3. Consider the system shown below.

elastic band

N

frictionless rod

X

m

i) Show that the equation of motion for the sliding block with linear spring (elastic band) is

(1)

-1/2

mx+kx(x² + z²)-¹/²[(x² + z²)¹/² -L] = 0./nThen show that the equation can be put in the following convenient form

ü+2u(u²+1)-¹/²[(u²2 +1)¹/² - 1] = 0,

where

u= x/2 and 1= L/z.

@² = k/2m.

(2)

The new independent variable is tot where

ii) Sketch motions in the phase plane for /<1, 1=1, and />1. What is the significance of 1=1?

iii) Expanding (2) for small u, show that

ü+u³+=0 when l = 1.

T ==

iv) Using (3), obtain the following exact relationship between the displacement u and time t:

2

du

T

7 = √² [ [u² − u²]³¹²

(3)

= F(-2,0)

where the motion begins from rest at u= uo. This expression can be put in a more

convenient form by letting u=-up cos. Using this, show that (4) becomes

dø

-j.

10[1-_-_sin² 01/2

(5)

where F is the elliptic integral of the first kind. Show that the period of motion is

T≈ 7.416/up.

(4)