Figure shows the total acceleration and velocity of a particle moving clockwise in a circle of radius $2.5 m$ at a given instant of time. At this instant, find:
(a) the radial acceleration,
(b) the speed of the particle and
(c) its tangential acceleration.

Open in App

Solution

$a = 25$ $m / s^{2}$
General form of velocity:
$\to V = ˙ r^r + r ˙ θ^θ$
where, a dot above denotes time derivative
General form of acceleration:
$\to a = (¨ r - r {˙ θ}^{2})^r + (r ¨ θ + 2 ˙ r ˙ θ)^θ$
Since, the particle is moving in a circle,
$˙ r = ¨ r = 0$
Now, $(a)$ radial acceleration:
$a_{r} = a cos θ$
$⟹ a_{r} = 25 \times cos 30^{o}$
$⟹ a_{r} = 25 \times \frac{\sqrt{3}}{2}$
$⟹ a_{r} = 21.650$
Also, $a_{r} = r {˙ θ}^{2} = 21.650$
Using magnitude:
${˙ θ}^{2} = \frac{21.650}{r}$
$⟹ {˙ θ}^{2} = \frac{21.650}{2.5}$
$⟹ {˙ θ}^{2} = 8.66$
$⟹ ˙ θ = 2.94$ $r a d / s$
$(b)$ Speed of the particle:
$V = r ˙ θ$
$⟹ V = 2.5 \times 2.94$
$⟹ V = 7.35$ $m / s$
$(c)$ Tangential acceleration:
$a_{θ} = a sin 30^{o}$
$⟹ a_{θ} = 25 \times \frac{1}{2}$
$⟹ a_{θ} = 12.5$ $m / s^{2}$

Suggest Corrections

Similar questions

Q. Figure. A.9 shows the direction of the total acceleration and velocity of a particle moving clockwise in a circle of radius

m

at in instant of time. Tangential acceleration at this instant is

5 m / s^{2}

. Which of the following statements is not correct?

Diagram shows the direction of the total acceleration and velocity of a particle moving clockwise in a circle of radius 5/√3 m at an instant of time. Tangential acceleration at this instant is 5 m/s². Which of the following statements is not correct?