Question

Two particles P and Q are moving in concentric circles of rarui r_p and r_Q respectively. If their period of revolutions are in ratio 2 : 3, then ratio of their centripetal acceleration is ____________.

Options

• `"r"_"p"^2/"r"_"Q"^2`

• `(4"r"_"p"^2)/(9"r"_"Q"^2)`

• `(9"r"_"p")/(4"r"_"Q")`

• `(4"r"_"Q")/(9"r"_"p")`

Answer 1

Two particles P and Q are moving in concentric circles of rarui r_p and r_Q respectively. If their period of revolutions are in ratio 2 : 3, then ratio of their centripetal acceleration is `(9"r"_"p")/(4"r"_"Q")`.

Explanation:

Centripetal acceleration is,

`"a" = "r"  omega^2`

`"but,"  omega = (2pi)/"T"`

`therefore "a"_"p" = "r" ((2pi)/"T"_"p")^2 = "r"_"p" [(4 pi^2)/"T"_"p"^2]`

`"Similarly, a"_"Q" = "r"_"Q" [(4 pi^2)/"T"_"Q"^2]`

`therefore "a"_"p"/a_"Q" = ("T"_"Q"^2 "r"_"p")/("T"_"p"^2 "r"_"Q")`

`"Given": "T"_"p"/"T"_"Q" = 2/3 Rightarrow "T"_"p"^2/"T"_"Q"^2 = 4/9`

`therefore "a"_"p"/"a"_"Q" = (9 "r"_"p")/(4"r"_"Q")`