Physic Labs

Problem 1

A mass mm slides from rest without friction down a plane of length LL inclined at angle α\alpha. Find its speed at the bottom and the travel time. Use an idealized laboratory-frame model: stated parameters remain constant during the process unless the motion itself makes a quantity vary. State a clear sign convention and use the defined symbols consistently. (a) Write the governing laws or constraints. (b) Derive the requested quantities symbolically and state their physical direction or meaning. (c) Check dimensions and examine a simple physical limit to validate the result. Finally, verify that every equation is dimensionally consistent and that the results satisfy the initial conditions. Since no numerical data are specified, the exact answers are expressions in the defined symbols.
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
v=2gLsin⁡α,t=2Lgsin⁡αΔU=−mgLsin⁡α,12mv2=mgLsin⁡α\boxed{v=\sqrt{2gL\sin\alpha},\quad t=\sqrt{\tfrac{2L}{g\sin\alpha}}}\qquad \boxed{\Delta U=-mgL\sin\alpha,\quad \tfrac12mv^2=mgL\sin\alpha}
problems.proof.analysis

The boxed results collect the derivation part by part while keeping notation and sign conventions consistent. Together they form the final answer to compare directly with each request.