Problem 1A-11***: Horizontal Spring With Friction

Similar set up as in Problem 1A-10, but now there is friction between the surface and the block. The equation of motion becomes

         spring on horizontal rough surface
Suppose the block is moved to an initial position x0 and is gently released at t=0.

(a) Determine x(t), v(t) and a(t).

(b) Draw graphs for x(t) and v(t).






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Solution:

(a)The equation of motion


as the general solution. We can demonstrate this by substituting Eq.(2) into Eq.(1). To simplify the calculation, we define

Since
          
we have
          
Substituting Eq.(3A) and Eq.(3B) into Eq.(1), we get

Substituting Eq.(2) and Eq.(3) into Eq.(4), dividing out the term e-αt from both sides, and rearranging terms according to sin(ωt) and cos(ωt), we get
          
Since sin(ωt) and cos(ωt) are independent for an arbitrary t, to let the above equation hold for all t, we must have

For C and D to have non-trivial solutions, we must have
          
Both terms on the left-hand side of the above equation are positive or zero, so to satisfy the equation we must have

Using the initial condition x(0) = x0 and Eq.(2), we get D = x0.
The condition of gently releasing the block at t = 0 implies
v(0) = 0. Thus from v(t) = dx/dt and Eq.(3A), we get
          
With C and D determined, we can summarize the results as

(b) To plot the graph of x(t), the form of x(t) as presented above is not convenient; with the superposition of a sine function and a cosine function it is difficult to see the properties of the function clearly. We make the following modification to x(t), leading to the introduction of the phase angle. We define

          
Then we can write x(t) as

It is always possible to find a ω0 such that it satisfies the following relations:
          
With this introduction of ω0, we can write
          
This ω0 is called the phase angle and plays the role of shifting the curve sin(ωt) by an amount of t = ω0/ω toward the right along the t axis. In the following plot of x(t) vs. t, we set
          


In the following plot of v(t) vs. t for x0 > 0, we set
          


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