Newton’s second law of motion lab report

1. Title and aim

Verification of Newton’s second law of motion: Force, mass, and acceleration on a linear track. Investigate the fundamental dynamic relation between applied net force F, system inertial mass M, and linear acceleration a on a precision air track or low-friction dynamic cart system, verifying Newton’s second law a = F / M at constant total mass M = 1.200 kg and extracting system mass from the inverse slope of the acceleration versus force graph.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Total system inertial mass of 1.2000 kg, differing by 0.000% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision Linear Dynamic Track and End Pulley: A rigid 2.0 m extruded aluminum guide track equipped with micro-adjusting leveling thumbscrews, metric scale, and an ultra-low-friction ball-bearing end pulley.
• Low-Friction Dynamic Cart with Mass Storage Posts: Precision wheeled cart (mass 800.0 g) with low-friction sapphire/ball-bearing axles, interrupter flag for optical gating, and brass mass mounting posts.
• Dual-Beam Infrared Photogate Timing Chronometer: Two optical photogates spaced at a fixed distance D = 0.500 m along the track, connected to a digital microsecond counter that computes instantaneous velocities and uniform acceleration.
• Precision Slotted Brass Weight Set and Tow Cord: Calibrated slotted masses (total 400.0 g) transferred between cart and lightweight mass hanger via an inextensible monofilament tow cord.
• Constant total system mass M_total = M_cart + m_hanger = 1.200 kg maintained by transferring slotted masses.
• Linear track leveled with friction compensation; dual photogate velocity timer spacing D = 0.500 m.
• Reference total inertial mass M_total = 1.200 kg; local gravity g = 9.80665 m s⁻².

4. Theory and graph choice

Sir Isaac Newton's Philosophiae Naturalis Principia Mathematica, published in 1687, established the mathematical laws governing the motion of material bodies under the action of forces. Newton's Second Law of Motion forms the central dynamic principle of classical mechanics: the net external force acting on a body is directly proportional to, and acts in the same direction as, the time rate of change of its linear momentum: F_net = d(p)/dt = d(m v)/dt. For a closed physical system of invariant mass m, this differential relation reduces to the celebrated algebraic formula F_net = m · a, establishing that the resulting acceleration a is directly proportional to the applied net force and inversely proportional to the body's inertial mass: a = F_net / m. Despite its conceptual simplicity, experimentally confirming Newton's Second Law requires meticulous experimental design to isolate the variables. In a typical laboratory setup consisting of a wheeled dynamic cart or air track glider connected via a light monofilament cord over a pulley to a hanging weight hanger, the accelerating force is supplied by the gravitational pull on the hanging mass: F = m_hanger · g. Crucially, this hanging mass accelerates alongside the cart, meaning that the total inertial mass set into motion is the combined mass of the entire system: M_total = M_cart + m_hanger. If an experimenter merely adds slotted weights to the hanger from an external supply, both the accelerating force and the total system mass increase simultaneously, masking the pure proportionality between force and acceleration. To maintain a strictly constant system mass M_total, slotted weights must be transferred exclusively between the cart body and the hanging hanger. Under this rigorous constraint of constant total mass, plotting acceleration a on the vertical axis against applied force F on the horizontal axis yields a straight line passing through the origin. The slope of this line S equals the reciprocal of the system mass, S = 1 / M_total, providing an absolute dynamic determination of the system's inertial mass M_total = 1 / S through ordinary least-squares regression.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units kg⁻¹.

5. Method

1. Inspect the aluminum track with a precision spirit level. Place the unweighted cart on the track and give it a gentle nudge. Adjust the leveling screw at the non-pulley end until the cart coasts smoothly along the entire track at constant velocity without speeding up or slowing down. This small angle of inclination exactly balances mechanical rolling friction.
2. Determine the tare mass of the dynamic cart (800.0 g) and the empty weight hanger (50.0 g) on a digital laboratory balance. Add slotted brass weights totaling 350.0 g to the cart's storage posts, bringing the combined system mass to exactly M_total = 800.0 + 50.0 + 350.0 = 1200.0 g (1.200 kg). Record this invariant reference mass.
3. Mount the first photogate at the 0.500 m mark and the second photogate at the 1.000 m mark, setting the inter-gate separation distance to D = 0.500 m. Measure the width w of the cart's interrupter flag using Vernier calipers (e.g., w = 0.0200 m). Set the digital timer to dual-gate acceleration mode.
4. Transfer slotted weights from the cart to the hanging hanger to achieve the first accelerating force F = 0.30 N (hanging mass m_h ≈ 30.6 g). Pass the monofilament cord over the low-friction end pulley, ensuring the cord runs parallel to the track surface.
5. Pull the cart back to the starting release latch (at least 0.20 m before the first photogate to allow smooth acceleration). Release the cart from rest without imparting any initial impulse. Catch the cart gently after it clears the second photogate before it strikes the end bumper. Record the acceleration a displayed on the digital timer.
6. Repeat the release sequence for applied forces of 0.60 N, 0.90 N, 1.20 N, and 1.50 N by transferring additional slotted masses from the cart to the hanger. Construct a graph of acceleration a versus applied force F. Perform an ordinary least-squares linear fit to evaluate the slope S, and compute the experimental system mass as M = 1 / S.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0000%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 1.6178e-32; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 7.7407e-17. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 7.7019e-17.

Propagating only the relevant fit coefficient gives a fit-only standard error of 1.1147e-16 kg. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Photogate transit timing resolution (±0.0001 s), flag width measurement uncertainty (±0.05 mm), residual rolling friction and air resistance along the track, cord stretching and pulley bearing drag, and the small unmodelled rotational inertia of the pulley wheel (equivalent mass ~0.005 kg) contribute uncertainty. The OLS inverse-slope standard error reflects random timing scatter across force settings, but does not include shared static scale mass calibration errors or slight track inclination zero offsets.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Verify that the monofilament tow cord runs exactly parallel to the track surface to ensure all cord tension acts in the direction of linear motion without introducing vertical lifting components.
• Maintain constant total system mass strictly by transferring masses between the cart and the hanger, never adding external weights from the bench.
• Calibrate track friction compensation before recording data so that the cart glides at constant speed when unweighted and unaccelerated.
• Arrest the cart with an energy-absorbing end cushion before it strikes the pulley support to prevent mechanical shock or derailment.

This virtual experiment assumes a massless, frictionless pulley, a perfectly inextensible cord of negligible mass, and exact friction compensation along the linear guide track. It neglects air resistance, cord elasticity, pulley bearing drag, and rotational inertia of the wheels and pulley. Generated readings reflect ideal Newtonian dynamics; physical experiments will exhibit small random timing scatter from photogate clock jitter. Theoretical acceleration: a = F / M_total. Linearized relation: a = (1 / M_total) · F. The ordinary least-squares regression extracts the slope S = 1 / M_total, from which total system mass is recovered as M_total = 1 / S. Reference total system mass M_total = 1.200 kg.

12. Conclusion

The worked example illustrates how to obtain Total system inertial mass from measurements, graph variables and a fitted coefficient. The estimate is 1.2000 kg. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.