Simple harmonic motion Vibration / Oscillation to-and-fro repeating movement 1 Simple harmonic motion S.H.M. A special kind of oscillation X O Y A X, Y : extreme points O: centre of oscillation / equilibrium position A: amplitude A special relation between the displacement and acceleration of the particle 2 Exploring the acceleration and displacement of S.H.M. a– x graph 1.5 0.3 acceleration displacement 1 0.2 acceleration 0.5 0.1 displacement 0 0 0 45 90 135 180 225 270 315 360 405 - 0.5 - 0.4 - 0.3 - 0.2 - 0.1 0 0.1 0.2 0.3 0.4 time - 0.1 - 0.5 - 0.2 -1 - 0.3 - 1.5 3 0.5 Exploring the acceleration and displacement of S.H.M. a – x graph a a ∝ -x x x Definition of Simple harmonic motion (S.H.M.) An oscillation is said to be an S.H.M if (1) the magnitude of acceleration is directly proportional to distance from a fixed point (centre of oscillation), and 4 Definition of Simple harmonic motion (S.H.M.) An oscillation is said to be an S.H.M if (1) the magnitude of acceleration is directly proportional to distance from a fixed point, and (2) the acceleration is always directed towards that point. a(-) a(-) a(-) a(+) a(+) a(+) a = 3k a = 2k a = k a = 0 a = k a = 2k a = 3k x -3 -2 -1 0 1 2 3 Note: For S.H.M., direction of acceleration and displacement is always opposite to each other. 5 For the projection P’ • Moves from X’ to O’ to Y’ and returns through O’ to X’ as P w 2 rw cos q completes each revolution. P Displacement a = rw2 rw2 sin q • Displacement from O q • x = r cos q = r cos w t O Acceleration Acceleration of P’ = component r of acceleration of P along the x x-axis O’ P’ X’ • a = -rw2 cos q (-ve means x directed towards O) ∴ a = -w2 x q = wt The motion of P’ is simple harmonic. Equations of S.H.M. Y’ 6 Equations of S.H.M. rw q w Period The period of oscillation of P’ = time for P to make one revolution T = 2p / angular speed ∴ T = 2p/w P rw sin q O rw cos q q r x Y’ O’ P’ X’ x Velocity Velocity of P’ = component of velocity of P along the x-axis v = -rw sin q = -rw sin w t 7 Equations of S.H.M. rw q w P rw sin q O rw cos q q A r Y’ O’ P’ X’ x Motion of P’ Amplitude of oscillation = Radius of circle ⇒A = r Displacement x: x = A cos q Velocity v: v = -wA sin q x Acceleration a: a = -w2A cos q 8 Relation between the amplitude of oscillation A and x, w, and v: v x sin q , cosq Aw A 2 2 v x 1 Aw A v 2 w 2 x 2 A2w 2 v 2 w 2 A2 x 2 Note: Maximum speed = wA at x = 0 (at centre of oscillation / equilibrium position). 9 Example 1 A particle moving with S.H.M. has velocities of 4 cm s-1 and 3 cm s-1 at distances of 3 cm and 4 cm respectively from its equilibrium. Find (a) the amplitude of the oscillation -1 4 ms Solution: By v2 = w2(A2 – x2) O when x = 3 cm, v = 4 cm s-1, 3 ms-1 -1 x = 4 cm, v = 3 cm s . 42 = w2(A2 – 32) --- (1) 32 = w2(A2 – 42) --- (2) (1)/(2): 16/9 = (A2 – 9) / (A2 – 16) 9A2 – 81 = 16 A2 - 256 A2 = 25 A = 5 cm ∴ amplitude = 5 cm 10 (b) the period, (c) the velocity of the particle as it passes through the equilibrium position. (b) Put A = 5 cm into (1) 42 = w2(52 – 32) w2 = 1⇒ w = 1 rad s-1 T = 2p/w = 2p s (c) at equilibrium position, x = 0 By v2 = w2(A2 – x2) v2 = 12(52 – 02) v = 5 cm s-1 11 Isochronous oscillations Definition: period of oscillation is independent of its amplitude. Examples: Masses on springs and simple pendulums 12 Phase difference of x-t, v-t and a-t graphs x v x A cosw t A t 0 T/4 T/2 3T/4 T y v wA sin w t wA t 0 T/4 T/2 3T/4 T w x v A wA w2A a a w 2 A cosw t w2A t 0 T/4 T/2 3T/4 T Vectors x, v and a rotate with the same angular velocity w. Their projections on the y-axis give the above x-t, v-t and a-t graphs. a 13 Phase difference of x-t, v-t and a-t graphs y w x v A wA w2A a Note: 1 a leads v by 90o or T/4. (v lags a by 90o or T/4) 2 v leads x by 90o or T/4. (x lags v by 90o or T/4) 3 a leads x by 180oor T/2. (a and x are out of phase or antiphase) 14 Energy of S.H.M. (Energy and displacement) From equation of S.H.M. v2 = w2(A2 – x2), ∴ K.E. = ½ mv2 = ½ mw2(A2 – x2) K.E. x -A 0 A Note: 1. K.E. is maximum when x = 0 (equilibrium position) 2. K.E. is minimum at extreme points (speed = 0) 15 Potential energy P.E. = ½ kx2 ∵ w2 = k/m ∴ P.E. = ½ mw2x2 Centre of oscillation ix P.E. x = -A x=A x -A 0 A P.E. is maximum at extreme points. (Spring is most stretched.) P.E. is minimum when x = 0. (Spring is not stretched) 16 Total energy = K.E. + P.E. 1 1 1 2 2 2 2 2 mw A x mw x mw 2 A2 (constant) 2 2 2 Energy ½ mw2A2 Total energy P.E. K.E. -A 0 A x 17 Energy and time From equation of S.H.M. x A cosw t and v wA sin w t 1 1 1 2 2 mv m w A sin w t mw 2 A 2 sin 2 w t K.E. = 2 2 2 1 1 1 2 2 2 2 m w x m w A cos w t mw 2 A 2 cos 2 w t P.E. = 2 2 2 Total energy = K.E. + P.E. 1 1 1 mw 2 A 2 sin 2 w t mw 2 A 2 cos 2 w t mw 2 A 2 sin 2 w t cos 2 w t 2 2 2 1 mw 2 A2 (constant) 2 18 Energy Total energy ½ mw2A2 P.E. = ½ mw2A2cos2wt K.E. = ½ mw2A2sin2wt 0 T/4 T/2 3T/4 T Time Centre of oscillation ix t = T/2 t=0 19 Examples of S.H.M. Mass on spring – horizontal oscillation Hooke’s law: F = kx where k is the force constant and x is the extension By Newton’s second law Centre of oscillation T = -ma kx = -ma a = -(k/m)x ix T which is in the form of a = -w2x Hence, the motion of the mass is simple harmonic, and Natural Extension w2 = k/m length (l) (x) Period of oscillation 2p m 2p w k 20 Mass on spring – vertical oscillation Natural length (l) Extension at e equilibrium Displacement x from equilibrium T’ T Centre of oscillation mg mg Displaced In Spring from unstretched equilibrium equilibrium At equilibrium, T’ = mg ke = mg Displaced from equilibrium, T – mg = -ma k(e + x) – mg = -ma mg k( x) mg ma k k a x m which is in the form of a = -w2x Hence, the motion of the mass is simple harmonic and w2 = k/m. Period of oscillation 2p m 2p 21 w k Effective mass of spring Not only the mass oscillates when it is released, but also the spring itself. The period of oscillation is affected by the mass of the spring. Hence, the equation T 2p m k should be rewritten as T 2p m ms k where ms is the effective mass of the spring. 22 Measurement of effective mass of spring To find the effective mass, we can do an experiment by using different masses m and measure the corresponding periods T. Use the results to plot a graph of T2 against m which is a straight line but it does not pass through the origin. T2 x Line of best fit x x x x m 23 ∴ ∴ m ms k m ms T 2 4p 2 k T 2p 2 2 4 p 4 p or T 2 m ms k k T2 slope = x x x 4p 2 k y-intercept = x x m 4p 2 ms k ∴ effective mass ms k y-intercept 2 4p In theory, effective mass of a spring is about ⅓ of the mass of string. Usually, we would neglect the effective mass for simplicity. 24 Combined Springs Oscillation Case 1: Springs in parallel F1 = k1x F2 = k2x F Let x be the common extension of the spring. ∵ the springs are in parallel, ∴ upward force F = F1 + F2 F = k1x + k2x = (k1 + k2)x Note: k1 + k2 is the equivalent force constant of the system. When the mass is set into vibration, the oscillation is simple harmonic. Period of oscillation m T 2p k where k = k1 + k2 25 Case 2: Springs in series F1 = k1x1 F2 = k2 x2 Let x1 and x2 be the extensions of the first and the second spring respectively. The total extension x = x1 + x2 ∵ the springs are in series, ∴ upward force F = F1 = F2 F = k 1 x1 = k2 x2 x1 F F 1 1 k1 k 2 and x2 x = x1 + x2 F F x k1 k2 or F kx where x ∵ F k1 F k2 1 1 1 k k1 k 2 26 Case 2: Springs in series Note: the equivalent force constant of the F1 = k1x1 system is k where 1 1 1 k k1 k 2 When the mass is set into vibration, the oscillation is simple harmonic. F2 = k2 x2 F Period of oscillation T 2p . where m k 1 1 1 k k1 k 2 27 Case 3: The mass is connected by two springs on both sides Equilibrium position Force Force constant k1 constant k2 T1 T2 Under extension x Suppose the springs are initially unstretched. Under compression When the mass is displaced to the right by x, 1st spring is extended but 2nd spring is compressed. Resultant force on the bob F = T1 + T2 ∴ F = k1x + k2x = (k1 + k2)x Note: k1 + k2 is the equivalent force constant of the system. The oscillation is simple harmonic. Period of oscillation T 2p m k where k = k1 + k2. 28 Simple pendulum Resolve tangentially (perpendicular to the string) mg sin q = -ma q where a is the acceleration along the arc l If q is small (i.e. <10o), sin q ≈ q and x ≈ lq , T mg sin q = -ma becomes mg q = -ma P x a = -g(x/l) = -(g/l)x O which is in the form of a = -w2x mg sin q Hence, the motion of the bob is simple 2 = g/l harmonic and w mg cos q 2p l mg Period of oscillation T 2p w g A 29 A simple pendulum has a period of 2 s and an amplitude of swing 5 cm. Calculate the maximum magnitudes of (a) velocity, and (b) acceleration of the bob. Solution: By T 2p w 2 2p w w p rad s 1 (a) maximum magnitude of velocity = wA = p(5) = 5p cm s-1 (b) maximum magnitude of velocity = w2A = 5p2 cm s-2 30