# 15.2 Energy in simple harmonic motion  (Page 2/8)

 Page 2 / 8 The transformation of energy in SHM for an object attached to a spring on a frictionless surface. (a) When the mass is at the position x = + A , all the energy is stored as potential energy in the spring U = 1 2 k A 2 . The kinetic energy is equal to zero because the velocity of the mass is zero. (b) As the mass moves toward x = − A , the mass crosses the position x = 0 . At this point, the spring is neither extended nor compressed, so the potential energy stored in the spring is zero. At x = 0 , the total energy is all kinetic energy where K = 1 2 m ( − v max ) 2 . (c) The mass continues to move until it reaches x = − A where the mass stops and starts moving toward x = + A . At the position x = − A , the total energy is stored as potential energy in the compressed U = 1 2 k ( − A ) 2 and the kinetic energy is zero. (d) As the mass passes through the position x = 0 , the kinetic energy is K = 1 2 m v max 2 and the potential energy stored in the spring is zero. (e) The mass returns to the position x = + A , where K = 0 and U = 1 2 k A 2 .

Consider [link] , which shows the energy at specific points on the periodic motion. While staying constant, the energy oscillates between the kinetic energy of the block and the potential energy stored in the spring:

${E}_{\text{Total}}=U+K=\frac{1}{2}k{x}^{2}+\frac{1}{2}m{v}^{2}.$

The motion of the block on a spring in SHM is defined by the position $x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$ with a velocity of $v\left(t\right)=\text{−}A\omega \text{sin}\left(\omega t+\varphi \right)$ . Using these equations, the trigonometric identity ${\text{cos}}^{2}\theta +{\text{sin}}^{2}\theta =1$ and $\omega =\sqrt{\frac{k}{m}}$ , we can find the total energy of the system:

$\begin{array}{cc}\hfill {E}_{\text{Total}}& =\frac{1}{2}k{A}^{2}{\text{cos}}^{2}\left(\omega t+\varphi \right)+\frac{1}{2}m{A}^{2}{\omega }^{2}{\text{sin}}^{2}\left(\omega t+\varphi \right)\hfill \\ & =\frac{1}{2}k{A}^{2}{\text{cos}}^{2}\left(\omega t+\varphi \right)+\frac{1}{2}m{A}^{2}\left(\frac{k}{m}\right){\text{sin}}^{2}\left(\omega t+\varphi \right)\hfill \\ & =\frac{1}{2}k{A}^{2}{\text{cos}}^{2}\left(\omega t+\varphi \right)+\frac{1}{2}k{A}^{2}{\text{sin}}^{2}\left(\omega t+\varphi \right)\hfill \\ & =\frac{1}{2}k{A}^{2}\left({\text{cos}}^{2}\left(\omega t+\varphi \right)+{\text{sin}}^{2}\left(\omega t+\varphi \right)\right)\hfill \\ & =\frac{1}{2}k{A}^{2}.\hfill \end{array}$

The total energy of the system of a block and a spring is equal to the sum of the potential energy stored in the spring plus the kinetic energy of the block and is proportional to the square of the amplitude ${E}_{\text{Total}}=\left(1\text{/}2\right)k{A}^{2}.$ The total energy of the system is constant.

A closer look at the energy of the system shows that the kinetic energy oscillates like a sine-squared function, while the potential energy oscillates like a cosine-squared function. However, the total energy for the system is constant and is proportional to the amplitude squared. [link] shows a plot of the potential, kinetic, and total energies of the block and spring system as a function of time. Also plotted are the position and velocity as a function of time. Before time $t=0.0\phantom{\rule{0.2em}{0ex}}\text{s,}$ the block is attached to the spring and placed at the equilibrium position. Work is done on the block by applying an external force, pulling it out to a position of $x=+A$ . The system now has potential energy stored in the spring. At time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s,}$ the position of the block is equal to the amplitude, the potential energy stored in the spring is equal to $U=\frac{1}{2}k{A}^{2}$ , and the force on the block is maximum and points in the negative x -direction $\left({F}_{S}=\text{−}kA\right)$ . The velocity and kinetic energy of the block are zero at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}\text{.}$ At time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s,}$ the block is released from rest. Graph of the kinetic energy, potential energy, and total energy of a block oscillating on a spring in SHM. Also shown are the graphs of position versus time and velocity versus time. The total energy remains constant, but the energy oscillates between kinetic energy and potential energy. When the kinetic energy is maximum, the potential energy is zero. This occurs when the velocity is maximum and the mass is at the equilibrium position. The potential energy is maximum when the speed is zero. The total energy is the sum of the kinetic energy plus the potential energy and it is constant.

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