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NEET-XII-Physics

07: Circular Motion

with Solutions - page 7
Qstn# iv-27 Prvs-QstnNext-Qstn
  • #27
    A block of mass m moves on a horizontal circle against the wall of a cylindrical room of radius R. The floor of the room on which the block moves is smooth but the friction coefficient between the wall and the block is μ. The block is given an initial speed v0. As a function of the speed v write (a) the normal force by the wall on the block, (b) the frictional force by wall and (c) the tangential acceleration of the block. (d) Integrate the tangential acceleration
    dvdt=vdvdsto obtain the speed of the block after one revolution. (a) the normal force by the wall on the block, (b) the frictional force by wall and (c) the tangential acceleration of the block. (d) Integrate the tangential acceleration
    dvdt=vdvdsto obtain the speed of the block after one revolution.
    Ans :
    Given:
    Radius of the room = R
    Mass of the block = m (a) Normal reaction by the wall on the block = N = `` \frac{m{v}^{2}}{R}`` (b) Force of frictional by the wall = `` \mu N=\frac{\mu m{v}^{2}}{R}`` (c) Let at be the tangential acceleration of the block.
    From figure, we get:
    `` -\frac{\mu m{v}^{2}}{R}=m{a}_{t}``
    `` \Rightarrow {a}_{t}=-\frac{\mu {v}^{2}}{R}`` (d) `` \,\mathrm{\,On \,}\,\mathrm{\,using \,}a=\frac{dv}{dt}=v\frac{dv}{ds},\,\mathrm{\,we \,}\,\mathrm{\,get \,}:``
    `` v\frac{dv}{ds}=\frac{\mu {v}^{2}}{R}``
    `` \Rightarrow ds=-\frac{R}{\mu }\frac{dv}{v}``
    `` \,\mathrm{\,Integrating \,}\,\mathrm{\,both \,}\,\mathrm{\,side \,},\,\mathrm{\,we \,}\,\mathrm{\,get \,}:``
    `` s=-\frac{R}{\mu }\,\mathrm{\,In \,}v+c``
    `` \,\mathrm{\,At \,},s=0,v={v}_{0}``
    `` So,c=\frac{R}{\mu }\,\mathrm{\,In \,}{v}_{0}``
    `` \Rightarrow s=-\frac{R}{\mu }\,\mathrm{\,In \,}\frac{v}{{v}_{0}}``
    `` \Rightarrow \frac{v}{{v}_{0}}={e}^{\mathit{-}\frac{\mu s}{R}}``
    `` \mathit{\Rightarrow }v={\,\mathrm{\,v \,}}_{0}{e}^{\mathit{-}\frac{\mathit{\mu s}}{R}}``
    `` \,\mathrm{\,For \,}\,\mathrm{\,one \,}\,\mathrm{\,rotation \,},\,\mathrm{\,we \,}\,\mathrm{\,have \,}:``
    `` s=2\pi r``
    `` \therefore v={v}_{0}{e}^{-2\pi \mu }``
    `` ``
    `` ``
    Page No 116: (a) Normal reaction by the wall on the block = N = `` \frac{m{v}^{2}}{R}`` (b) Force of frictional by the wall = `` \mu N=\frac{\mu m{v}^{2}}{R}`` (c) Let at be the tangential acceleration of the block.
    From figure, we get:
    `` -\frac{\mu m{v}^{2}}{R}=m{a}_{t}``
    `` \Rightarrow {a}_{t}=-\frac{\mu {v}^{2}}{R}`` (d) `` \,\mathrm{\,On \,}\,\mathrm{\,using \,}a=\frac{dv}{dt}=v\frac{dv}{ds},\,\mathrm{\,we \,}\,\mathrm{\,get \,}:``
    `` v\frac{dv}{ds}=\frac{\mu {v}^{2}}{R}``
    `` \Rightarrow ds=-\frac{R}{\mu }\frac{dv}{v}``
    `` \,\mathrm{\,Integrating \,}\,\mathrm{\,both \,}\,\mathrm{\,side \,},\,\mathrm{\,we \,}\,\mathrm{\,get \,}:``
    `` s=-\frac{R}{\mu }\,\mathrm{\,In \,}v+c``
    `` \,\mathrm{\,At \,},s=0,v={v}_{0}``
    `` So,c=\frac{R}{\mu }\,\mathrm{\,In \,}{v}_{0}``
    `` \Rightarrow s=-\frac{R}{\mu }\,\mathrm{\,In \,}\frac{v}{{v}_{0}}``
    `` \Rightarrow \frac{v}{{v}_{0}}={e}^{\mathit{-}\frac{\mu s}{R}}``
    `` \mathit{\Rightarrow }v={\,\mathrm{\,v \,}}_{0}{e}^{\mathit{-}\frac{\mathit{\mu s}}{R}}``
    `` \,\mathrm{\,For \,}\,\mathrm{\,one \,}\,\mathrm{\,rotation \,},\,\mathrm{\,we \,}\,\mathrm{\,have \,}:``
    `` s=2\pi r``
    `` \therefore v={v}_{0}{e}^{-2\pi \mu }``
    `` ``
    `` ``
    Page No 116:
  • #27-a
    the normal force by the wall on the block,
    Ans : Normal reaction by the wall on the block = N = `` \frac{m{v}^{2}}{R}``
  • #27-b
    the frictional force by wall and
    Ans : Force of frictional by the wall = `` \mu N=\frac{\mu m{v}^{2}}{R}``
  • #27-c
    the tangential acceleration of the block.
    Ans : Let at be the tangential acceleration of the block.
    From figure, we get:
    `` -\frac{\mu m{v}^{2}}{R}=m{a}_{t}``
    `` \Rightarrow {a}_{t}=-\frac{\mu {v}^{2}}{R}``
  • #27-d
    Integrate the tangential acceleration
    dvdt=vdvdsto obtain the speed of the block after one revolution.
    Ans : `` \,\mathrm{\,On \,}\,\mathrm{\,using \,}a=\frac{dv}{dt}=v\frac{dv}{ds},\,\mathrm{\,we \,}\,\mathrm{\,get \,}:``
    `` v\frac{dv}{ds}=\frac{\mu {v}^{2}}{R}``
    `` \Rightarrow ds=-\frac{R}{\mu }\frac{dv}{v}``
    `` \,\mathrm{\,Integrating \,}\,\mathrm{\,both \,}\,\mathrm{\,side \,},\,\mathrm{\,we \,}\,\mathrm{\,get \,}:``
    `` s=-\frac{R}{\mu }\,\mathrm{\,In \,}v+c``
    `` \,\mathrm{\,At \,},s=0,v={v}_{0}``
    `` So,c=\frac{R}{\mu }\,\mathrm{\,In \,}{v}_{0}``
    `` \Rightarrow s=-\frac{R}{\mu }\,\mathrm{\,In \,}\frac{v}{{v}_{0}}``
    `` \Rightarrow \frac{v}{{v}_{0}}={e}^{\mathit{-}\frac{\mu s}{R}}``
    `` \mathit{\Rightarrow }v={\,\mathrm{\,v \,}}_{0}{e}^{\mathit{-}\frac{\mathit{\mu s}}{R}}``
    `` \,\mathrm{\,For \,}\,\mathrm{\,one \,}\,\mathrm{\,rotation \,},\,\mathrm{\,we \,}\,\mathrm{\,have \,}:``
    `` s=2\pi r``
    `` \therefore v={v}_{0}{e}^{-2\pi \mu }``
    `` ``
    `` ``
    Page No 116: