Potential Energy for Mass on Spring
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Exercise:
Show that the potential energy of a mass on a spring system i.e. the of elastic and gravitational potential energy can be written as sscEpot frac k y^t where yt is the displacement from the equilibrium position and k the elastic constant of the spring. vspacemm Remark: Consider that the gravitational potential energy is only defined up to an arbitrary constant which can be chosen appropriately.
Solution:
A spring with elastic constant k is stretched when a mass m is susped from it. The displacement in the equlibrium position is see figure below y_ fracmgk labeleq:equilibrium center includegraphicswidthtextwidth#image_path:potential-energy-mass-on-spring-# center For a displacement yt from the equilibrium position the elastic energy sscEel of the spring and the gravitational potential energy sscEG are sscEel frac k lefty_-ytright^ sscEG m g yt + E_ with an arbitrary constant E_ that corresponds to the gravitational potential energy at the equlibrium position. vspacemm Using refeq:equilibrium for the equilibrium elongation of the spring we find sscEel+sscEG frac k y_^ - k y_ yt + frac k y^t + m g yt + E_ frac k fracm gk y_ - k fracm gk yt + frac k y^t + m g yt + E_ frac m g y_ + frac k y^t + E_ We can choose the constant to be E_ -frac m g y_ to get sscEpot sscEel + sscEG frac k y^t quad square
Show that the potential energy of a mass on a spring system i.e. the of elastic and gravitational potential energy can be written as sscEpot frac k y^t where yt is the displacement from the equilibrium position and k the elastic constant of the spring. vspacemm Remark: Consider that the gravitational potential energy is only defined up to an arbitrary constant which can be chosen appropriately.
Solution:
A spring with elastic constant k is stretched when a mass m is susped from it. The displacement in the equlibrium position is see figure below y_ fracmgk labeleq:equilibrium center includegraphicswidthtextwidth#image_path:potential-energy-mass-on-spring-# center For a displacement yt from the equilibrium position the elastic energy sscEel of the spring and the gravitational potential energy sscEG are sscEel frac k lefty_-ytright^ sscEG m g yt + E_ with an arbitrary constant E_ that corresponds to the gravitational potential energy at the equlibrium position. vspacemm Using refeq:equilibrium for the equilibrium elongation of the spring we find sscEel+sscEG frac k y_^ - k y_ yt + frac k y^t + m g yt + E_ frac k fracm gk y_ - k fracm gk yt + frac k y^t + m g yt + E_ frac m g y_ + frac k y^t + E_ We can choose the constant to be E_ -frac m g y_ to get sscEpot sscEel + sscEG frac k y^t quad square