Simple Pendulum & Spring Constant Experiments Simple Pendulum & Spring Constant Experiments In school labs, a pendulum and a spring–mass system are the two friendliest oscillators. Each swings or bounces back and forth under a linear restoring force, so their time period is predictable and can be used to determine a constant of nature or of the system. With a simple pendulum, the time period depends only on the effective length and local gravity when the swing is small, so you can estimate g accurately by timing several oscillations and plotting T² against L. With a vertical spring, the oscillation period depends on mass m and spring stiffness k; so by measuring T for a known mass, you can compute k. Both are clean, high-score experiments if you follow a disciplined procedure: ensure small amplitude, measure lengths from the right reference, time many oscillations to reduce reaction error, and analyze data graphically to suppress random fluctuations. This lesson builds the intuition first—why a longer pendulum is slower and a stiffer spring is faster—then moves to equations, careful measurements, error handling, and NEET-style numericals. Along the way, we flag classic traps: L includes the bob’s radius and hook, T is for a complete oscillation (to and fro), and the spring’s own mass slightly modifies the effective oscillating mass. By the end, you should be able to plan the experiment, predict outcomes, justify approximations (like “small-angle”), and extract precise values with uncertainties from your data. remember Everyday feel: Longer swings (like a child’s long swing) move slower; a short swing is quicker. A soft spring bounces slowly, a stiff spring bounces quickly. That’s the heart of the two experiments. Simple pendulum A heavy, small bob suspended by a light, inextensible string from a fixed, frictionless support, oscillating with small angular amplitude. Distance from the point of suspension (knife-edge/pivot) to the center of the bob. Include hook and bob radius. Effective length (L) Time for one complete oscillation (mean position → extreme → back to the same mean position in the same direction). Time period (T) Small-angle approximation For small angular amplitude ( 10 to 15 ), (in radians), making the pendulum motion simple harmonic. Spring constant (k) Measure of stiffness of a spring; restoring force F = -k x . Larger k means stiffer spring. If the spring has significant mass m s , the oscillating mass is approximately m + m s/3 (for a uniform light spring). Effective mass of spring–mass system Gradual loss of amplitude due to air resistance and internal friction; ideally kept small in these experiments. Damping Valid for small angular amplitude and a light, inextensible string with a frictionless pivot. Pendulum time period This period calculation is valid only for small angular displacements, requiring the approximation that sine of theta equals theta. Spring–mass time period For a light spring with negligible mass; include m s/3 if spring mass m s is non-negligible. The period of oscillation for a mass attached to a spring depends only on the mass and the spring's stiffness. Small angular displacement: (in radians). Light, inextensible string; frictionless pivot; negligible air drag. Bob considered as a point mass at distance L from pivot. T = 2 L g , g = 4 2 L T 2 Time period of a simple pendulum and relation for g Use measured L and T to compute local g. Plotting T² vs L gives slope = 4π²/g. Clamp stand on a stable table; tie a light string with a small heavy bob. Measure L from the knife-edge/pivot to center of bob (include hook and radius). Displace by a small angle (≤ 10–15°), release gently without push. Use a stopwatch: record time for N (e.g., 20 or 30) oscillations thrice; compute mean. Compute T = (mean time)/N for each L; repeat for 5–7 different L values. Plot T² on y-axis vs L on x-axis; draw best-fit straight line through the origin. Calculate slope m = Δ(T²)/ΔL; then g = 4π²/m. Pendulum: Quick procedure control T² dependent derived slope m = 4π²/g Origin (L=0, T²=0) m L General point (slope m = 4π²/g) s² T² (s²) Straight line through origin; slope decreases if g is larger. custom T² vs L for a simple pendulum is linear; slope m = 4π²/g. Length L (m) 2D PLOT T² vs length (simple pendulum) Tsq = (4 pi 2/g) L Tsq 0.50 20 28.6 1.430 2.045 0.70 20 33.6 1.680 2.822 0.90 20 37.8 1.890 3.572 1.10 20 41.8 2.090 4.368 Sample pendulum data (illustrative) Trial L (m) N (oscillations) Time t (s) T = t/N (s) T² (s²) T is for one complete oscillation. A swing from mean → extreme → back to mean in the same direction. Counting only mean → extreme is half period. neet-alert Measure L from the pivot to the center of the bob. Add the bob’s radius and hook length to the naked string length. This fixes the most common systematic error. tip