Calculate the Period and its Square

For each measurement, you record the time for 10 complete oscillations. To find the period ($T$) for a single oscillation, divide the total time by 10. Then, square the period to get $T^2$.

For the single spring system (Fig. 1(a)):

$$T = \frac{t}{10}$$ $$T^2 = \left(\frac{t}{10}\right)^2$$

For the two identical springs system (Fig. 1(b)):

$$T_0 = \frac{t_0}{10}$$ $$T_0^2 = \left(\frac{t_0}{10}\right)^2$$

Organize your recorded times and calculated periods and squared periods into two separate tables, one for each spring setup.

For the single spring system (Fig. 1(a)):

Mass $m$ (g)	Time $t$ for 10 oscillations (s)	Period $T$ (s)	$T^2$ (s$^2$)
150	(record value)	(calculate)	(calculate)
200	(record value)	(calculate)	(calculate)
250	(record value)	(calculate)	(calculate)
300	(record value)	(calculate)	(calculate)
350	(record value)	(calculate)	(calculate)

For the two identical springs system (Fig. 1(b)):

Mass $m$ (g)	Time $t_0$ for 10 oscillations (s)	Period $T_0$ (s)	$T_0^2$ (s$^2$)
150	(record value)	(calculate)	(calculate)
200	(record value)	(calculate)	(calculate)
250	(record value)	(calculate)	(calculate)
300	(record value)	(calculate)	(calculate)
350	(record value)	(calculate)	(calculate)

Plot a graph with $m$ on the vertical axis and $T^2$ on the horizontal axis. You will plot two sets of data points, one for the single spring system and one for the two-spring system.

• Horizontal Axis ($x$-axis): Label this axis "$T^2$ (s$^2$)" and choose an appropriate scale for the range of your $T^2$ values.
• Vertical Axis ($y$-axis): Label this axis "$m$ (g)" and choose an appropriate scale for the range of your mass values (150 g to 350 g).
• Plot the data points from your tables.
• Draw a line of best fit through the points for each set of data. You should observe a linear relationship, as $m$ is directly proportional to $T^2$ for a simple harmonic oscillator.