Question

A disc arranged in a vertical plane has several grooves directed along the chords drawn from a point A (topmost point) as shown in the figure. Several bodies begin to slide down the respective grooves from A simultaneously. The ratio of their times of slide will be: (neglect friction and air resistance)

• A. AB : AC : AD : AE
• B. 1 : 1 : 1 : 1
• C. AE : AD : AC : AB
• D. 1 : 2 : 3 : 4

Answer 1

The correct option is B 1 : 1 : 1 : 1

Suppose a chord makes an angle $\theta$ with the vertical line.
A body constrained to move along this line will experience the component of acceleration due to gravity along this line which is $-g\cos \theta$.
The length of this chord is $2r\cos \theta$.
In the equation: $s=ut+\frac{1}{2}a{t}^2$, substituting u $=$0, $s=2r\cos \theta$ and $a=-g\cos \theta$,
We get: $2r\cos \theta =\frac{1}{2}gcos\theta t^2$
$t=\sqrt{\frac{4r}{g}}$
We find that the time is independent of $\theta$.
Hence, the time taken will be same for any chord.