1) SIMPLE HARMONIC MOTION/OSCILLATIONS

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1 1) SIMPLE HARMONIC MOTION/OSCILLATIONS 1.1) OSCILLATIONS Introduction: - An event or motion that repeats itself at regular intervals is said to be periodic. Periodicity in Space is the regular appearance of a given "object" in space; for example, the location of lines on a ruler or the location of atoms in a crystalline structure of a solid. Periodicity in Time is the recurrence of the same event after a time interval; for example, the position of block in an oscillating block-spring system, or of a bob in a moving pendulum, or a piston in a car engine. - During a periodic motion, the object moves regularly around a certain equilibrium location from which is calculated the displacement. The graph of displacement vs. time contains all information about the motion details (fig.1). By definition, if an object s displacement from equilibrium follows a pure harmonic (sine or cos) function of time, one says that it is performing a Simple Harmonic Motion (SHM). Figure 1: This graph shows an object s displacement as a function of time in SHM. Depending on the time, the displacement of object may be either positive (above the equilibrium location) or negative (below the equilibrium location). The equilibrium location is shown by the dashed line. SHM Phasor, Functions & Equation: In a SHM, the displacement is a harmonic function of time. The phasor model (fig. 2) is used to introduce time into a harmonic function of angle. The phasor is a vector that rotates at a constant angular frequency ω while its tail is fixed at the origin of an (x,y) frame. The angle φ of phasor to Ox axe is called the phase and the ω values are taken as positive for counter-clockwise rotation. 1 Figure 2: The phasor model allows to visualize the phase value and displacement at any moment. If the angle (phase) of this vector with respect to the + -axis is φ 0 at t = 0, then at time t it will be φ(t) = φ 0 + ωt (1) The magnitude of the vector in the phasor model is taken

2 equal to the maximum shift of block from the equilibrium position. This parameter is called the amplitude (A) of oscillations. With the use of trigonometry, one may find that the y- component of phasor is y = Asinφ (2) Then, by substituting (1) into (2), one obtains the following harmonic function of time for displacement : which is mostly written as x(t) = Asin(ωt+ φ 0 ) (3) The other component of the phasor, i.e. the cosine component x(t) = Acos(ωt+ φ 0 ), can be also used to describe displacement of a SHM. However, in this course we will refer to the sine component. Thus, a vector with amplitude A (maximum displacement) rotating at a constant angular frequency ω is known as the phasor model. It is the basic model used in physics to study all oscillations and waves. Let s examine how to define the parameters A, ω, and φ 0 for a SHM. The A- parameter is defined straightforward from the maximum value of shift from equilibrium. To define ω - value one may use the period (T), which in figure 2 represents the time taken by the phasor to complete one full oscillation (one revolution). During this time, the phasor rotates by the angle 2π rad. So, from (1), one gets: Φ = 2π = ωt (4) The real frequency of oscillations is defined as: (5) From (4) and (5), one finds out the relationship between the angular (ω) and the real (f) frequencies: (6) Knowing the value of the displacement at t = 0, one may define φ 0 as:. Generally, one uses angles between 0 and 2π for φ 0. Note that using φ 0 = 3π/2 is the same as using φ 0 = -π/2. If the initial displacement of object from equilibrium position is zero, the phase constant may be zero or π ; but the phase constant is not zero if the initial displacement is not zero. Notations & Dimensions: y Displacement (depending on time, it may be positive, negative, or zero); A Amplitude of oscillations (always positive); φ Phase (radians); φ 0 Phase constant (radians); T Period (seconds); f Natural frequency (Hz = s -1 ) ω Angular frequency (rad/s). A SHM is an oscillation with: a) Constant amplitude and constant period(see next sections). b) Harmonic (sine or cosine) time dependence. c) Constant energy(see next sections). 2

3 - Differential equation for SHM (7) (8) (9) The last relation can be presented as: y(t) (10) The equation (10) is known as SHM equation ; the expression (7) is the SHM function. Since y is the displacement, the second derivative (y ) gives the acceleration (a), i.e. y (t) = a(t). Then, from (9) it comes out that: a(t) = -ω 2 y(t) (11) Let s take another look at (9). The maximum possible value of sin(φ 0 + ωt) = 1 (since -1 sin(x) 1). This means that the maximum value of acceleration is a max = Aω 2. Similarly, (8) is the first derivative of the displacement and gives the velocity as a function of time. Likewise, as -1 cos(x) 1, it comes out that υ max = Aω. Note: The relations (10) and (11) hold only for a SHM and can be used as a criterion for determining whether or not a given oscillation is a SHM. Remember: cos(x) = sin(x + π/2) and sin(x) = sin(x + π). Using these trigonometric identities, we may transform (8) into: y (t) = Aωsin[( ) + π/2] = υ(t) (8 ) and (9) into y (t) = Aω 2 sin[( ) + π] = a(t) (9 ) Now, (8 ) and (9 ) are the y-components of two new phasors, the velocity phasor with magnitude Aω and the acceleration phasor with magnitude Aω 2. So, the three phasors (displacement, velocity, acceleration) rotate with the same angular frequency (ω). From (8 ) one may notice that the velocity phasor is always ahead of the displacement phasor by π/2 rad. Also, from (9 ) we see that the acceleration phasor is π rad ahead of the displacement phasor (figure 3). Figure 3: Here the three phasors are shown with their corresponding phase shifts. The velocity phasor is π/2 rad ahead of the displacement phasor. The acceleration phasor is π rad ahead of the displacement phasor. All three phasors rotate simultaneously with the same ω (animation 2). 3

