Oscillations Pham Tan Thi, Ph.D Department of Biomedical Engineering Faculty of Applied Science Ho Chi Minh University of Technology What is an Oscillation? • Any motion that repeats itself • Described with reference to an equilibrium position where the net force is zero, and a restoring force which acts to return object to equilibrium • Characterize by: - Periodic (T) or frequency (f) or angular frequency (ω) - Amplitude (A) 14.2 Model for periodic motion When the body is displaced from its equilibrium position at x = 0, the spring exerts a restoring force back toward the equilibrium position Simple Harmonic Oscillation/Motion Total force exerts to object given by Hooke’s law F = kx (a) x 0: glider displaced to the right from the equilibrium position Newton’s second law of motion: y d2 x F =m dt d x = dt ax y x Fx x Fx n x mg then Fx , 0, so ax , 0: stretched spring pulls glider toward equilibrium position (b) k x= m !2 x x 0: The relaxed spring exerts no force on the glider, so the glider has zero acceleration y y General solution for motion x = Acos(!t + ) Phase of the motion k where ! = Angular frequency m Velocity v = A!sin(!t + ) Acceleration a= A! cos(!t + ) x x mg (c) x , 0: glider displaced to the left from the equilibrium position Fx 0, so ax 0: compressed spring pushes glider toward equilibrium position ax y y x n O n Fx x mg Fx x It’s simplest rium position, x-component o change in the exerts on the ax = Fx>m Figure 14.2 Whenever the tends to restor restoring forc ing to return th Let’s analyz right to x = A (Fig 14.2a) Th When the body its motion it ov rium position t tion are to the r stop We will s The body then the starting poi If there is no fr this motion re toward the equ In different librium in diff force that tend Amplitude, Here are some The amplit displacement f positive If the motion is 2A T complete roun through O to Period and Frequency Period (T): is the time to complete one full cycle, or one oscillation [s] Frequency (f): is the number of cycles per second [s-1] The relation between Frequency and Period: 14.2 Simple Harmonic Motion s A and gives an 445 = v and 14.13 How f x-velocity x-acceleration a vary during Tone cycle of SHM or x x x x 2A x x 5A (14.18) v0 x We’ll sketch 14); then divide ) The right side lt is (14.19) a nonzero initial ment That’s reae velocity v0 x , it 2A 2A/2 A/2 ax ! 2amax vx ax vx ax vx 2vmax ax vx x x x x ax amax vx ax vx x x ax vx vmax ax vx ax 2amax vx I: Describing Motion A x x x T = Simple Harmonic Motion f Angular Frequency The oscillation frequency is measured in cycles per second, Hertz We may also define an angular frequency ω, in radians per second, to describe the oscillation 2⇡ !(in rad/s) = = 2⇡f T The position of an object oscillating with simple harmonic motion can then be written as x(t) = Acos!t then, the maximum speed of this object is vmax 2⇡A = = 2⇡f A = !A T Displacement as function of time in SHM x = Acos(!t + ) k ! = m Changing m, A or k changes the graph of x versus t: change f Mechanical Energy in Simple Harmonic Motion Potential Energy, U Consider the oscillation of a spring as a SHM, the potential energy of the spring is given by U = kx where k = m! 2 U = m! x The potential energy in terms of time, t, is given by U = m! x2 where x = Asin(!t + ) U = m! A2 sin2 (!t + ) Mechanical Energy in Simple Harmonic Motion Kinetic Energy, K The kinetic energy of an object in SHM is given by K = mv 2 where v2 = ! K = m! (A2 p A2 x2 x2 ) The kinetic energy in terms of time, t, is given by K = m! (A2 x2 ) where v = A!cos(!t + ) 2 K = m! A cos (!t + ) Mechanical Energy in Simple Harmonic Motion Total Energy, E The total energy of a body in SHM is the sum of its kinetic energy, K and its potential energy, U E =K +U From the principle of conservation of energy, this total energy is always constant in a closed system E = K + U = constant The equation of total energy in SHM is given by E = m! (A2 2 E = m! A x ) + m! x2 2 OR E = kA vmax = Am A = vA (14.23) This agrees with Eq.Energy (14.15): vx oscillates betweenHarmonic -vA and +vA Motion Mechanical in Simple Interpreting E, and U in SHM Conservation of K, Energy Figure 14.14 shows the energy quantities E, K, and U at x = 0, x = ! A>2, and is a graphical display of Eq (14.21); energy (kinetic, x = ! A Figure 14.15 max potential, and total) is plotted vertically and the coordinate x is plotted horizontally 1 1 E = mv + kx = kA = mv 2 2 (a) The potential energy U and total mechanical energy E for a body in SHM as a function of displacement x The total mechanical energy E is constant Energy E U5 kx2 (b) The same graph as in (a), showing kinetic energy K as well At x 6A the energy is all potential; the kinetic energy is zero At x the energy is all kinetic; the potential energy is zero Energy K U U 2A O x E5K1U A K x 2A O A At these points the energy is half kinetic and half potential x 14.15 Kin energy U, and as functions o value of x the U equals the show that the half potential ... principle of conservation of energy, this total energy is always constant in a closed system E = K + U = constant The equation of total energy in SHM is given by E = m! (A2 2 E = m! A x ) + m! x2 2 OR... Asin(!t + ) U = m! A2 sin2 (!t + ) Mechanical Energy in Simple Harmonic Motion Kinetic Energy, K The kinetic energy of an object in SHM is given by K = mv 2 where v2 = ! K = m! (A2 p A2 x2 x2 ) The... rium position t tion are to the r stop We will s The body then the starting poi If there is no fr this motion re toward the equ In different librium in diff force that tend Amplitude, Here are some