我知cos(x) = sin(x+(π/2))
意思就係話... general 就係Asin(ωt) 而in diff. case, 我地可以derive Acos(ωt) = =?
thanks a lot: )
there maybe still a lot of questions to seek help from you = =...~
- 天同Lv 79 年前最愛解答
The simple answer is: which equation to be used depends on the initial condition of the system. That is, the displacement (value of x) at t = 0.
The equation: x=Asin(ωt+θ) is a general equation for a simple harmonic motion (SHM). It suits all conditions. If we take a simple pendulumn for example, when the pendulumn bob is displaced to the right most extreme position and then released, the initial condition is thus t=0, x = A. Substitute these values into the equation, we get, A=Asin(θ). This gives θ = pi/2 radians.
The equation for such system is therefore represented by the equation:
x = Asin(ωt + pi/2) which is x =A.cos(ωt)
Assume the pendulum bob is at the equilibrium position (x=0) at t=0 (these are the initial condition for such system) moving towards the right, when we put x = 0, t=0 into the general equation, we get: 0 =Asin(θ), i.e. θ = 0. Hence, such system follows the equation: x =A.sin(ωt)
Becuase the two systems described above are of different initial conditions, it is apparent that they have different velocities (and acceleration too) at different time. For example, after a quarter of a period, the first system will be at the equilibrium position and have the max velocity. The velocity direction during this first quarter period is towards the left, hence -ve in sign. For the second system, it will be at the extreme position and have zero velocity.
As a further example, if the pendulumn bob is first displaced to the left most extreme position and released, then the initial conditions are t=0, x = -A. The general equation now becomes: -A=A.sin((θ), i.e. θ = -pi/2
The equation of this system is: x = A.sin(ωt -pi/2) = A.sin[-(pi/2-ωt)] = -A.sin(pi/2-(ωt) = -A.cos(ωt)
2012-03-07 21:15:17 補充：
Please refer to further explanation on the velocity equations in the "opinion(意見)" section.
2012-03-07 21:20:53 補充：
The general equation x=A.sin(wt+pi/2) is equivalent to x=Acos(wt)
Differentiate the general equation for velocity v
v = A.d[sin(wt+pi/2)]dt = Aw.cos(wt+pi/2) = -Aw.sin(wt)
2012-03-07 21:23:25 補充：
This is the same as obtained by differentiating the special equation x = A.cos(wt)
Hence, no matter which equation (general form or special form) you use, you should get the same result for velocity and acceleration.