Module 05: Waves and Optics ยท quiz worth 40 pts
This lesson leans on Newton's second law (a = F/m) without ever naming it, because it assumes you already had that from an earlier mechanics course. If your course skipped straight to Waves and Optics, here's the one line you need.
Acceleration is how fast something speeds up or changes direction. A bigger push (force) gives more acceleration. A bigger mass resists that push more, so it gives less acceleration for the same force. That's the whole relationship: more force speeds things up faster, more mass slows that speeding-up down.
A heavier bob feels a bigger pull from gravity (more force), but it's also harder to accelerate (more mass). Those two changes are proportional and cancel exactly, so the bob's acceleration, and the whole rhythm of the swing, ends up identical to a lighter bob on the same length string. Nothing about the swing changes, even though the force on it did.
Picture pushing a shopping cart versus pushing a loaded one down the same ramp. Gravity pulls the loaded cart down the ramp harder, but it's also heavier and harder to speed up. On a frictionless ramp, both carts would actually reach the bottom at the same time, for the exact same reason a heavy pendulum bob keeps the same period as a light one.
Physics Classroom: Pendulum Motion โ Optional refresher on period and restoring motion. Use it to check the idea, not in place of collecting the assigned lab data.
Got it now? Scroll back up and re-read the section โ it should land differently.
Simple harmonic motion is motion that repeats itself over and over on a fixed rhythm, like a pendulum swinging or a mass bobbing on a spring. A pendulum's bob is the hanging mass. Its length, L, is measured from the point of attachment to the center of the bob, not its bottom edge. Its period, T, is the time in seconds for one complete swing: over and back to the starting point. Its amplitude is how far it is pulled from rest before release.
This one is worth reasoning through rather than memorizing, because the same pattern reappears in free fall. Gravity pulls a heavier bob with more force, so a heavier bob has a bigger restoring force pulling it back toward the bottom of the swing. But a heavier bob also has more mass to accelerate, which resists that force more. Newton's second law says acceleration a = F/m. Both the force F and the mass m scale up together for a heavier bob, so the acceleration, and therefore the whole motion and its timing, comes out the same. The bob's weight cancels out of its own swing.
This is the one idea in the whole lesson worth being able to say out loud, since it's exactly what the lab asks you to explain.
Heavier bob โ bigger restoring force (gravity pulls harder). Heavier bob โ bigger inertia (harder to accelerate). Both scale with mass the same way, so mass cancels out of the final motion.
"A heavier bob has more restoring force, but also more mass to accelerate, so its acceleration and period stay the same as a lighter bob's."
Got it now? Scroll back up and re-read the section โ it should land differently.
A wider swing has to cover more distance, but it also starts with a bigger restoring force pulling it back. For swings smaller than about 10ยฐ from vertical, those effects cancel closely enough that period stays essentially constant. Past 10ยฐ, the period begins to lengthen compared with the small-angle model. The opening activity compares two modest pull-back distances. The assigned virtual-lab option separately requires a 45ยฐโ60ยฐ release, so follow that procedure and expect the small-angle formula to be a pattern check rather than an exact prediction for those trials.
T in seconds, L in meters (measured to the center of the bob), g = 9.8 m/sยฒ near Earth's surface. This is exactly the relationship your own length-vs-period data should show; the source lab asks you to find it by graphing rather than stating it up front.
This formula never appears in the source lesson, only the pattern it produces. It's here so you can check your own lab data against something concrete.
T = 2ฯโ(L/g). Longer L under the square root means a bigger number means a longer period. g is fixed near Earth's surface, so length is the only thing you're actually changing in this lab.
The lab wants your timed data. The virtual option also uses a 45ยฐโ60ยฐ release, outside the formula's small-angle condition, so compare the trend and expect some difference instead of replacing measurements with predictions.
Optional interactive: PhET Pendulum Lab lets you isolate length, mass, and gravity. It is a practice model, not a substitute for the procedure and data table assigned in this lesson.
Got it now? Scroll back up and re-read the section โ it should land differently.
Predict the period, then compare it to a measured value.
T โ 1.27 s, matching the roughly 1.267 s that ten real timed swings (12.67 s รท 10) should give.
Predict the period for the longer pendulum.
T โ 1.55 s, matching the roughly 1.553 s a real 60 cm pendulum gives. A longer pendulum swings slower, never faster.
Your reaction time on the stopwatch (roughly 0.2โ0.3 s of error) is the same whether you time one swing or ten. Spreading that fixed error across ten swings, then dividing by 10, shrinks its effect on the period you report by a factor of ten.
The restoring force acts as if all the bob's mass were at one point, its center. Measuring to the bottom edge of a stack of washers overstates L and will make your predicted period run long.
Adding washers to an existing string changes where the center of mass sits, so the effective length changes too. Re-measure; do not assume the string length alone still equals L.
Release without a push. The virtual option calls for 45ยฐโ60ยฐ; use that range consistently. If you compare those results with T = 2ฯโ(L/g), note that this formula is a small-angle model and can predict a slightly shorter period.
The source lesson calls this out directly: length is to the center of the bob, not its lower edge. A stack of washers has its center partway up the stack, not at the bottom washer.
It's a natural guess and it's wrong. More weight means more restoring force, but also more inertia resisting that force; they cancel, and period stays the same.
Stacking on extra washers instead of rebuilding the pendulum shifts the true length. Always re-measure to the new center of mass.
T versus L on its own is curved, not a straight line, because the relationship is a square root. Graphing Tยฒ against L straightens it into a line, which is what actually lets you pull out g from the slope.
Work each question with the formula before checking the answer. These mirror what your own lab data should show.
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