44 A spring of mass 2 kg and stiffness 1
kN/m is dropped from a height of 2 m on
the ground. Calculate the maximum
distance through which the spring is
compressed
8: A
Answers
Answer:
A spring of mass 2kg and stiffness 1 KN/m is dropped from a height of 2 m on the ground.
To find:
Max compression?
Calculation:
Let max compression be "x".
Now, we can say:
Initial PE = mg(2 + x)
Final PE = ½kx²
Now, as per law of conservation of energy:
\rm \: mg(2 + x) = \dfrac{1}{2} k {x}^{2}mg(2+x)=
2
1
kx
2
\rm \implies \: 2 \times 10 \times (2 + x) = \dfrac{1}{2} \times 1000 \times {x}^{2}⟹2×10×(2+x)=
2
1
×1000×x
2
\rm \implies \: 20 (2 + x) = 500{x}^{2}⟹20(2+x)=500x
2
\rm \implies \: 40 + 20x = 500{x}^{2}⟹40+20x=500x
2
\rm \implies \: 500{x}^{2} - 20x - 40 = 0⟹500x
2
−20x−40=0
\rm \implies \: 50{x}^{2} - 2x - 4= 0⟹50x
2
−2x−4=0
\rm \implies \: 25{x}^{2} - x - 2= 0⟹25x
2
−x−2=0
After solving the quadratic equations:
\rm \implies \: x = 0.3 \: m⟹x=0.3m
So , max compression of spring is 0.3 metres.