In the List-I below, four different paths of a particle are given as functions of time. In these functions, and are positive constants of appropriate dimensions and . In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: is the linear momentum, is the angular momentum about the origin, is the kinetic energy, is the potential energy and is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

Options

A. A-PQRST, B-Q,T, C-Q,R,S,T, D-T
B. A=P,Q,R,S,T, B=R,T, C=C,Q,R,S,T, D=Q,T
C. A=Q,R,S, B=T, C=P,Q,S, D=Q,T
D. A=P,Q,R,T, B=Q,T, C=Q,R,S,T, D=Q,T

Answer

A

Explanation

A. `vecr = alphathati + betathatj`
`vecv = alpha hati + betahatj`
`veca = 0 rArr vecF = 0`
`therefore A-P, Q-R, S,T`
B. `vecr = alpha cos omegathetai + betasin omegat hatj`, `vecv=-alpha omegasin omegat hati + beta omega cos omega t hatj`
`veca =-alpha omega^(2) cos omegat hai - beta omega^(2) sin omega t hatj, veca = -omega^(2)vecr rArr` Force is towards centre
`veca.vecv=(alpha^(2) - beta^(2))omega^(3) sin omegat cos omegat ne 0`
`rArr` Work is done by the force and kinetic energy of particle will not be conserved.
`therefore B-Q, T`
C. `vecr = alpha cos omega t hati + alpha sin omega t hatj`
`rArr` Uniform circular motion, `therefore` C,Q,R,S,T
D. `vecr = alphat hati + beta/2t^(2) hatj`, `vecv = alpha hati + beta t hatj`
`veca = betahatj rArr` force is not passing through origin `therefore` D-T