b)A cylinder has a mass of 20kg and is released from rest when h=0 as shown in the figure.Determine its speed when h=3m.The springs have an unstretched length of 2 m.Take k=40 N/m.

**Given** : m=20 kg

h=0

k=40 N/m**To find **: Speed when h=3m

#### Solution

**POSITION 1**

Un-stretched length of spring = 2 m

Extension (x_{1}) of spring = 0

Spring energy =ES_{1} =`1/2 kx _1^2`

=0

PE_{1}=mgh

=0 J

KE_{1}=0 J**AT POSITION II: **

Let h=-3m

PE_{2} = mgh = 20 x 9.81 x (-3)

=-588.6 J

`KE_2 = 1/2 xx 20v^2`

=10v^{2}

In△ABD,**By Pythagoras theorem**

`AD=sqrt(2^2+3^2)`

=3.6056 m

Extension(x_{2}) of spring = 3.6056 - 2=1.6056 m

`E_(S2) = 1/2 Kx 2/2 = 1/2 xx 40 xx 1.6056^2`

=51.5559 J**Applying work-energy principle**

`U_(1-2) = KE_2 - KE_1`

`PE_1 - PE_2 + E_(S1) - E_(S2) = KE_2 - KE_1`

588.6 - 51.5559 = 10v^{2}

v = 7.3283 m/s

**Speed when h=3m is 7.3283 m/s**