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

``` ```
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  <-Previous page