easy Compute T = t/N, then use g = 4 2 L / T 2 . L = 0.80 ; m N = 20 t = 36.0 ; s m/ s 2 A pendulum of effective length L = 0.80 m makes 20 oscillations in 36.0 s. Estimate g. Why the graph method is preferred: Any single reading of T may carry random timing error due to human reaction. But if you plot many L–T² pairs and draw a best-fit line, the slope averages out random scatter. A proper line must pass through the origin because T → 0 as L → 0 in the small-angle model. If your line intercept is non-zero beyond uncertainty, suspect a length offset error or a large-angle deviation. Always state uncertainties with your final g, especially when asked in practical exams. SWING: Small angle, Wire light, Inextensible string, No push at release, Gauge length to the center of bob. Now the vertical spring: Attach a hanger and add known masses. The static extension helps you estimate k via k = mg , but oscillations give a more robust value through time period. Pull down slightly (small amplitude) and release; time N oscillations. For a light spring, T = 2 m k . If the spring’s mass m s is not negligible, replace m with m + m s/3 . The oscillation method reduces reading error because T is a derived quantity from a long timing interval. T = 2 m k ( or 2 m + m s/3 k if spring mass is not negligible ) Time period of a spring–mass system Hooke’s law holds: F = -k x in the linear regime. Small oscillations about equilibrium; horizontal motion negligible (vertical setup). Spring is light; if not, include effective mass correction. Spring: Quick procedure Suspend a vertical spring from a rigid support; attach a light mass hanger. Add a known mass m; note the equilibrium mark for reference. Pull down by a small displacement (1–2 cm), release gently. Record time for N oscillations thrice; compute mean T = t/N. Compute k from k = 4 2 m T 2 (or replace m by m + m s/3 if required). Repeat for different m and verify T ∝ √m. Compute T = t/N , then k = 4 2 m T 2 . medium m = 0.200 ; kg N = 25 t = 22.5 ; s N/m A 200 g mass oscillates vertically on a light spring. Time for 25 oscillations is 22.5 s. Find the spring constant k. tip If the spring has noticeable mass m s , use T = 2 m + m s/3 k . This reduces the systematic overestimation of k. Heavier bob makes a pendulum swing slower, so T increases with mass. For small-angle oscillations, T = 2 L/g is independent of bob mass. Mass affects damping slightly, not the ideal T. Length L is just the string length from the knot. Effective length is measured from the pivot to the center of the bob. Add hook length and the bob’s radius to the naked string. Stokes method: measure terminal velocity of a sphere in viscous fluid to compute η. = 2 r 2 ( - ) g 9 v Viscosity by terminal velocity (Stokes’ law) Laminar flow (low Reynolds number). Sphere of radius r; liquid density , sphere density . Large container to avoid wall effects. At null deflection in potentiometer: E 1/E 2 = l 1/l 2 . E 1 E 2 = l 1 l 2 EMF comparison using potentiometer (E symbols) Uniform potentiometer wire with constant current. Null method: no current drawn from the cell at balance. Same polarity connections. Same principle with ε notation: 1/ 2 = l 1/l 2 . EMF comparison using potentiometer (ε symbols) 1 2 = l 1 l 2 Uniform wire, steady current. Null deflection: no current through test cell. Positive terminals common. Although viscosity and potentiometer belong to other experiments in the same unit, their high-yield formulas often appear in the same practicals section. Remember: Stokes’ method requires laminar flow (small balls in viscous fluid), and potentiometer is a null method (no current drawn at the balance point), making it more accurate than a moving-coil voltmeter for EMF comparison. Using a T²–L graph, the best-fit line has slope m = 0.405 s²/m. Determine g. Slope m = 0.405 ; s 2/m m/ s 2 hard For T²–L graph, m = 4 2/g g = 4 2/m . Trap-based data sanity check neet-alert Sanity check slopes: For Earth, m = 4π²/g ≈ 4.03 s²/m. If your graph slope is far from 3–5 s²/m, suspect a length zero error or large-angle swings. Uncertainties and error budget for pendulum: The dominant random error is human reaction in starting/stopping the stopwatch. Reduce it by timing many oscillations and using the mean of repeated trials. Systematic errors include (i) length offset because you measured to the bottom of the bob instead of its center, (ii) large-angle swing violating the SHM approximation (T becomes slightly larger), and (iii) air drag and pivot friction, both increasing T marginally. State each suspected source and whether it makes g over- or under-estimated: e.g., if T is overestimated, g = 4π²L/T² is underestimated. Keep amplitude ≤ 10°; beyond ~15°, the small-angle approximation breaks down. Release without a push; hold the bob gently and let go. Measure