Question

A particle moves in a circle of radius 1.0 cm at a speed given by v = 2.0 t where v is cm/s and t in seconds.
(a) Find the radial acceleration of the particle at t = 1 s.
(b) Find the tangential acceleration at t = 1 s.
(c) Find the magnitude of the acceleration at t = 1 s.

Answer 1

Speed is given as a function of time. Therefore, we have:
v = 2t
Radius of the circle = r = 1 cm
At time t = 2 s, we get:
(a) Radial acceleration
$a=\frac{v^2}{r}=\frac{2{}^2}{1}=4\mathrm{cm}/{\mathrm{s}}^2$
(b) Tangential acceleration
$$\begin{gathered} a=\frac{dv}{dt} \\ =\frac{d}{dt}\left(2t\right)=2\mathrm{cm}/{\mathrm{s}}^2 \end{gathered}$$
(c) Magnitude of acceleration
$$\begin{gathered} a=\sqrt{4^2+2^2} \\ =\sqrt{20}\mathrm{cm}/{\mathrm{s}}^2 \end{gathered}$$