Physic Labs

Problem 1

A particle of mass mm slides without friction from rest on a wedge of mass MM and incline angle α\alpha. The wedge rests on a smooth horizontal floor. Find the wedge's acceleration and the particle's acceleration relative to the wedge; gravity is gg downward. (a) State the assumptions and establish the governing equation for the requested quantity. (b) Solve it in terms of the given symbols, identifying the direction, sign, or physical nature of the result. (c) Check a simple limiting case and state when the model remains valid. Use SI units consistently, retain the symbols in the setup, and enforce all geometric constraints and initial conditions.
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
X=tọa độ ngang của neˆm,s=qua˜ng đường hạt đi xuoˆˊng dọc mặt nghieˆng,T=12(M+m)X˙2−mX˙s˙cos⁡α+12ms˙2,U=−mgssin⁡α,(M+m)X¨−ms¨cos⁡α=0,−X¨cos⁡α+s¨=gsin⁡α,s¨=(M+m)gsin⁡αM+msin⁡2α,X¨=mgsin⁡αcos⁡αM+msin⁡2α\boxed{X=\text{tọa độ ngang của nêm},\quad s=\text{quãng đường hạt đi xuống dọc mặt nghiêng},\quad T=\frac12(M+m)\dot X^2-m\dot X\dot s\cos\alpha+\frac12m\dot s^2,\quad U=-mgs\sin\alpha,\quad (M+m)\ddot X-m\ddot s\cos\alpha=0,\quad -\ddot X\cos\alpha+\ddot s=g\sin\alpha,\quad \ddot s=\frac{(M+m)g\sin\alpha}{M+m\sin^2\alpha},\quad \ddot X=\frac{mg\sin\alpha\cos\alpha}{M+m\sin^2\alpha}}
problems.proof.analysis

The results are listed in the order requested and use one consistent sign convention. Their dimensions, initial conditions, and model constraints must all agree.