question_answer 53)

                  In Fig. a body A of mass \[m\] slides on plane inclined at angle \[{{\theta }_{1}}\] to the horizontal and \[{{\mu }_{1}}\] is the coefficient of friction between A and the plane. A is connected by a light string passing over a frictionless pulley to another body B, also of mass \[m,\] sliding on a frictionless plane inclined at angle \[{{\theta }_{2}}\] to the horizontal. Which of the following statement are true?                 (a) A will never move up the plane.                 (b) A will just start moving up the plane when                 \[\mu =\,\frac{\sin \,{{\theta }_{2}}\,-\,\sin \,{{\theta }_{1}}}{\cos {{\theta }_{1}}}\]                 (c) For A to move up the plane, \[{{\theta }_{2}}\] must always be greater than \[{{\theta }_{1}}\].                 (d) B will always slide down with constant speed.

Answer:

                  (b, c) For motion of A                 \[ma=mg\text{ }\sin {{\theta }_{1}}-T-\mu \,mg\,\cos \,{{\theta }_{1}}\]           ? (i)                                 For motion of B                 \[ma=Tmg\text{ }\sin \,{{\theta }_{2}}\]                        ...(ii)                 Adding                 \[2\,ma\,\,=\,\,mg\,\,\sin {{\theta }_{1}}-\,\mu \,\,mg\,\,\cos \,\,{{\theta }_{1}}-\,mg\,\,\sin \,{{\theta }_{2}}\]                 \[2\,ma\,\,=\,\,\,mg\,\,(\sin {{\theta }_{1}}-\,\sin \,\,{{\theta }_{2}})-\mu \,\,mg\,\,\cos \,{{\theta }_{1}}\]                 \[a=\frac{g}{2}\,(\sin \,{{\theta }_{1}}-\,\sin \,{{\theta }_{2}})\,\,-\,\frac{\mu g}{2}\,\cos \,{{\theta }_{1}}\]                 In both are at rest                 \[\mu \,=\,\frac{(\sin \,{{\theta }_{1}}-\,\sin \,{{\theta }_{2}})}{\cos {{\theta }_{1}}}\]