4.1.1 Newton's Second Law & Spring Forces

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Generalizations: 4.1 Scalar-Vector Multiplication

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Newton's Second Law

Let's return to our our Hooke's Law example where we measured how mass moved when hung on the end of a spring. We can calculate the spring forces using Newton's Second Law.

Solution. We have our mass vector $\vec{m}$ . Through Newton's Second Law, we know:

\underset{force (N)}{f} = \underset{mass (kg)}{m} \cdot \underset{acceleration (\frac{m}{s^{2}})}{a}

The springs are being pulled down by gravity, which accelerates at $9.8 \frac{m}{s^{2}}$ . We can plug our mass vector $\vec{m}$ and the acceleration of gravity into the force equation to get our vector of spring forces.

\vec{f} = \vec{m} \cdot 9.8

Multiplying $\vec{m} \cdot 9.8$ is 4.1 Scalar-Vector Multiplication because $9.8$ is a scalar value. We now have a way of calculating the forces!

\vec{f} = [\begin{matrix} m_{1} \\ m_{2} \\ ⋮ \\ m_{n} \end{matrix}] \cdot 9.8 = [\begin{matrix} m_{1} \cdot 9.8 \\ m_{2} \cdot 9.8 \\ ⋮ \\ m_{n} \cdot 9.8 \end{matrix}]