Simple harmonic motion 5 shm hooke s law shm describes any periodic motion that results from a restoring force f that is proportional to the displacement x of an object from its equilibrium position. Phys 200 lecture 17 simple harmonic motion open yale. Using newtons second law of motion f ma,wehavethedi. If the system is disturbed from its equilibrium position, it will start to oscillate back and forth at a certain natural frequency, which depends on. Sketch of a pendulum of length l with a mass m, displaying the forces acting on the mass resolved in the tangential direction relative to the motion. Simple harmonic motion and wave mechanics 1 the motion c is not periodic. A particular and useful kind of periodic motion is simple harmonic motion shm. We then have the problem of solving this differential equation. Damped simple harmonic motion exponentially decreasing envelope of harmonic motion shift in frequency. Simple harmonic motion one degree of freedom massspring, pendulum, water in pipes, rlc circuits damped harmonic motion 2. This leads to a differential equation of familiar form, although with.
Forced oscillations this is when bridges fail, buildings collapse, lasers oscillate, microwaves cook food, swings swing. Then newtons second law gives thus, instead of the homogeneous equation 3, the motion of the spring is now governed. What is the general equation of simple harmonic motion. This relationship is known as hookes law after the seventeenth century english physicist robert hooke. How to properly solve the harmonic motion differential. Forced harmonic oscillators amplitudephase of steady state oscillations transient phenomena 3. In this case study simple harmonic motion and its applica tions. The motion of the pendulum is a particular kind of repetitive or periodic motion called simple harmonic motion, or shm. With the free motion equation, there are generally two bits of information one must have to appropriately describe the masss motion. Simple harmonic motion university of texas at austin. This is a second order homogeneous linear differential equation, meaning that the highest derivative appearing is a second order one, each term on the left contains. A simple realization of the harmonic oscillator in classical mechanics is a particle which is. Damped simple harmonic motion department of physics.
The following physical systems are some examples of simple harmonic oscillator mass on a spring. Simple harmonic motion shm is a periodic vibration or oscillation having the following characteristics. Dynamics of simple harmonic motion many systems that are in stable equilibrium will oscillate with simple harmonic motion when displaced by from equilibrium by a small amount. With a trigonometric identity, i can combine those two terms cosine and sine into one. Simple harmonic motion occurs when the restoring force is proportional to the displacement. Simple harmonic motion blockspring a block of mass m, attached to a spring with spring constant k, is free to slide along a horizontal frictionless surface. Simplest model for harmonic oscillatormass attached to one end of spring while other end is held fixed. Introduction simple harmonic motion let us reexamine the problem of a mass on a spring see sect. Ordinary differential equationssimple harmonic motion. Second order differential equations and simple harmonic motion. Consider a mass which slides over a horizontal frictionless surface. Linear simple harmonic motion is defined as the motion of a body in which the body performs an oscillatory motion along its path.
Single variable, this course provides a brisk, entertaining treatment of differential and integral calculus, with an emphasis on conceptual understanding and applications to the engineering, physical, and social sciences. Di fferent ap plications problems are solved analytically with exact equation of simple harmonic motion. Lectures on differential equations uc davis mathematics. Simple harmonic motion differential equations youtube. A mass m attached to a spring of spring constant k exhibits simple harmonic motion in closed space. Institute for theoretical physics events xwrcaldesc. We will illustrate this with a simple but crucially important model, the damped harmonic. Differential equation for harmonic motion mathematics. The force acting on the object and the magnitude of the objects acceleration are directly proportional to the displacement of the object from its equilibrium position. The restoring force exerted by the spring is f k x, f k x. Correct way of solving the equation for simple harmonic motion. The above equation is known to describe simple harmonic motion or free motion.
The solution of this equation of motion is where the angular frequency is determined by the mass and the spring constant. This is the generic differential equation for simple harmonic motion. Oscillatory motion is simple harmonic motion if the magnitude of the restoring force f r is linearly proportional to the magnitude of the displacement x from equilibrium. Write and apply formulas for finding the frequency f, period t, velocity v, or acceleration acceleration ain terms of displacement displacement xor time t. Initially the mass is released from rest at t 0 and displacement x 0. The wire defines the rotation axis, and the moment of inertia i about this axis is known. Finding the period and frequency for simple harmonic motion. Pdf a case study on simple harmonic motion and its. An alternative definition of simple harmonic motion is to define as simple harmonic motion any motion that obeys the differential equation 11. If a particle repeats its motion about a fixed point after a regular time interval in such a way that at any moment the acceleration of the particle is directly proportional to its displacement from the fixed point at that moment and is always dir. We call this ordinary differential equation the damped harmonic oscillator equation.
