Physic Labs

Problem 1

Two masses m1,m2m_1,m_2 are joined by a taut, massless inextensible string over a fixed frictionless pulley, so their accelerations have equal magnitude. Mass m1m_1 slides on a frictionless incline of angle α\alpha; m2m_2 hangs freely. Take gg constant and assume m2>m1sin⁡αm_2>m_1\sin\alpha, so m2m_2 descends. Ignore the sizes of the bodies and all drag. (a) Draw the force diagrams, choose positive directions, and write Newton’s equation for each mass. (b) Determine the acceleration aa and tension TT. (c) Check the near-equilibrium limit when the driving forces almost balance.
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a=gm2−m1sin⁡αm1+m2  (m2↓),T=m1m2g(1+sin⁡α)m1+m2;m2→m1sin⁡α: a→0, T→m2g\boxed{a=g\frac{m_2-m_1\sin\alpha}{m_1+m_2}\;(m_2\downarrow),\qquad T=\frac{m_1m_2g(1+\sin\alpha)}{m_1+m_2};\qquad m_2\to m_1\sin\alpha:\ a\to0,\ T\to m_2g}
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Collect the requested results in one final line. A positive acceleration corresponds to the chosen downward direction of m2m_2.