Question

The displacement of the point of the wheel initially in contact with ground when the wheel roles forward half revolution will be (radius of wheel is R)​

Answer 1

$\red{\bigstar}\underline{\underline{\textsf{\textbf{ Given :- }}}}$

• There is a point on the body of a wheel .
• The radius of the wheel is R .
• The wheel completes half a revolution .

$\red{\bigstar}\underline{\underline{\textsf{\textbf{ To Find :- }}}}$

• The displacement after the half revolution.

$\red{\bigstar}\underline{\underline{\textsf{\textbf{ Solution :- }}}}$

Given that , there is a wheel of radius R and it completes half revolution .And we need to find out its displacement .We know that displacement is the minimum distance between the initial and the final position .

Now refer to the attachment . We can see that the minimum distance is the length AC , which is the hypotenuse of a right angle triangle ABC.

❒ Using Pyhthagoras Theorem :-

$\sf\dashrightarrow AC^2 = AB^2 + AC^2 \\\\\\\sf\dashrightarrow AC^2 = (\pi R)^2 +(2R)^2 \\\\\\\sf\dashrightarrow AC^2 = \pi^2R^2 + 4R^2 \\\\\\\sf\dashrightarrow AC^2 = R^2( \pi^2 + 4 ) \\\\\\\sf\dashrightarrow AC =\sqrt{ R^2(\pi^2+4)} \\\\\\\sf\dashrightarrow \underset{\blue{\sf Required \ Displacement }}{\underbrace{\boxed{\pink{\frak{ AC (Displacement)= R \sqrt{ 4 + \pi^2} }}}}}$

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• Also let's find out the angle .

❒ In ∆ ABC :-

$\sf\dashrightarrow tan\theta = \dfrac{BC}{BA} \\\\\\\sf\dashrightarrow tan\theta =\dfrac{2R}{\pi R} \\\\\\\sf\dashrightarrow tan\theta =\dfrac{ 2}{\pi } \\\\\\\sf\dashrightarrow \underset{\blue{\sf Required \ Angle }}{\underbrace{\boxed{\pink{\frak{ \theta \quad =\quad tan^{-1}\bigg(\dfrac{2}{\pi}\bigg) }}}}}$

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