4 Horizontal Block-Spring System: Consider a block with mass m tied to the end of an extended spring without mass. Assume that the block is moving without friction on a horizontal plane. In these circumstances, the gravitation force is cancelled by normal force and the net force applied on the block is equal to the force exerted on it by the spring (fig. 4). The displacement (x) of the block from the equilibrium position is equal to the spring extension. The spring's force on block is equal to elastic force of spring which is given by Hooke s law where k is the elasticity constant of the spring. (12) From Newton s second law, the net force on the block has only an x-component F x = ma x (13) Next, by the use of (12), we get: Fig.4 Figure 4 (14) As k and m are positive quantities we can assign (15) and the relation (14) takes the form a = -ω 2 2 d x 2 x, which is the same as (11) or x 2 dt which is the same as (10). We conclude that the block (or the system block-spring) is moving in a SHM. From expression (15) one can derive the formulas for angular frequency and the period of this SHM as Note: T depends on k and m, but not on A (SHM requirement). Energy in a horizontal spring-block SHM In a SHM there is no friction (ideal model). Therefore, in general, there are three forces acting on the block; the elastic force, the weight and the normal force. The elastic force and weight are conservative forces and one can define " a system ". The potential energy of this system is U = U el + U g = ½ kx 2 + mgh 4

5 The normal force is the only external force acting on this system. As its work is zero, W ext = ΔE = 0 and we deal with an isolated system. The mechanical energy of this system is E = U el + U g + K where U g doesn t change during the oscillations. If one fix U g = 0 at support level, the total mechanical energy of the system becomes E = U el + K and, as ΔE = 0, this quantity must remain constant during oscillations. Now, let s use the SHM expressions for displacement and velocity to calculate E. (18) (19) and (20) Fig.5 As seen from equations (18-20), U el and K are positive and oscillate in time as shown in graph 5. But, their sum, i.e. total energy of system spring-block, E = ½ ka 2 remains constant in time. The potential energy U of this system is a parabolic function of displacement x (see Fig.5). One says that SHM is characterized by a parabolic potential well. The form of the potential well is defined by the force at its origin. The parabolic form of potential is met whenever a restoring (elastic) force is exerted on the system. The parabolic wells and restoring forces are used as a first approximation for the study of oscillations. Other types of forces may act on a system and the potential wells may not be parabolic. Simple Harmonic Oscillations In a Simple Harmonic Motion (SHM), the displacement is a real shift in space. Meanwhile, any physical parameter (electric current, temperature, pressure, etc ) may oscillate in time as a pure harmonic function. In this case, one says that it performs a Simple Harmonic Oscillation (SHO). Therefore a SHM is in fact a SHO where the displacement is a real shift in space. 5

6 When deriving the differential equation (10), we didn t mention any special requirement for the physical nature of y(t). Actually, equation (10) is valid for any SHO and is known as the SHO equation. SHO equation If the " displacement from equilibrium value " of a physical quantity obeys to a differential equation of type (10), one may affirm straight away that this parameter is performing a SHO. - There are three main types of harmonic oscillations: a) Simple Harmonic Oscillations (SHO); no energy loss = ideal model b) Damped Harmonic Oscillations (DHO); energy loss = real life model c) Forced Harmonic Oscillations (FHO); external force compensates for lost energy = real life model We will study DHM and FHM characteristics in the next section. Note that the results are valid for any DHO and FHO because the form of the mathematical expressions is the same. Actually, almost all the results that we obtain from studying SHM are also valid for SHO. Also, the total energy for any SHO has the same for as the expression (20), except that instead of an elasticity constant k, there is another restoring constant. Important: In any SHO, the total energy E A 2. Animation 1: Animation 2: 6

Periodic Motion. Periodic motion is motion of an object that. regularly repeats

Periodic Motion. Periodic motion is motion of an object that. regularly repeats Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A special kind of periodic motion occurs in mechanical systems