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:
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
$v\left(t\right)=\text{\u2212}A\omega \text{sin}\left(\omega t+\varphi \right)$ . 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:
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
${E}_{\text{Total}}=(1\text{/}2)k{A}^{2}.$ 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
$\left({F}_{S}=\text{\u2212}kA\right)$ . 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.
Questions & Answers
find the angle of projection at which the horizontal range is twice the maximum height of a projectile
I have another problem,what is the difference between pure chemistry and applied chemistry ?
lasitha
l stands for linear displacement and l>>r
anand
r stands for radius or position vector of particle
anand
pure chemistry is related to classical and basic concepts of chemistry while applied Chemistry deals with it's application oriented concepts and procedures.
anand
what is the deference between precision and accuracy?
ShAmy
I think in general both are same , more accuracy means more precise
anand
and lessor error
anand
friends,how to find correct applied chemistry notes?
lasitha
what are the main concepts of applied chemistry ?
lasitha
anand,can you give an example for angular displacement ?
lasitha
when any particle rotates or revolves about something , an axis or point ,then angle traversed is angular displacement
according to John Dalton all the matter was composed of atom and indivisible, indestructible building blocks. while the atom of different elements have different mass and size.
Rakesh
two parallel wires are 5.80cm apart and carry currents in opposite direction , as shown in figure .find the magnitude and direction of the magnetic field at point p due to two 1.50mm segments of wire that are opposite each other and each 8.00 cm from p.