= Photogate measurement of acceleration down an incline
{title2=$a=dv/dt$}
A modern teaching experiment slows gravity-driven motion enough to resolve it with a <photogate>. PASCO scientific's https://cdn.pasco.com/product_document/Inclined-Plane-Dynamics-Set-Manual-ME-6966.pdf[Inclined Plane Dynamics Set, ME-6966, Instruction Manual 012-10874A] describes “Acceleration Down an Incline” on printed pages 15–16. Its apparatus is a GOcar on a PAStrack with an adjustable incline support, a bracket-mounted <photogate>, a cart-mounted picket fence, and an interface with recording software. For the interface method, the fence's $1\,\mathrm{cm}$ pattern is aligned to interrupt the beam. Recording starts before release from rest; the cart is caught at the bottom bumper. The procedure specifies three runs per <angle>, starting at $14^\circ$ and reducing by $2^\circ$, with the release <position> fixed. The readout is <velocity> against <time>; its slope estimates <acceleration>. Mean <accelerations> are plotted against $\sin\theta$, testing the ideal slope $g$. An added-<mass> comparison is also requested. These are protocol settings, not observed results.<Inclined Plane Dynamics Set, ME-6966, Instruction Manual 012-10874A by PASCO scientific>[Printed pp. 15–16, equipment, procedure steps 1–8, analysis steps 1–3 and question 2; interface timing note]{ref}
The cart undergoes <rolling without slipping> at its wheels, whereas the worked block undergoes <contact slip>. Even ideal wheel <moment of inertia> changes the cart prediction. For total cart <mass> $M$, wheel radii $r_i$ and axial <moments of inertia> $I_i$, rigid wheels with fixed axles and no losses give
$$
a=\frac{g\sin\theta}{1+\displaystyle\sum_i I_i/(Mr_i^2)}.
$$
To derive it, combine translational <kinetic energy> $Mv^2/2$ with wheel <rotational kinetic energy> $\sum_i I_i(v/r_i)^2/2$, then equate their rate of change to the release of <gravitational potential energy> $Mg\sin\theta\,v$. This is the multiwheel counterpart of <rolling acceleration with rotational inertia>. Neglecting wheel <moment of inertia> yields $a\simeq g\sin\theta$; treating a cart as a solid rolling cylinder would give a different correction.
Useful additional controls are repeated releases without a push, checking the fence alignment and spacing, and checking that the recorded <velocity> trace is straight over the fitted interval. A gently moving cart on a level track can expose losses or unintended tilt. These checks help distinguish an <angle> or timing error from neglected wheel <moment of inertia>, bearing <friction>, or <drag force>. The three-run spread measures repeatability but cannot by itself reveal a shared calibration error. The inspected protocol makes the gravity component measurable; it does not measure the sliding block's <coefficient of kinetic friction>.
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