Point $A$ on a wheel of radius $r$ touches the horizontal plane at point $P$. It rolls without slipping, till point $A$ is at the highest position in the first turn. What is the final distance $AP$?
Let's solve the problem step by step to find the distance \(AP\).
Step 1: Understanding the Motion
The wheel rolls without slipping, starting with point \(A\) at the bottom. As it rolls, point \(A\) completes a half-circle and ends up at the highest point on the wheel. This corresponds to a half circumference travel of the wheel on the ground.
Step 2: Calculate the Half-Circumference of the Wheel
The half-circumference of the wheel is given by:
\(\frac{1}{2} \times 2\pi r = \pi r\)
where \(r\) is the radius of the wheel.
Step 3: Geometric Considerations
Initially, point \(A\) touches the ground at point \(P\). After rolling, point \(A\) reaches the highest position on the wheel. At this point, it is vertically above the center of the wheel by a height \(2r\) (diameter of the wheel) above point \(P\).
Step 4: Calculate the Distance \(AP\)
Using the Pythagorean theorem, the distance \(AP\) forms a right triangle with the half-circumference and the diameter as follows:
\(AP = \sqrt{(\pi r)^2 + (2r)^2}\)
This simplifies to:
\(AP = \sqrt{\pi^2 r^2 + 4r^2} = r \sqrt{\pi^2 + 4}\)
This matches the correct answer.
Conclusion:
The final distance \(AP\) is \(r\sqrt{4+\pi^2}\).
Important Questions from Geometry (Notes)
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