Simple Harmonic Motion 的问题, STPM的, 请高手解答

tanflash1314

When a swimmer is standing at the end of a diving board, the end of the board is depressed a distance of 0.30m. If he jumps slowly at the end of the board, what is the period of the simple harmonic motion produced?

我不会算啊, 只给了一个 amplitude, 要怎样得到 period, T 呢？

kensai

你还有一个g=9.81

puangenlun

“x(t) =A0 sin (wt) + (V/w) cos (wt)”

请问后面的(V/w) cos (wt)是怎样出现的？

北方戰士

抱歉，用错equation。
实际的equation应该是
x = x0 cos (wt) + (V/w) sin (wt)


derivation 是这样:

assume
x(t) = A sin (wt + b) where b is phase angle
elaborate
x(t) = A{sin(wt)cos(b)+cos(wt)sin(b)}
at t=0, x(t) = x(0) = x0

x(t) = x0 cos(wt) + A cos(b) sin(wt)

observe that dx(0)/dt = (Aw)cos(b) = V0 =>initial velocity

therefore
x(t) = x0 cos(wt) + (V0/w) sin(wt)

北方戰士

用simple pendulum solve。
I'm going to show you the fundamental concept of oscillatory motion, I hope you enjoy it.

for any single degree of freedom system (i.e. 1 mass, 1 stiffness, 1 direction of motion possible)
in this case, the board itself contain mass of m, it has a stiffness of k and the motion is only free to move up and down. so it can be best represented by a 1 degree of freedom model.

assume x(t) = A sin (wt + b)          (1)
where w = frequency of oscillation, A is amplitude, b is phase angle

equation of motion is
kx - m(w^2)x = 0        (2)

which tells, stiffness force + inertia force = 0
in other words it means, no external force is acting on the function。

from eq. (2) we can see
natural frequency of the system, w is
sqrt (k/m)       * sqrt = square root

from the fact that F = kx
then mg/x = k                    (3)

sub (3) into (2) we have
w = sqrt (g/x)

and

w= 2*pi*f
f=2*pi*sqrt (g/x)
T=1/f

now back to your question.
f = 2*pi*sqrt (g/0.3)
T = 1/f
=0.0278 sec

it's analogous to simple pendulum problem!

puangenlun

楼上的推理不知道有没有错
但是结果很吓人

我的计算是这样
k=mg/x
T=2*pi*sqrt(m/k)=2*pi*sqrt(x/g)=1.09877s
不知道对不对

"x(t) = A sin (wt + b)"
题目没有牵涉initial condition
所以我认为可以直接let b=0
不知道对不对

北方戰士

楼上的推理不知道有没有错
但是结果很吓人

我的计算是这样
k=mg/x
T=2*pi*sqrt(m/k)=2*pi*sqrt(x/g)= ...
puangenlun 发表于 16-1-2012 01:51 AM

    你的答案是对的。应该是T=2*pi*sqrt(x/g)。
derivation出现了小小的错误。
b 的出现是一个generalization。因为我们不懂oscillation在哪一个点被开始计算。
当然我们可以把它控制为0。结果还是一样没有差。

我很惊讶stpm有这样的问题出现。在他们还未了结什么是fundamental of vibration - single degree of freedom system之前，这不应该出现