Question

A particle is moving with speed $v=2t^2$ on the circumference of a circle of radius $R$. Match the quantities given in Table-1 with corresponding results in Table-2.

Table-1	Table-2
(A) Magnitude of tangential acceleration of particle	(P) decreases with time
(B) Magnitude of centripetal acceleration of particle	(Q) increases with time
(C) Magnitude of angular speed of particle with respect to centre of circle	(R) remains constant
(D) Angle between the total acceleration and centripetal acceleration of particle	(S) depends on the value of radius

• A. (A) - Q; (B) - Q; (C) - Q; (D) - P,S
• B. (A) - Q; (B) - Q,S; (C) - Q,S; (D) - P,S
• C. (A) - Q; (B) - Q,S; (C) - Q,S; (D) - P
• D. (A) - Q; (B) - P,S; (C) - P,S; (D) - Q,S

Answer 1

The correct option is B (A) - Q; (B) - Q,S; (C) - Q,S; (D) - P,S

Given $v=2t^2$
Tangential acceleration, $a_t=\frac{dv}{dt}=4t$
$\Rightarrow a_t$ increases with time

Centripetal acceleration, $a_c=\frac{v^2}{R}=\frac{4t^4}{R}$
​​​​​​​$\Rightarrow a_c$ increases with time and depends on radius

Angular speed $\omega =\frac{v}{R}=\frac{2t^2}{R}$
​​​​​​​$\Rightarrow \omega$ increases with time and depends on radius

$\tan \theta =\frac{a_t}{a_c}=\frac{4t}{\frac{4t^4}{R}}=\frac{R}{t^3}$
$\theta ={\tan }^{-1}\left(\frac{R}{t^3}\right)$
$\Rightarrow \theta$ decreases with time and depends on radius