Question

7. A particle is moving on a circular path with constant
speed v. It moves between two points A and B,
which subtends an angle 60° at the centre of circle.
The magnitude of change in its velocity and change
in magnitude of its velocity during motion from A to
B are respectively
(1) Zero, Zero
(2) V, 0
(3) 0, v
(4) 2V, v​

Answer 1

Answer:

$\rm (2) \ v, 0$

Explanation:

Velocity of particle at point A:

$\rm \overrightarrow{v}_A = v \hat{j}$

Velocity of particle at point B:

$\rm \overrightarrow{v}_B = -v \ sin\theta \hat{i} + v \ cos \theta \hat{j}$

Change in velocity:

$\rm \leadsto \Delta \overrightarrow{v} = \overrightarrow{v}_B - \overrightarrow{v}_A \\ \\ \rm \leadsto \Delta \overrightarrow{v} = - v \: sin \theta \hat{i} + v \: cos \theta \hat{j} - v \hat{j} \\ \\ \rm \leadsto \Delta \overrightarrow{v} = - v \: sin \theta \hat{i} - v (1 - cos \theta) \hat{j} \\ \\ \rm \leadsto | \Delta \overrightarrow{v} | = \sqrt{(- v \: sin \theta) ^{2} + [ - v ( 1 - cos \theta)] ^{2} } \\ \\ \rm \leadsto \Delta v = \sqrt{ {v}^{2} ( sin ^{2} \theta + 1 + cos ^{2} \theta - 2cos \theta) } \\ \\ \rm \leadsto \Delta v = \sqrt{ {v}^{2} ( 2 - 2cos \theta) } \\ \\ \rm \leadsto \Delta v = \sqrt{ 2{v}^{2} ( 1 - cos \theta) } \\ \\ \rm \leadsto \Delta v = \sqrt{4 {v}^{2} {sin}^{2} \dfrac{ \theta}{2} } \\ \\ \rm \leadsto \Delta v = 2v \: sin \dfrac{ \theta}{2}$

For $\rm \theta = 60\degree$:

$\rm \leadsto \Delta v = 2v \: sin \bigg(\dfrac{ 60 \degree}{2} \bigg) \\ \\ \rm \leadsto \Delta v = 2v \: sin 30 \degree \\ \\ \rm \leadsto \Delta v = \cancel{2}v \times \dfrac{ 1}{ \cancel{2}} \\ \\ \rm \leadsto \Delta v = v$

$\therefore$ Magnitude of change in velocity (∆v) = v

Change in magnitude of its velocity during motion from A to B:

$\rm \implies | \overrightarrow{v}_B | - | \overrightarrow{v}_A | \\ \\ \rm \implies v - v \\ \\ \rm \implies 0$

Answer 2

Answer:

Explanation:

A particle is moving on a circular path with constant

speed v. It moves between two points A and B,

which subtends an angle 60° at the centre of circle.

The magnitude of change in its velocity and change

in magnitude of its velocity during motion from A to

B are respectively

(1) Zero, Zero

(2) V, 0

(3) 0, v

(4) 2V, v​