Question

A particle covers 3/4th of a circle of radius R. The distance and displacement of the particle will be

Answer 1

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✺Answer:

♦️GiveN

• A partcle covers 3/4th of a circle of Radius R

♦️To FinD

• Distance
• Displacement of the particle...?

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✺Concept Behind the Q.

Before solving the question, we need to know certain basic things which are as follows:

♠️ Distance

The length of the actual path between the initial and final position of a moving body in a given time interval i known as distance travelled by the body.

• Distance is length of actual path.
• It is a scalar quantity. Always positive for a moving body.
• Measured by Odometer in Vehicles.
• SI unit of distance is metre.

♠️Displacement

The shortest distance between the initial and final position of a moving body in a given interval of time is known as displacement.

• Length of the shortest path, i.e a straight line between initial and final position.
• It is a vector quantity. Can be positive, negative or zero for a moving body.
• Displacement = Xf - Xi (final position- initial position)
• SI unit of displacement is same metre.

♦️Point to Remember

$\large{ \boxed{ \rm{ \green{distance \: \geqslant |displacement|}}}}$

Distance travelled by an object is either equal to or greater than the magnitude of displacement of object.

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✺Explanation of Q.

▶️ Refer to the attachment

Let the object started its journey from B and after 3/4 th Revolution, he reached A. Then initial point is B and final point is A. Distance covered by the body is 3/4th of the perimeter of circle (actual path).

And displacement would be the shortest path i.e. a straight line Drawn from B to A. So, we can find displacement by Pythagoras theoram as we have perpendicular = base = R

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✺Solution

Distance travelled by the body

= 3/4 th of the circumference of the circle.

We know,

$\large{ \rm{circumference \: of \: circle = 2\pi R}}$

Distance travelled by the body

$\large{ \rm{ \rightarrow \: \frac{3}{4} \times 2\pi R \: units}} \\ \\ \large{ \rm{ \rightarrow \: \frac{3\pi R}{2} \: units}}$

If we join A and B , we get a right angled triangle with hypotenuse and other two sides r each .

We know, Pythagoras theoram.

$\large{ \rm{ {p}^{2} + {b}^{2} = {h}^{2} }}$

Displacement of the body

$\large{ \rm{ \rightarrow \: {R}^{2} + {R}^{2} = {h}^{2} }} \\ \\ \large{ \rm{ \rightarrow \: 2 {R}^{2} = {h}^{2}}} \\ \\ \large{ \rm{ \rightarrow \: h = \sqrt{2 {R}^{2} } }} \\ \\ \large{ \rm{ \rightarrow \: h = \sqrt{2} R}}$

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✺So Final Answer:

$\Large{ \rm{ \pink{ \therefore{distance \: = \frac{3\pi R}{2} }}}} \\ \Large{ \rm{ \blue{ \therefore{displacement \: = \sqrt{2} R}}}}$

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