitational potential energy is only defined up to an arbitrary constant which can be chosen appropriately.

Solution:
For a displacement yt from the equlibrium position see figure of the pulum mass the gravitational potential energy is sscEpot G m g yt+E_ with an arbitrary constant E_ which corresponds to the gravitational potential energy at the equilibrium position. vspacemm The elastic potential energy for the same displacement is sscEpot el frac klefty_-ytright^ center includegraphicswidthtextwidth#image_path:potential-energy-mass-on-spring-# center The extension y_ of the spring at the equilibrium position is given by F_k F_G Longrightarrow k y_ m g Longrightarrow y_ fracm gk It follows for the total potential energy sscEpot sscEpot G+sscEpot el m g yt+E_ + frac klefty_-ytright^ m g yt+E_ + frac kleftfracm gk-ytright^ m g yt+E_ + fracm^ g^k+frack y^t-m g yt E_+fracm^ g^k+frack y^t By choosing the reference level for the gravitational potential energy such that E_ -fracm^ g^k we can rewrite the potential energy as sscEpot frack y^t Graphing the gravitational and the elastic potential energy and the kinetic energy for the mass on a spring over one oscillation period reveals the conservation of energy. center includegraphicswidthtextwidth#image_path:potential-energy-mass-on-spring-#center
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Harmonic Oscillations
Tags
mass on spring, oscillation energy, potential energy, simple harmonic motion
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(3, default)
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0 (default)
Language
ENG (English)
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