As shown in figure, a non-stretchable string of length 2 ℓ has one end fixed to a nail at point A, and a small ball is tied to the other end. The small ball can move in the vertical plane. Initially,

Question

As shown in figure, a non-stretchable string of length $2 ℓ$ has one end fixed to a nail at point A, and a small ball is tied to the other end. The small ball can move in the vertical plane. Initially, the ball is on the same horizontal line with A, and is launched with an initial velocity $v_{0}$ straight downward. To the left of point A, at a distance $ℓ$ from A , there is another nail at point B . When the string passes over point B , the ball will move upward. The minimum value of the initial speed $v_{0}$ required for the ball to eventually collide with the nail at point A is $α g ℓ$ . The value of $α$ is :

TestHub · Accepted Answer

After the string catches on B the ball moves on a circle of radius $ℓ$ about B, and nail A lies on this circle. Let the string go slack when the radius makes angle $β$ with BA: $v^{2} = g ℓ sin β$ (T = 0).

Requiring the subsequent projectile, launched tangentially from the slack point, to pass through A gives $cos β = - \frac{1}{2}$ , i.e. $β = 12 0^{\circ}$ .

Energy from the start to the slack point: $v_{0}^{2} = v^{2} + 2 g ℓ sin β$

$= 3 g ℓ sin 12 0^{\circ} = \frac{3 3}{2} g ℓ$ .

So $α = \frac{3 3}{2} \approx 2.60$ - within the official range 2.45-2.65.

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