Question

Your lab instructor has asked you to measure a spring constant using a dynamic method—letting it oscillate—rather than a static method of stretching it. You and your lab partner suspend the spring from a hook, hang different masses on the lower end, and start them oscillating. One of you uses a meter stick to measure the amplitude, the other uses a stopwatch to time $10$oscillations.

Your data are as follows:

Use the best-fit line of an appropriate graph to determine the spring constant.

Short answer

The spring constant$k=6.4\mathrm{N}/\mathrm{m}$

Step-by-step solution

Step :1 Introduction 

In simple harmonic motion, the period of an oscillator is given by

$T=2\pi \sqrt{\frac{m}{k}}$

Note that $T$does not depend on the amplitude but only on the mass $m$and the force constant $k$.

To determine the spring constant, we must utilise the best-fit line of an acceptable graph.

Step : 2 The oscillator's period 

Equation left is used to calculate the oscillator's period:

$T=2\pi \sqrt{\frac{m}{k}}$

Squaring both sides we obtain

$T^2=4{\pi }^2\frac{m}{k}$

$T^2=\left(\frac{4{\pi }^2}{k}\right)m$

A graph of $T^2$vs $m$ is a straight line starting at the origin with a slope equal to , according to this equation.

Step :3  New table 

Make a new table for $T^2$and the associated $m$, remembering that the times in the given table are for 10 oscillations, therefore we must divide them by$10$in the new table.

Step :4 Graph 

Using graphing software, plot $T^2$versus $m$and calculate the equation of the best-fit line:

$T^2=\left(6.1253{\mathrm{s}}^2/\mathrm{kg}\right)m$

Comparing Equations (1) and (2), we obtain:

$\frac{4{\pi }^2}{k}=6.1253{\mathrm{s}}^2/\mathrm{kg}$

Solve for $k$

localid="1650122061136" $\begin{array}{r}k=\frac{4{\pi }^2}{6.1253{\mathrm{s}}^2/\mathrm{kg}}\\ =6.4\mathrm{N}/\mathrm{m}\end{array}$