**Problem 1A-10*:** Horizontal Spring Without Friction

A tiny block attached to a spring is lying on a very smooth horizontal surface
with one end of the spring fastened to a wall. The line on the surface aligned
to the spring is denoted as the x-axis. When the spring is totally relaxed,
the position of the tiny block is taken as x = 0. The position of the block at
time t can be expressed as a function x(t). The block is moved to a position
x=x_{0} and is gently released at t=0. The equation of motion for the
block is

where K is a positive constant.

(a) Find x(t), its speed v(t), and the acceleration a(t).

(b) What is the amplitude of the oscillatory motion? What is the maximum speeds of the block and are reached when and where?

(c) Draw graphs of x(t) and v(t).

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**Solution**

(a) The equation of motion

is satisfied by a general solution of the type

Since

Substituting d

Furthermore the initial condition x

xThe condition that the block is released gently at t=0 means that the speed at t=0 is zero. Sincee v(t)=dx/dt. Eq.(1) becomes_{0}= x(0) = Csin(0) + Dcos(0) = D .

0 = v(0) = ω{Ccos(0) - Dsin(0)} = ωC ⇒ C = 0 .Thus we have

x(t) = x_{0}cos(ωt) , ω^{2}= K .

(b) From x(t)=x_{0}cos(ωt), we see that the amplitude of the oscillation is x_{0}. The
absolute value of the speed, |v(t)|, becomes maximum when

Since sin(ωt)=±1 and cos(ωt)=0 imply x(t)=0, the maximum speed is reached at

when the block passes the position x=0.

(C) In the following figure, a=π/2ω, b=x_{0}, and c=ωx_{0}.

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