L to the center of the bob; recheck after each change of L. Ensure the plane of oscillation is fixed and away from the stand to avoid grazing. Use a sharp knife-edge support to minimize friction; align the string vertically at rest. Avoid cross-breezes; close windows or do the experiment in a calm corner. Use a low least-count stopwatch; time ≥ 20 oscillations. Precautions that actually matter Data strategy for spring k: If you measure T for several different masses m, then plotting T² vs m gives a straight line with slope 4 2/k . This approach averages errors and lets you diagnose if a single measurement was off. A non-zero intercept might indicate spring mass or friction effects; if the intercept is positive, consider replacing m with m + m s/3 to see if the line hits near the origin. Illustrative spring oscillation data Trial m (kg) t (s) T (s) T² (s²) 0.10 25 15.8 0.632 0.399 0.15 25 19.4 0.776 0.602 0.20 25 22.5 0.900 0.810 0.25 25 25.1 1.004 1.008 A light spring (mass negligible) is tested with masses 0.10, 0.15, 0.20 kg. Measured T are 0.630 s, 0.775 s, 0.900 s respectively. Estimate k by the best two-point slope method on T² vs m. For T²–m graph, slope S = Δ(T²)/Δm = 4π²/k ⇒ k = 4π²/S. medium N/m T(0.10) = 0.630 ; s T(0.20) = 0.900 ; s Boundary checks: For the pendulum, as L → 0, T → 0 in the ideal formula; physically, below some L the bob is no longer a point mass and pivot friction dominates. As L → ∞, T → ∞, but air drag and finite lab height cap what you can test. For the spring, as k → ∞ (very stiff), T → 0; in practice, measurement is limited by reaction time. As m → 0, T → 0 ideally, but the hanger mass and spring mass set a lower bound. remember Units sanity: If L is in metres and T in seconds, g comes out in m/s² automatically. If you accidentally use cm for L, convert to m before computing g. Least count and timing: If your stopwatch least count is 0.1 s and you time only one oscillation of ~1.5 s, the relative error is about 7%. Time 20 oscillations (~30 s) to shrink it to ~0.3%. Similarly, measuring several lengths and spacing them across the graph range improves slope accuracy. Worked measurement example (pendulum with multiple lengths): Suppose you measure L = 0.50, 0.70, 0.90, 1.10 m and get T = 1.43, 1.68, 1.89, 2.09 s (from 20 swings). Compute T², plot T² vs L, and use two well-separated points to estimate slope. For example, using (L, T²) = (0.50, 2.045) and (1.10, 4.368), slope m ≈ (4.368 − 2.045)/(1.10 − 0.50) ≈ 3.88 s²/m. Then g = 4π²/m ≈ 39.478 / 3.88 ≈ 10.2 m/s². The slight overestimate suggests either under-timing (pressed stop late) or length slightly short (forgot hook or radius). A full least-squares fit would refine this toward ~9.8 m/s². Common practical questions answered Why small angle? It linearizes sinθ so the motion becomes SHM with a constant period. Does mass matter for pendulum T? No; mass cancels in the SHM equation. What if amplitude is large? Period increases slightly; data curve bends above the ideal line. Why graph? To average random errors, reveal trends, and extract slope robustly. Why N oscillations? To suppress stopwatch reaction error. Damping and corrections: Light damping does not change T much over a few tens of oscillations; amplitude decays but period remains nearly constant for small angles. If significant, record the time window near the middle swings to avoid start/stop transients. For spring oscillations, friction at the top hook and air drag can increase T slightly; if you see systematic drift, re-lubricate the hook, realign the spring, or reduce amplitude. Static and dynamic measurements of k should agree within uncertainty; differences hint at spring mass or friction effects. Static vs dynamic k This comparison is valid for a spring vertically attached to a mass 'm' undergoing Simple Harmonic Motion (SHM), where the static constant i Reporting answers: Always state the value with appropriate significant figures and units, e.g., g = (9.78 ± 0.10) m/s². If a graph is involved, include the best-fit line, the slope calculation, and a brief comment on uncertainties (e.g., length least count 1 mm, stopwatch 0.1 s). In viva, be ready to justify small-angle and light-string assumptions. Worked example with spring mass correction: Suppose the spring mass m s = 60 g and you hang m = 200 g. Time for 25 oscillations is 22.5 s, so T = 0.900 s. If you ignore spring mass, k ≈ 9.75 N/m as computed earlier. Including correction, effective mass is m eff = m + m s /3 = 0.200 + 0.060/3 = 0.220 kg. Then k = 4 2 m eff /T 2 = 39.478 0.220/0.810 ≈ 10.73$ N/m. The corrected k is higher, as expected, because extra mass makes the system slower; to match the observed T, the spring must be stiffer. Do not confuse time for N swings with N×T where T is unknown. Always compute T = (measured total time)/N. Using