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.

Verified step by step guidance

1. Step 1: Understand the relationship between the spring constant (k) and the oscillation period (T). The formula for the period of a mass-spring system is T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant.
2. Step 2: Rearrange the formula to isolate k. Squaring both sides gives T² = (4π²m)/k. Rearranging for k gives k = (4π²m)/T².
3. Step 3: Use the data provided in the table. For each mass (m) and corresponding time (T), calculate T² and plot a graph of m versus T². The slope of the best-fit line will be proportional to 1/k.
4. Step 4: Determine the slope of the best-fit line from the graph. The slope (S) is equal to T²/m, and k can be calculated using the relationship k = 4π²/S.
5. Step 5: Use the slope obtained from the graph to calculate the spring constant (k). Ensure units are consistent throughout the calculation (e.g., mass in kilograms, time in seconds).

Key Concepts

Spring Constant (k)

The spring constant, denoted as 'k', is a measure of a spring's stiffness. It is defined by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from the equilibrium position, expressed as F = -kx. A higher spring constant indicates a stiffer spring that requires more force to stretch or compress it.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In the context of springs, when a mass is attached to a spring and displaced, it will oscillate back and forth in a sinusoidal manner. The period of oscillation depends on the mass and the spring constant, following the formula T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant.

Graphing and Best-Fit Line

Graphing is a crucial method for visualizing relationships between variables. In this experiment, plotting the square of the period (T²) against the mass (m) allows for the determination of the spring constant through the slope of the best-fit line. The relationship is linear, where the slope is related to the spring constant, enabling the calculation of 'k' from the graph.