Lab: Linearizing Pendulum Period Data

The period of a pendulum follows T = 2 π √(L/g). Plot T against L directly and you get a curve, because T depends on the square root of L, not L itself. Square both sides and you get T² = (4 π² / g) L. That equation says T² is directly proportional to L, with no square root in sight. So if you plot T² on the vertical axis and L on the horizontal axis, the data falls on a straight line through the origin. This trick, squaring one variable to kill off a square root, is called linearizing data, and it turns a curve you cannot easily measure into a line you can.

Once the graph is a line, its slope is 4 π² / g, straight from the equation. Measure the slope from your best-fit line, then solve for g by dividing 4 π² by that slope. If the slope reads 4.0, g comes out to 4 π² divided by 4.0, which is π², about 9.87 m/s², close to the accepted 9.8 m/s^2. Any gap between your value and 9.8 points to real experimental error: a mistimed swing, a stretched string, or a release angle too large for the small-angle approximation this whole formula depends on.

This lab ties directly to the big idea of the unit: period depends only on length and g, never on amplitude or mass, which is the signature of simple harmonic motion. It also connects to measurement technique. Timing twenty swings instead of one and dividing by twenty spreads a fixed reaction-time error over a much bigger total, shrinking the uncertainty in each period value and tightening the line you fit.