Physics 2 - Lecture 1: Oscillations - Huynh Quang Linh
Driven Oscillations and Resonance
To keep a damped system going, energy must be
put into the system. When this is done, the
oscillator is said to be driven or forced. If you put
energy into the system faster than it is dissipated,
the energy increases with time, and the amplitude
increases. If you put energy in at the same rate it is
being dissipated, the amplitude remains constant
To keep a damped system going, energy must be
put into the system. When this is done, the
oscillator is said to be driven or forced. If you put
energy into the system faster than it is dissipated,
the energy increases with time, and the amplitude
increases. If you put energy in at the same rate it is
being dissipated, the amplitude remains constant
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- OSCILLATION Tran Thi Ngoc Dung – Huynh Quang Linh – Physics A2 HCMUT 2016
- 1.1 Simple harmonic motion Spring pendulum O Newton’s 2nd law: x ma P N fspring (1) f d2x s (1)/x : m 2 kx dt x 2 d x 2 x ox 0 o k / m dt 2 x : distance from A: amplitude. equilibrium x Acos(ot ) (ω t + ): phase of the motion v A sin( t ) Velocity phase constant. o o The phase constant depends on the 2 Acceleration: a Ao cos(ot ) choice at t = 0. 1 2 1 2 2 Kinetic Energy KE mv kA sin (o t ) 2 2 1 1 Potential Energy PE kx 2 kA 2 cos2 ( t ) 2 2 o The total energy in simple harmonic 1 motion is proportional to the square of W KE PE kA 2 const the amplitude. 2 Average Kinetic Energy =Average 1 2 Potential Energy KE PE kA 4
- Damped Oscillations O Let to itself, a spring or a pendulum eventually stops oscillating because the mechanical energy is dissipated by x f frictional forces. The oscillation is damped drag v ma P N fspr fdrag (1) x fspring d2x x m kx - bv dt 2 d2x b dx k b x 0 ωo k/m; 2 dt 2 m dt m m 2 d x dx 2 2 ox 0 dt 2 dt t 1 m 2 2 x Ae cos(t ) τ : decay time T 2β b 2 2 2 o 2 2 β ωo :overdamped o o 1 T>To=2 /o o β ωo :critically damped When is greater than or equal to o the system does not oscillate. If = o the system is said to be critically damped; it returns to equilibrium with no oscillation in the shortest time possible. When >o the system is overdamped.
- Driven Oscillations and Resonance O To keep a damped system going, energy must be put into the system. When this is done, the x oscillator is said to be driven or forced. If you put fdrag energy into the system faster than it is dissipated, v the energy increases with time, and the amplitude x f F increases. If you put energy in at the same rate it is x spring driven being dissipated, the amplitude remains constant over time. Fdrivent=Focost We will discuss the general solution of ma P N fspring fdrag Fdriven (1) Equation * qualitatively. 2 It consists of two parts, the transient d x solution and the steady-state solution. m kx - bv Focost dt 2 The transient part of the solution is identical to that for a damped oscillator d2x b dx k F x o cost Over time, this part of the solution dt 2 m dt m m becomes negligible because of the exponential decrease of the amplitude. We b Fo ωo k/m; 2 ; fo are then left with the steady− state m m solution. 2 d x dx 2 x Ce t cos(t ) 2 ox focost trans dt 2 dt xsteady Acos(t ) x Acos(t )
- Resonance curves x Acos(t ) f =0.05o A o 2 2 2 2 2 (o ) 4 2 tan 2 2 o 2 2 0 resonance o 2 =0.25o f A o fo 2 Aresonance o 2 2 2 o
- 2. A particle is in simple harmonic motion along the x axis. The amplitude of the motion is xm. At one point in its motion its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at x = 0) is U = 3 J. When it is at x = xm, the kinetic and potential energies are: A. K = 5 J and U = 3J B. K = 5 J and U = −3J C. K = 8 J and U = 0 ans: D D. K = 0 and U = 8J E. K = 0 and U = −8J 3. Two uniform spheres are pivoted on horizontal axes that are tangent to their surfaces. The one with the longer period of oscillation is the one with: A. the larger mass B. the smaller mass C. the larger rotational inertia D. the smaller rotational inertia E. the larger radius ans: E