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(D) Let the initial position be $A$ at an angle $	heta$ with the vertical. The ball is projected horizontally with $u = 3 \, m/s$. Let the string be

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$37$

Vedclass · Answer

(D) Let the initial position be $A$ at an angle $	heta$ with the vertical. The ball is projected horizontally with $u = 3 \, m/s$. Let the string be taut again at the lowest point $B$ (vertical position). The height of $A$ from the lowest point $B$ is $h = L - L \cos 	heta = L(1 - \cos 	heta)$. Using the law of conservation of energy between $A$ and $B$: $\frac{1}{2} m u^2 + mgh = \frac{1}{2} m v_B^2$ $\frac{1}{2} u^2 + gL(1 - \cos 	heta) = \frac{1}{2} v_B^2$ $v_B^2 = u^2 + 2gL(1 - \cos 	heta) = 3^2 + 2(10)(1)(1 - \cos 	heta) = 9 + 20(1 - \cos 	heta) = 29 - 20 \cos 	heta$. For the string to be just taut at the lowest point,the tension $T$ must be at least zero. However,the problem implies the string becomes taut at the vertical position. This occurs when the horizontal displacement matches the arc length or when the projectile motion path intersects the vertical line. Given the standard constraint for this type of problem,the velocity at the vertical position must satisfy the condition for circular motion or projectile trajectory. Solving for $	heta$ where the path intersects the vertical: Using $x = u t = L \sin 	heta$ and $y = \frac{1}{2} g t^2 = L(1 - \cos 	heta)$. $t = \frac{L \sin 	heta}{u} \implies \frac{1}{2} g (\frac{L^2 \sin^2 	heta}{u^2}) = L(1 - \cos 	heta)$. $\frac{g L \sin^2 	heta}{2 u^2} = 1 - \cos 	heta \implies \frac{10(1) (1 - \cos^2 	heta)}{2(3^2)} = 1 - \cos 	heta \implies \frac{10(1 - \cos 	heta)(1 + \cos 	heta)}{18} = 1 - \cos 	heta$. Assuming $1 - \cos 	heta 
eq 0$,we get $1 + \cos 	heta = \frac{18}{10} = 1.8$. $\cos 	heta = 0.8$. Therefore,$	heta = \cos^{-1}(0.8) = 37^o$.

$A$ ball is held in the position shown with a string of length $L = 1 \, m$ just taut and then projected horizontally with a velocity of $u = 3 \, m/s$. If the string becomes taut again when it is vertical,the angle $\theta$ is given by ........ $^o$.

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