MHT CET · Physics · Motion In Two Dimensions

A wheel of radius 1 m rolls through \(180^{\circ}\) over a plane surface. The magnitude of the displacement of the point of the wheel initially in contact with the surface is.

A \(2 \pi\)
B \(\pi\)
C \(\sqrt{\pi^2+4}\)
D \(3 \pi\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\sqrt{\pi^2+4}\)

Step-by-step Solution

Detailed explanation

Distance travelled by the wheel in half revolution \(=\frac{\mathrm{C}}{2}=\frac{2 \pi \mathrm{r}}{2}=\pi \mathrm{r}\)
Where C is the circumference of the wheel.
\(\therefore\) From figure,
Displacement of initial point of contact after half revolution \(=\mathrm{AB}\)
\(\therefore A B^2=A D^2+\mathrm{DB}^2 \)
\( A B^2=(\pi r)^2+(2 r)^2=r^2\left(\pi^2+4\right) \)
\( \therefore A B=r \sqrt{\left(\pi^2+4\right)} \)
\( \therefore A B=\sqrt{\left(\pi^2+4\right)}\)
...(given \(r=1 \mathrm{~m}\) )

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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