k m Summary: 02m If the body is subject to: 平衡位置 Fig.15-1 A restoring force that is proportional to the displacement but opposite in sign(F=-kx)i or The acceleration is proportional to its displacement and of opposite sign( a==0x)i or The motion equation is x=Acos(@t+)
Summary: If the body is subject to: A restoring force that is proportional to the displacement but opposite in sign( F = −kx ) ; or The acceleration is proportional to its displacement and of opposite sign( a x ) ; 2 = − or The motion equation is ; x = Acos(t +) k m x Fig.15 −1
Example 14-1: prove the system is a simple harmonic oscillator as shown in fig. 15-2 k k 0Fig.15-2 0 r m g L=-Y
Example 14-1:prove the system is a simple harmonic oscillator as shown in Fig.15-2. l 0 x 0 l x k m k mg l 0 = F = −kx l 0 x 0 l k m Fig.15 − 2
Example 14-2: In Fig. 15-3, prove the !!!L simple pendulum(单摆) is a simple harmonic oscillator when e is small e Prove: (1) Displacement: e (2) Restoring torque(恢复力矩): Fig.15-3V mg M=-mgl sin e ≈=mgl6f0rsml16 (3) Angular(角量) acceleration: B=n=-80=02a=18r=m2 Note:x=10 0的2sm2x( Neary 线量)
Example 14-2: In Fig.15-3, prove the simple pendulum(单摆） is a simple harmonic oscillator when is small. Prove: (1) Displacement: or x (2)Restoring torque(恢复力矩）: M = −mgl sin (3) Angular（角量） acceleration: = = − = − l g I M = I = ml l g Note: x = l x dt d x a = = − (linear线量) x Fig.15−3 −mgl for small
2. The velocity and acceleration of SHM: x= Acos( at+p) k m:x Its velocity is given by: mmm 平衡位置 dx-a4sin(ot+φ Fi.15-1 dt and its acceleration is equal to: a=5=-O2Ac0s( at+)(or=-o2x)
2. The velocity and acceleration of SHM: x = Acos(t + ) Its velocity is given by: Asin( t ) dt dx v = = − + and its acceleration is equal to: Acos( t ) ( or x ) dt d x a = = − + = − k m x Fig.15 −1
A Note: (1)Maximal values: Mmx =@A a=024 (2)The curves of x(), v(t)and at): 0,a 0A T 0 A 0a 24 (q=0) a<0 00 <0 >0 0 减速加速 减速加速 (3) 看图:x、 y and a
Note: (1) Maximal values: a A v A x A max max max = = = (2) The curves of : x(t)、v(t) and a(t) o T t x、 、a x 2A > 0 < 0 < 0 > 0 a < 0 < 0 > 0 > 0 减速 加速 减速 加速 A A -A - A -2A a ( = ) (3)看图：x、v and a