The phasor representation gives us two independent solutions, even though we might only want to use only one of them to describe the motion. Home differential equation of a simple harmonic oscillator and its solution a system executing simple harmonic motion is called a simple harmonic oscillator. Simple harmonic motion a system can oscillate in many ways, but we will be. The mathematics of harmonic oscillators simple harmonic motion in the case of onedimensional simple harmonic motion shm involving a spring with spring constant k and a mass m with no friction, you derive the equation of motion using newtons second law. Simple harmonic motion and introduction to problem solving.
A mass m 100 gms is attached at the end of a light spring which oscillates on a friction less horizontal table with an amplitude equal to 0. In a wind instrument like a trumpet, the vibrations are caused by the players lips while the sound is caused by exciting the air molecules by blowing across the opening in a flute. Simple harmonic motion can be used to describe the motion of a mass at the end of a linear spring without a damping force or any other outside forces acting on the mass. In a percussion instrument like the triangle, the vibrations occur when the instrument is struck. Elastic limit if exceeded, the spring does not return to its original shape. Its best thought of as the motion of a vibrating spring. At t 0 the blockspring system is released from the equilibrium position x 0 0 and with speed v 0 in the negative xdirection.
Second order differential equations are typically harder than. Coupled harmonic oscillators massessprings, coupled pendula, rlc circuits 4. Deriving equation of simple harmonic motion physics forums. With complex notation we combine two equations into one. Equation 1 is a second order linear differential equation, the solution of which provides the displacement as a function of time in the form. Now we have to find the displacement x of the particle at any instant t by solving the differential equation 1 of the simple harmonic oscillator. What follows are my lecture notes for a first course in differential equations, taught at the hong kong university of. This is confusing as i do not know which approach is physically correct or, if there is no correct approach, what is the physical. Harmonic motion part 2 calculus video khan academy. Differential equations and linear algebra mit math. In simple harmonic motion, the force acting on the system at any instant, is directly proportional to the displacement from a fixed point in its path and the direction of this force is. The equation for describing the period shows the period of oscillation is independent of both the amplitude and gravitational acceleration, though in practice the amplitude should be small.
Uniform circular motion the projection onto the yaxis is also a solution to the differential equation. So the net force on our simple harmonic oscillator sho is. In most cases students are only exposed to second order linear differential equations. Professor shankar gives several examples of physical systems, such as a mass m attached to a spring, and explains what happens when such systems are disturbed. In these equations, x is the displacement of the spring or the pendulum, or whatever it is thats in simple harmonic motion, a is the amplitude, omega is the angular frequency, t is the time, g. Equation 1 is known as differential equation of simple harmonic oscillator. The block is free to slide along the horizontal frictionless surface. We then focus on problems involving simple harmonic motioni. We test whether acoswt can describe the motion of the mass on a spring by substituting into the differential equation fkx.
Simple harmonic motion and circular motion chapter 14. Our prototype for shm is a mass attached to a spring. The damped, driven oscillator is governed by a linear differential equation section 5. The focus of the lecture is simple harmonic motion. Simple harmonic motion has important properties, for example, the period of oscillation does not depend on the amplitude of the motion and lots of systems do undergo simple harmonic motion even if sometimes it is an approximation. Flash and javascript are required for this feature. A good example of the difference between harmonic motion and simple harmonic motion is the simple pendulum. Applying newtons second law of motion, where the equation can be written in terms of and derivatives of as follows. The canonical example of simple harmonic motion is the motion of a massspring system illustrated in the figure on the right.
Describe the motion of pendulums pendulums and calculate the length required to produce a given frequency. Suppose that the mass is attached to a light horizontal spring whose other end is anchored to an immovable object. Near equilibrium the force acting to restore the system can be approximated. In mechanics and physics, simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement simple harmonic motion can serve as a mathematical model for a variety of motions, such as the oscillation of a spring.
Simple harmonic motion concepts introduction have you ever wondered why a grandfather clock keeps accurate time. In other words, it will oscillate around the equilibrium point in a sinusoidal manner as a function of time. Second order equations involve the second derivative d2ydt2. Differential equation of a simple harmonic oscillator and. Here are some examples of periodic motion that approximate simple harmonic motion. Theres no real general way to necessarily derive a solution to a general differential equation, so ill give some fairly general informal derivations which assume that we are looking for a nice analyti.
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