Question

For each natural number $k$, let $C_k$ denote the circle with radius $kcm$, and centre at the origin. On the circle $C_k$ a particle moves $kcm$, in the center-clockwise direction. After completing its motion on $C_k$, the particle moves to $C_{k+1}$ in the radial direction. The motion of the particle continuous in this manner. The particle starts at $(1,0)$. If the particle crosses the positive direction of the $x-$axis for the first time on the circle $C_k$, then find $n$.

Answer 1

Solution:-

It is given that on the circle $C_k$ of radius $kcm$, the particle moves $kcm$.

Therefore, angular displacement $\left(\theta \right)$ on $k^{th}$ circle is given by-

$\theta =\frac{\text{distance travelled by the particle}}{\text{radius}}=\frac{k}{k}\text{radian}=1\text{radian}$

Thus angular displacement on each circle is $1$ radian.

If the particle crosses the x-axis for the first time on circle $C_n$, then-

Total angular displacement $=n\text{radians}$, i.e.,

We have to move to the circle with radius $n$ to cover the angle of $\frac{n}{2\pi }$.

$\therefore n>2\pi$

Since $n$ is the natural number such that $n>2\pi$.

$\therefore n=7$

Hence the required answer is $n=7$.