Question

A body completed 1*3/4th of a circular path of radius R , then the ratio of distance travelled to its displacement is :-
a.11:√2
b.11:4
c. 11:2√2
d.22:7

Answer 1

Anѕwєr :

$\huge \boxed{\red{\sf { 11\ratio \sqrt{2} } }}$

Explαnαtion:

Suppose that body starts from point A so after 1¾th round it'll first cover 1 round and come back to A again after 1 round then it'll cover ¾th of the circular path and will come to B.

Now, in order to find the ratio of distance travelled to its displacement, firstly we'll calculate distance and displacement.

$\Large {\underline { \sf {Distance \; travelled :}}}$

Let the distance covered be D. Distance covered by it will be the 1¾th of the total length of the circular path. The total length of the circular path is the circumference and circumference is calculated by using the formula 2πr. So,

$\implies \sf { D = 1\dfrac{3}{4} \Bigg \{ 2\pi r\Bigg \} }$

Here, the radius is R.

$\implies \sf { D = \dfrac{7}{4} \Bigg \{ 2\times \dfrac{22}{7} \times R \Bigg \} }$

$\implies \sf { D = \dfrac{7}{4} \Bigg \{ \dfrac{44}{7} \times R \Bigg \} }$

$\implies \sf { D = \dfrac{7}{4} \times \dfrac{44}{7}R}$

$\implies \sf { D = \dfrac{1}{4} \times 44R}$

$\implies \boxed{\sf { D = 11R}}\dots \; \bf{( 1 )}$

$\rule{200}2$

$\Large {\underline { \sf { Displacement :}}}$

Let the displacement be S.

Displacement will be the shortest distance from its initial to final position. Here, the initial position is A and the final position is B. So, the displacement is AB. Now, we have to calculate the length of AB. AO is the radius, drop OB as a perpendicular on point O. It'll also be considered as the radius, so its length is R too. Now, by using Pythagoras Property,

$\implies \sf { S = AB}$

$\implies \sf { S = \sqrt{(AO)^2 + (OB)^2}}$

$\implies \sf { S = \sqrt{(R)^2 + (R)^2}}$

$\implies \sf { S = \sqrt{R^2 + R^2}}$

$\implies \sf { S = \sqrt{2R^2}}$

$\implies \boxed{\sf { S = \sqrt{2}R}}\dots \; \bf{( 2 )}$

$\rule{200}2$

$\Large {\underline { \sf {Required \; ratio :}}}$

Here,

$\implies \sf {Ratio = D \ratio S}$

$\implies \sf {Ratio = \dfrac{D}{S} }$

Substitute the values from ( 1 ) and ( 2 ).

$\implies \sf {Ratio = \dfrac{11R}{ \sqrt{2}R} }$

$\implies \sf {Ratio = \dfrac{11}{ \sqrt{2}} }$

$\implies \boxed{\red{\sf { Ratio = 11\ratio \sqrt{2} } }}$

∴ The ratio of distance travelled to its displacement is 11:√2. The correct option is Option A.

$\rule{200}2$