question_answer 52)

                  Mass \[{{m}_{1}}\] moves on a slope making an angle with the horizontal and is attached to mass \[{{m}_{2}}\]  by a string passing over a frictionless pulley as shown in Fig. The co-efficient of friction between nil and the sloping surface is \[\mu \] .                 Which of the following statements are true?                 (a) If \[{{m}_{2}}>{{m}_{1}}\sin \,\theta ,\] the body will move up the plane.                 (b) If \[{{m}_{2}}>{{m}_{1}}(\sin \,\theta +\mu \cos \theta ),\] the body will move up the plane.                 (c) If \[{{m}_{2}}<{{m}_{1}}(\sin \,\theta +\mu \cos \theta ),\] the body will move up the plane.                 (d) If \[{{m}_{2}}<{{m}_{1}}(\sin \,\theta -\mu \cos \theta ),\] the body will move down the plane.

Answer:

                  (b, d) Let body moves up the plane, then                 \[{{m}_{1}}a=T{{m}_{1}}\,\,g\text{ }\sin \theta \mu R\]                 \[{{m}_{1}}a=T{{m}_{1}}g\text{ }\sin \theta \mu \text{ }{{m}_{1}}g\text{ }\cos \theta \]            ?. (i)                 Also \[{{m}_{2}}a={{m}_{2}}gT\]                                            ? (ii)                 \[\therefore \] \[({{m}_{1}}+\,{{m}_{2}})\,a=\,{{m}_{2}}\,g\,-{{m}_{1}}\]\[(g\,\sin \,\theta +\,\mu \,g\,\cos \theta )\]                 \[=\,[{{m}_{2}}-{{m}_{1}}\,(\sin \theta \,+\,\mu \,\cos \theta )]g\]                 \[a=\,\frac{[{{m}_{2}}-{{m}_{1}}\,(\sin \,\theta \,+\mu \,\cos \theta )]\,g}{({{m}_{1}}+\,{{m}_{2}})}\]                 The body will move up if \[{{m}_{2}}>{{m}_{1}}\]                 \[(\sin \,\theta \,\,+\,\mu \,\cos \theta )\]                 Let body moves down the plane, then                 \[{{m}_{1}}a=\,{{m}_{1}}g\,\,\sin \,\theta -\,T-\mu R\]                 \[={{m}_{1}}\,g\,\sin \,\theta \,-T-\,\mu {{m}_{1}}\,g\,\cos \theta \]                      Also \[{{m}_{2}}\,a=\,T-\,{{m}_{2}}g\]                 \[\therefore \]\[({{m}_{1}}+\,{{m}_{2}})\,a\]\[=\,[{{m}_{1}}\,(\sin \,\theta \,-\mu \,\cos \theta )-{{m}_{2}}]g\]                 or \[a=\,\frac{[{{m}_{2}}\,-{{m}_{1}}]\,(\sin \,\theta +\,\mu \,\cos \theta )-\,{{m}_{2}}]\,g}{({{m}_{1}}+\,{{m}_{2}})}\]                  Thus, body will move down the plane, if                         \[{{m}_{2}}<{{m}_{1}}(\sin \,\theta -\,\mu \,\,\cos \theta )\]