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A-block-of-mass-M-is-placed-on-smooth-ground-Its-upper-surface-is-smooth-and-vertical-surface-is-rough-with-coefficient-of-friction-A-block-of-mass-m-1-is-placed-on-its-horizontal-surface-and-tie

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Question Number 21564 by Tinkutara last updated on 27/Sep/17
A block of mass M is placed on smooth ground. Its upper surface is smooth and vertical surface is rough with coefficient of friction μ. A block of mass m_1 is placed on its horizontal surface and tied with a massless inextensible string passing over smooth pulley. Its other end is connected to another block of mass m_2 , which touches the vertical surface of block M. Now a horizontal force F starts acting on it. Q1. Which of the following is incorrect about above system? (1) There exists a value of F at which friction force is equal to zero (2) When F = 0, the blocks cannot remain stationary (3) There exists two limiting values of F at which the blocks m_1 and m_2 will remain stationary w.r.t. block of mass M (4) The limiting friction between m_2 and M is independent of F Q2. In the above case, let m_1 − μm_2 be greater than 1. Choose the incorrect value of F for which the blocks m_1 and m_2 remain stationary with respect to M (1) (M + m_1 + m_2 )((m_2 g)/m_1 ) (2) ((m_2 (M + m_1 + m_2 ))/((m_1 − μm_2 )))g (3) (((M + m_1 + m_2 )m_2 g)/((m_1 + μm_2 ))) (4) (M + m_1 + m_2 )(g/μ) Q3. Let vertical part of block M be smooth. Choose the correct alternative (1) There exist two limiting values for system to remain relatively at rest (2) For one unique value of F, the blocks m_1 and m_2 remain stationary with respect to block M (3) The blocks m_1 and m_2 cannot be in equilibrium for any value of F (4) There exists a range of mass M, for which system remains stationary
$$\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{M}\:\mathrm{is}\:\mathrm{placed}\:\mathrm{on}\:\mathrm{smooth} \\ $$$$\mathrm{ground}.\:\mathrm{Its}\:\mathrm{upper}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{smooth}\:\mathrm{and} \\ $$$$\mathrm{vertical}\:\mathrm{surface}\:\mathrm{is}\:\mathrm{rough}\:\mathrm{with}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:\mathrm{friction}\:\mu.\:\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{m}_{\mathrm{1}} \:\mathrm{is}\:\mathrm{placed} \\ $$$$\mathrm{on}\:\mathrm{its}\:\mathrm{horizontal}\:\mathrm{surface}\:\mathrm{and}\:\mathrm{tied}\:\mathrm{with} \\ $$$$\mathrm{a}\:\mathrm{massless}\:\mathrm{inextensible}\:\mathrm{string}\:\mathrm{passing} \\ $$$$\mathrm{over}\:\mathrm{smooth}\:\mathrm{pulley}.\:\mathrm{Its}\:\mathrm{other}\:\mathrm{end}\:\mathrm{is} \\ $$$$\mathrm{connected}\:\mathrm{to}\:\mathrm{another}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{m}_{\mathrm{2}} , \\ $$$$\mathrm{which}\:\mathrm{touches}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{surface}\:\mathrm{of} \\ $$$$\mathrm{block}\:{M}.\:\mathrm{Now}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{force}\:{F} \\ $$$$\mathrm{starts}\:\mathrm{acting}\:\mathrm{on}\:\mathrm{it}. \\ $$$$\boldsymbol{\mathrm{Q}}\mathrm{1}.