A block of mass m is kept on a smooth moving wedge If the acceleration of the wedge is equal to a, then match the following:

`{:(,"Column -I",,"Column -II"),("(a)","Acceleration of m",,(p) Ma+m(g cos theta+ a sin theta)sin theta),("(b)",underset("relative to the wedge")"Acceleration of m",,(q)(g sin theta-a cos theta)),("(c)",underset("exerted by ground")"Force on the wedge",,(r)(sqrt(g^(2)+a^(2)))sin theta),("(d)",underset("exerted by external agent")"Force on the wedge",,(s) (M+mcos^(2)theta)g+ma sin thetacos theta):}`

Answer

`a rarr r; b rarr q; c rarr s; d rarrp`

Explanation

From the fig

`a_(r)=gsintheta-acos theta`
`:.(b)rarr(q)`
`a_("net")=sqrt(a^(2)+a_(r)^(2)+2a(a_(r))cos theta)`
`=sqrt(a^(2)+(gsintheta-acos theta)^(2)+2a(gsintheta-acos theta)cos theta)`
`=sqrt(a^(2)+g^(2)sin^(2)theta+a^(2)cos^(2)theta-2gasin theta costheta)`
`+2gasin theta cos theta-2a^(2)cos^(2)theta`
`=sqrt(a^(2)+g^(2)sin^(2)theta-a^(2)cos^(2)theta)`
`=(sqrt(a^(2)+g^(2)))sin theta`
`:.(a) rarr(r)`
(c) Also, `N=mgcos theta+masin theta`
`:.N_(1)=Mg+Ncos theta`
`=Mg(mgcos theta+masintheta)cos theta`
`=Mg+mg cos^(2)theta+masin theta cos theta`
`:. (c) rarr (s)`
(d) `F_(ext)-Nsin theta=Ma`
`F_(ext)=Ma+(mgcos theta+masin theta)sin theta`
`=Ma+mg sin theta cos theta +ma sin^(2)theta`
`:. (d) rarr (p)`