Which one of the following operations is valid?
a) Vector multiplied by scalar
b) Vector added to scalar
c) Vector subtracted from scalar
d) Vector divided by vector

Answer

A, B, C, D

Explanation

a. Radius =R
Horizontal speed=v
From the free body diagram ure
N=normal force
`=mg-(mv^2)/R`
b. When the particle is givenmaximum velocity so that the centrifugal force balances the weight, the particle does not slip on the sphere,
`rarr (mv^2)/R=mg`
`rarr v=sqrt(gR)`
c if the body is given velocity `v_1` at the top such that
`v_1=sqrt(gR)/2`
`v_1^2=(gR)/4` Let the velocity be `v_2` when it leaves contact with the surface, ure

So, (mv_2^2)/R=mgcostheta`
`rarrv_2^2=Rgcostheta`........i
`Again `(1/2)mv_2^2-(1/2)mv_1^2`
`=mgR(1-costheta)`
`rarr v_2^2=v_1^2+2gR(1-costheta)`............ii
From equatioon i and equation ii
`Rgcostheta=((Rg)/4)+2gR(1-costheta)`
`rarr costheta=(1/4+2-2costheta)`
`rarr 3costheta=(9/4)`
`rarr theta=cos^-1(3/4)`