In mechanics and physics, simple harmonic motion is a special type of
periodic motion or oscillation motion where the restoring force is
directly proportional to the displacement and acts in the direction
opposite to that of displacement.
Contents 1 Introduction 2 Dynamics 3 Energy 4 Examples 4.1
5 See also 6 Notes 7 References 8 External links Introduction[edit] The motion of a particle moving along a straight line with an acceleration which is always towards a fixed point on the line and whose magnitude is proportional to the distance from the fixed point is called simple harmonic motion [SHM].[citation needed]
In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, the spring exerts a restoring elastic force that obeys Hooke's law. Mathematically, the restoring force F is given by F = − k x , displaystyle mathbf F =-kmathbf x , where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m−1), and x is the displacement from the equilibrium position (m). For any simple mechanical harmonic oscillator: When the system is displaced from its equilibrium position, a
restoring force that obeys
Once the mass is displaced from its equilibrium position, it
experiences a net restoring force. As a result, it accelerates and
starts going back to the equilibrium position. When the mass moves
closer to the equilibrium position, the restoring force decreases. At
the equilibrium position, the net restoring force vanishes. However,
at x = 0, the mass has momentum because of the acceleration that the
restoring force has imparted. Therefore, the mass continues past the
equilibrium position, compressing the spring. A net restoring force
then slows it down until its velocity reaches zero, whereupon it is
accelerated back to the equilibrium position again.
As long as the system has no energy loss, the mass continues to
oscillate. Thus simple harmonic motion is a type of periodic motion.
Dynamics[edit]
In Newtonian mechanics, for one-dimensional simple harmonic motion,
the equation of motion, which is a second-order linear ordinary
differential equation with constant coefficients, can be obtained by
means of
F n e t = m d 2 x d t 2 = − k x , displaystyle F_ mathrm net =m frac mathrm d ^ 2 x mathrm d t^ 2 =-kx, where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is a constant (the spring constant for a mass on a spring). (Note that in reality this is in fact an approximation, only valid for speeds that are small compared to the speed of light.) Therefore, d 2 x d t 2 = − k m x , displaystyle frac mathrm d ^ 2 x mathrm d t^ 2 =- frac k m x, Solving the differential equation above produces a solution that is a sinusoidal function. x ( t ) = x 0 cos ( ω t ) + v 0 ω sin ( ω t ) displaystyle x(t)=x_ 0 cos left(omega tright)+ frac v_ 0 omega sin left(omega tright) This equation can be written in the form: x ( t ) = A cos ( ω t − φ ) , displaystyle x(t)=Acos left(omega t-varphi right), where ω = k m , A = c 1 2 + c 2 2 , tan φ = c 2 c 1 , displaystyle omega = sqrt frac k m ,qquad A= sqrt c_ 1 ^ 2 + c_ 2 ^ 2 ,qquad tan varphi = frac c_ 2 c_ 1 , In the solution, c1 and c2 are two constants determined by the initial conditions, and the origin is set to be the equilibrium position.[A] Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position), ω = 2πf is the angular frequency, and φ is the phase.[B] Using the techniques of calculus, the velocity and acceleration as a function of time can be found: v ( t ) = d x d t = − A ω sin ( ω t − φ ) , displaystyle v(t)= frac mathrm d x mathrm d t =-Aomega sin(omega t-varphi ), Speed: ω A 2 − x 2 displaystyle omega sqrt A^ 2 -x^ 2 Maximum speed: ωA (at equilibrium point) a ( t ) = d 2 x d t 2 = − A ω 2 cos ( ω t − φ ) . displaystyle a(t)= frac mathrm d ^ 2 x mathrm d t^ 2 =-Aomega ^ 2 cos(omega t-varphi ). Maximum acceleration: Aω2 (at extreme points) By definition, if a mass m is under SHM its acceleration is directly proportional to displacement. a ( x ) = − ω 2 x . displaystyle a(x)=-omega ^ 2 x. where ω 2 = k m displaystyle omega ^ 2 = frac k m Since ω = 2πf, f = 1 2 π k m , displaystyle f= frac 1 2pi sqrt frac k m , and, since T = 1/f where T is the time period, T = 2 π m k . displaystyle T=2pi sqrt frac m k . These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion). Energy[edit] Substituting ω2 with k/m, the kinetic energy K of the system at time t is K ( t ) = 1 2 m v 2 ( t ) = 1 2 m ω 2 A 2 sin 2 ( ω t + φ ) = 1 2 k A 2 sin 2 ( ω t + φ ) , displaystyle K(t)= tfrac 1 2 mv^ 2 (t)= tfrac 1 2 momega ^ 2 A^ 2 sin ^ 2 (omega t+varphi )= tfrac 1 2 kA^ 2 sin ^ 2 (omega t+varphi ), and the potential energy is U ( t ) = 1 2 k x 2 ( t ) = 1 2 k A 2 cos 2 ( ω t + φ ) . displaystyle U(t)= tfrac 1 2 kx^ 2 (t)= tfrac 1 2 kA^ 2 cos ^ 2 (omega t+varphi ). In the absence of friction and other energy loss, the total mechanical energy has a constant value E = K + U = 1 2 k A 2 . displaystyle E=K+U= tfrac 1 2 kA^ 2 . Examples[edit] An undamped spring–mass system undergoes simple harmonic motion. The following physical systems are some examples of simple harmonic
oscillator.
T = 2 π m k displaystyle T=2pi sqrt frac m k shows the period of oscillation is independent of both the amplitude
and gravitational acceleration. The above equation is also valid in
the case when an additional constant force is being applied on the
mass, i.e. the additional constant force cannot change the period of
oscillation.
Uniform circular motion[edit]
The motion of an undamped pendulum approximates to simple harmonic motion if the angle of oscillation is small. In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length l with gravitational acceleration g displaystyle g is given by T = 2 π l g displaystyle T=2pi sqrt frac l g This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due to gravity, g displaystyle g , therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational field strength. Because the value of g displaystyle g varies slightly over the surface of the earth, the time period will vary slightly from place to place and will also vary with height above sea level. This approximation is accurate only for small angles because of the expression for angular acceleration α being proportional to the sine of the displacement angle: − m g l sin θ = I α , displaystyle -mglsin theta =Ialpha , where I is the moment of inertia. When θ is small, sin θ ≈ θ and therefore the expression becomes − m g l θ = I α displaystyle -mgltheta =Ialpha which makes angular acceleration directly proportional to θ,
satisfying the definition of simple harmonic motion.
Scotch yoke[edit]
Main article: Scotch yoke
A
Newtonian mechanics
Small-angle approximation
Isochronous
Uniform circular motion
Complex harmonic motion
Damping
Harmonic oscillator
Notes[edit] ^ The choice of using a cosine in this equation is arbitrary. Other valid formulations are: x ( t ) = A sin ( ω t + φ ′ ) , displaystyle x(t)=Asin left(omega t+varphi 'right), where tan φ ′ = c 1 c 2 , displaystyle tan varphi '= frac c_ 1 c_ 2 , since cos θ = sin(π/2 − θ). ^ The maximum displacement (that is, the amplitude), xmax, occurs when cos(ωt ± φ) = 1, and thus when xmax = A. References[edit] Walker, Jearl (2011). Principles of
External links[edit] Simple Harmonic Motion from HyperPhysics Java simulation of spring-mas |