Question

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

Answer 1

 

POSITION 1
Un-stretched length of spring = 2 m
Extension (x₁) of spring = 0 
Spring energy =ES₁ =`1/2 kx _1^2` 
                                =0
PE₁=mgh 
      =0 J
KE₁=0 J
AT POSITION II: 

Let h=-3m
PE₂ = mgh = 20 x 9.81 x (-3)
      =-588.6 J
`KE_2 = 1/2 xx 20v^2`

=10v²

In△ABD,
By Pythagoras theorem
`AD=sqrt(2^2+3^2)`
      =3.6056 m
Extension(x₂) 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 = 10v²
v = 7.3283 m/s 

Speed when h=3m is 7.3283 m/s