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=x0 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 d2x/dt2 into the equation of motion yields
         
Furthermore the initial condition x0=x(0) leads to
         x0 = x(0) = Csin(0) + Dcos(0) = D .
The 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 = v(0) = ω{Ccos(0) - Dsin(0)} = ωC  ⇒  C = 0 .
Thus we have
         x(t) = x0cos(ωt) ,   ω2= K .

         

         

(b) From x(t)=x0cos(ωt), we see that the amplitude of the oscillation is x0. 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=x0, and c=ωx0.




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