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

 Page 2 / 8

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.

derivation of simple harmonic equation
if an equation is dimensionally correct does this mean that equation must be true?
how do I calculate angular velocity
w=vr where w, angular velocity. v; velocity and r; radius of a circle
michael
sorry I meant Maximum positive angular velocity of
Priscilla
Can any one give me the definition for Bending moment plz...
I need a question for moment
what is charge
An attribution of particle that we have thought about to explain certain things like Electomagnetism
Nikunj
please what is the formula instantaneous velocity in projectile motion
A computer is reading from a CD-ROM that rotates at 780 revolutions per minute.What is the centripetal acceleration at a point that is 0.030m from the center of the disc?
change revolution per minute by multiplying from 2pie and devide by 60.and take r=.030 and use formula centripital acceleration =omega sqare r.
Kumar
OK thank you
Rapqueen
observation of body boulded
a gas is compressed to 1/10 0f its original volume.calculate the rise temperature if the original volume is 400k. gamma =1.4
the specific heat of hydrogen at constant pressure and temperature is 14.16kj|k.if 0.8kg of hydrogen is heated from 55 degree Celsius to 80 degree Celsius of a constant pressure. find the external work done .
Celine
hi
shaik
hy
Prasanna
g
what is imaginary mass and how we express is
what is imaginary mass how we express it
Yash
centre of mass is also called as imaginary mass
Lokmani
l'm from Algeria and fell these can help me
Many amusement parks have rides that make vertical loops like the one shown below. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if: (a) The car goes over the top at faster than this speed? (b) The car goes over the top at slower than this speed?
how can I convert mile to meter per hour
1 mile * 1609m
Boon
hey can someone show me how to solve the - "Hanging from the ceiling over a baby bed ...." question
i wanted to know the steps
Shrushti
sorry shrushti..
Rashid
which question please write it briefly
Asutosh
Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of 3.7 m/s2 .
what is summit
Asutosh
highest point on earth
Ngeh