a guessed T and multiplying up is circular and wrong. neet-alert Pivot and plane: Ensure the bob oscillates in a single vertical plane. Twisting motion (torsion) introduces a different restoring torque and corrupts the time period. If the knot causes lateral friction, re-tie or use a smooth hook on a knife-edge to approximate a frictionless pivot. Small lateral slips can make timing inconsistent across trials. Choosing L values: Spread L across a wide range (e.g., 0.5 m to 1.2 m) for a well-scaled T²–L plot. Avoid values too close together; closely spaced points magnify coordinate reading errors when finding slope. After plotting, use the largest clean triangle to compute slope from well-separated points on the best-fit line. Environmental effects on g: The local value of g varies slightly with latitude and altitude. Your lab value may be 9.76–9.83 m/s² and still be reasonable. If your result is consistently low, check for large-angle swings or length underestimation; if high, check for timing underestimation or length overestimation. Dimensional cross-checks: For the pendulum, T = 2 L/g has dimension [T], since [L]/[L T⁻²] = [T²]. For the spring, T = 2 m/k has [M]/[M T⁻²] = [T²]. Dimensional checks are a quick way to catch algebra slips before plotting. Alternative linearizations: Instead of T² vs L for the pendulum, some labs plot T vs √L. Both give straight lines, but T²–L avoids plotting radicals and keeps errors additive. For springs, T² vs m is the best linear plot; T vs √m is also linear but harder to construct precisely. Finite amplitude correction (advanced): For larger amplitudes, the pendulum period is T( 0) T 0 [1 + 0 2 16 + ] for 0 in radians. This shows why you must keep amplitude small; a 20° swing (0.349 rad) increases T by about 0.76%. Choice of bob: A dense, small spherical bob reduces air drag (smaller area) and makes it easier to identify the center for measuring length. A large, light bob increases air resistance and damping, biasing T upward. Always check that the knot or hook does not shift during the run; re-verify L after reties. Stopwatch technique: One student can call out counts while another handles the stopwatch. Start counting from zero at a chosen mean position; press start when the bob crosses that mark in the agreed direction, count 1 at the next crossing in the same direction, and stop at N. Keeping the direction consistent avoids half-period mistakes. Viva-style justifications: If asked why the line must pass through the origin, say: because T² ∝ L in the SHM model and for L = 0 there is no restoring torque arm, hence T = 0. If asked why mass does not affect T, show how m cancels in m x + (mg/L) x = 0 for the pendulum and appears only with k for springs, not for pendulums. Safety and setup: Clamp stands can tip if pulled too far. Keep the stand’s base loaded and ensure the bob’s arc is clear of the support. For spring work, ensure the mass hanger is secure; never load beyond the elastic limit—Hooke’s law must hold, else your k estimate fails. Calibration thought: If your lab g is known (e.g., 9.80 m/s²), you can invert the pendulum method to calibrate a meter scale’s zero shift: the observed intercept on T²–L indicates an effective length offset. Similarly, with a known accurate mass, dynamic k measurements can check the stopwatch or reveal frictional losses. Comparing static and dynamic k (worked check): Using earlier spring data, suppose static extension for m = 0.20 kg is Δℓ = 0.020 m. Then k static = mg/ = (0.20 9.8)/0.020 = 98/2 = 49 N/m. This is far from the dynamic ~9.8 N/m, so the assumed Δℓ is unrealistic for a ‘soft’ spring. Always sanity-check: a spring with k ≈ 10 N/m would extend ≈ 0.20 m under 0.20 kg, not 0.02 m. Worked uncertainty sketch: If your stopwatch uncertainty per run is ±0.1 s and you time 20 oscillations with T ≈ 1.8 s (total ≈ 36 s), the fractional timing uncertainty is ~0.1/36 ≈ 0.28%. Propagation gives g/g L/L + 2 , T/T . With L measured to ±1 mm around 0.80 m, δL/L ≈ 0.125%. Then δg/g ≈ 0.125% + 2×0.28% ≈ 0.69%. A note on zero errors: If the meter scale has a zero error, correct L before plotting. A constant positive offset in L shifts all points right; the best-fit line may not pass through the origin if uncorrected. Similarly, an unaccounted stopwatch lag adds a constant to times, bowing the plot subtly; using long timing intervals mitigates this. Key terms recap period Time period (T) Time for one complete oscillation (to and fro). Distance from pivot to center of bob including hook and radius. Effective length (L) Local gravitational acceleration near Earth’s surface. Acceleration due to gravity (g) Stiffness of a spring in Hooke’s law. Spring constant (k) Small-angle approximation Use (radians) for 15 . Damping Loss of amplitude due to resistive forces; keep small for accurate T.