When a mass of 2kg is hung off spring, the spring stretches by 4cm. What is spring's constant? a)...

When a mass of {eq}\displaystyle 2kg {/eq} is hung off spring, the spring stretches by {eq}\displaystyle 4cm. {/eq} What is spring's constant?

{eq}\displaystyle a) 5 N/cm \\b) 5 N/m \\c) 5 cm/N \\d) 5 m/N {/eq}

We are given:

• The mass hung from the spring, {eq}m=2\;\rm kg {/eq}

• The extension in the spring, {eq}\Delta x=4\;\rm cm {/eq}

At the stretched length, the restoring force, {eq}F {/eq}, of the spring must be able to balance the weight, {eq}W {/eq}, of the mass.

{eq}F_r=W {/eq}.

The weight of an object is given by the following equation:

{eq}W=mg {/eq}

• where {eq}m {/eq} is the mass and {eq}g=9.8\;\rm m/s^2 {/eq} is the acceleration due to gravity.

According to Hooke's Law, the magnitude of the restoring force developed by a spring when it is stretched/compressed by a length, {eq}\Delta x {/eq}, is given by the equation:

{eq}F=K\Delta x {/eq}

Here,

• {eq}K {/eq} is the spring constant.

For the given system, we have:

{eq}\begin{align*} F_r&=W\\ \Rightarrow K\Delta x&=mg\\ \Rightarrow K&=\dfrac{mg}{\Delta x}\\ &=\dfrac{2\;\rm kg \times 9.8\;\rm m/s^2 }{4\;\rm cm}\\ &=\boxed{\mathbf{4.9\;\rm N/cm}} \end{align*} {/eq}