Understanding the Formulas

The period of oscillation ($T$) is the time taken for one complete oscillation. If $N$ oscillations take a total time $t$, then the period is:

$$T = \frac{t}{N}$$

For a mass-spring system, the theoretical period is given by:

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

where $m$ is the mass and $k$ is the spring constant. Squaring both sides gives:

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

This shows that $T^2$ is directly proportional to $m$.

1(a))

• Setup: Suspend a single spring from a retort stand. Attach a mass hanger to the free end of the spring.
• Masses: Use masses $m = 150 \, \text{g}, 200 \, \text{g}, 250 \, \text{g}, 300 \, \text{g},$ and $350 \, \text{g}$ (convert to kg for calculations).
• Oscillation: For each mass, pull it down slightly and release it to start vertical oscillations.
• Timing: Use a stopwatch to measure the time $t_1$ for 10 complete oscillations. Start the stopwatch when the mass passes a reference point (e.g., equilibrium position) in one direction, and stop it after 10 complete oscillations when it passes the same reference point in the same direction.
• Calculations: For each mass, calculate the period $T_1 = \frac{t_1}{10}$ and then $T_1^2$.

1(b))

• Setup: Suspend two identical springs in parallel from the retort stand. Attach a mass hanger to the combined free ends of the springs.
• Masses: Use the same masses as in Step 2: $m = 150 \, \text{g}, 200 \, \text{g}, 250 \, \text{g}, 300 \, \text{g},$ and $350 \, \text{g}$.
• Oscillation: For each mass, pull it down slightly and release it to start vertical oscillations.
• Timing: Use a stopwatch to measure the time $t_2$ for 10 complete oscillations, similar to Step 2.
• Calculations: For each mass, calculate the period $T_2 = \frac{t_2}{10}$ and then $T_2^2$.

Organize all your measured and calculated data in a table. Ensure units are consistent (e.g., mass in kg, time in s).

Mass $m$ (kg)	Time for 10 oscillations $t_1$ (s) (Single Spring)	Period $T_1$ (s) (Single Spring)	$T_1^2$ (s$^2$) (Single Spring)	Time for 10 oscillations $t_2$ (s) (Parallel Springs)	Period $T_2$ (s) (Parallel Springs)	$T_2^2$ (s$^2$) (Parallel Springs)
0.150
0.200
0.250
0.300
0.350

Plot a graph with $m$ on the vertical axis and $T^2$ on the horizontal axis.

• Axes: Label the vertical axis as $m$ (kg) and the horizontal axis as $T^2$ (s$^2$).
• Data Points: Plot the $(T_1^2, m)$ points for the single spring system and $(T_2^2, m)$ points for the parallel spring system. You can plot both sets of data on the same graph, perhaps using different symbols or colors.
• Line of Best Fit: Draw a straight line of best fit through the plotted points for each system.
• Expected Outcome: According to the formula $m = \frac{k}{4\pi^2} T^2$, the graph should yield a straight line passing through the origin. The gradient of this line will be $\frac{k}{4\pi^2}$, from which the effective spring constant $k$ for each system can be determined.