\:\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{incorrect} \\ $$$$\mathrm{about}\:\mathrm{above}\:\mathrm{system}? \\ $$$$\left(\mathrm{1}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{value}\:\mathrm{of}\:{F}\:\mathrm{at}\:\mathrm{which} \\ $$$$\mathrm{friction}\:\mathrm{force}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{zero} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{When}\:{F}\:=\:\mathrm{0},\:\mathrm{the}\:\mathrm{blocks}\:\mathrm{cannot} \\ $$$$\mathrm{remain}\:\mathrm{stationary} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{two}\:\mathrm{limiting}\:\mathrm{values}\:\mathrm{of} \\ $$$${F}\:\mathrm{at}\:\mathrm{which}\:\mathrm{the}\:\mathrm{blocks}\:{m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \:\mathrm{will} \\ $$$$\mathrm{remain}\:\mathrm{stationary}\:\mathrm{w}.\mathrm{r}.\mathrm{t}.\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass} \\ $$$${M} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{The}\:\mathrm{limiting}\:\mathrm{friction}\:\mathrm{between}\:{m}_{\mathrm{2}} \\ $$$$\mathrm{and}\:{M}\:\mathrm{is}\:\mathrm{independent}\:\mathrm{of}\:{F} \\ $$$$\boldsymbol{\mathrm{Q}}\mathrm{2}.\:\mathrm{In}\:\mathrm{the}\:\mathrm{above}\:\mathrm{case},\:\mathrm{let}\:{m}_{\mathrm{1}} \:−\:\mu{m}_{\mathrm{2}} \:\mathrm{be} \\ $$$$\mathrm{greater}\:\mathrm{than}\:\mathrm{1}.\:\mathrm{Choose}\:\mathrm{the}\:\mathrm{incorrect} \\ $$$$\mathrm{value}\:\mathrm{of}\:{F}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{blocks}\:{m}_{\mathrm{1}} \:\mathrm{and} \\ $$$${m}_{\mathrm{2}} \:\mathrm{remain}\:\mathrm{stationary}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to} \\ $$$${M} \\ $$$$\left(\mathrm{1}\right)\:\left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right)\frac{{m}_{\mathrm{2}} {g}}{{m}_{\mathrm{1}} } \\ $$$$\left(\mathrm{2}\right)\:\frac{{m}_{\mathrm{2}} \left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right)}{\left({m}_{\mathrm{1}} \:−\:\mu{m}_{\mathrm{2}} \right)}{g} \\ $$$$\left(\mathrm{3}\right)\:\frac{\left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right){m}_{\mathrm{2}} {g}}{\left({m}_{\mathrm{1}} \:+\:\mu{m}_{\mathrm{2}} \right)} \\ $$$$\left(\mathrm{4}\right)\:\left({M}\:+\:{m}_{\mathrm{1}} \:+\:{m}_{\mathrm{2}} \right)\frac{{g}}{\mu} \\ $$$$\boldsymbol{\mathrm{Q}}\mathrm{3}.\:\mathrm{Let}\:\mathrm{vertical}\:\mathrm{part}\:\mathrm{of}\:\mathrm{block}\:{M}\:\mathrm{be} \\ $$$$\mathrm{smooth}.\:\mathrm{Choose}\:\mathrm{the}\:\mathrm{correct}\:\mathrm{alternative} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{There}\:\mathrm{exist}\:\mathrm{two}\:\mathrm{limiting}\:\mathrm{values}\:\mathrm{for} \\ $$$$\mathrm{system}\:\mathrm{to}\:\mathrm{remain}\:\mathrm{relatively}\:\mathrm{at}\:\mathrm{rest} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{For}\:\mathrm{one}\:\mathrm{unique}\:\mathrm{value}\:\mathrm{of}\:{F},\:\mathrm{the}\:\mathrm{blocks} \\ $$$${m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \:\mathrm{remain}\:\mathrm{stationary}\:\mathrm{with} \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{block}\:{M} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{The}\:\mathrm{blocks}\:{m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \:\mathrm{cannot}\:\mathrm{be}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}\:\mathrm{for}\:\mathrm{any}\:\mathrm{value}\:\mathrm{of}\:{F} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{range}\:\mathrm{of}\:\mathrm{mass}\:{M},\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{system}\:\mathrm{remains}\:\mathrm{stationary} \\ $$
Commented by Tinkutara last updated on 27/Sep/17

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