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A ball is attached to a pivot, and hangs down vertically due to gravity. The length of the string connecting the ball to the pivot is L, and it is assumed that the string is ideal. The ball is given an impulse such that the velocity of the ball is $\sqrt{3gL}$, where g is the acceleration due to gravity. Find the maximum height attained by the ball. Any external factors (air resistance, friction etc.) are not to be taken into account.

I tried attempting it as follows: Using conservation of mechanical energy we have that when the ball attains maximum height, its kinetic energy should be all converted to potential energy, so, $mgh = \frac{1}{2} mv^2$, putting v = $\sqrt{3gL}$, I get: $h = \frac{3L}{2}$

But the answer given is $\frac{40L}{27}$! The next question asks, "What would be the maximum height if the string were to be replaced with a rod?", and the answer is indeed the expected $\frac{3L}{2}$.

Why is there a slight decrease in the maximum height when the connecting object is a string? I fail to understand this. And how am I supposed to account for this missing difference?

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Depending on the initial speed, the string or rod may have to provide an outward force at some point during the circular motion, to oppose the weight of the object (see Note below). The rigid rod can provide an outward push, the string cannot - it buckles when compressed.

As a result, the object on the rod continues along the circle until its KE is zero. The object on the string leaves the circle. It becomes a projectile and is still moving (horizontally) at its highest point. It still has some KE at this point, so it has less PE than if it stayed on the circle, because KE+PE is conserved. Less PE means that the object on the string reaches a lower maximum height.


Note

The tension in the string is a reactive force. It takes whatever value is necessary to keep the object a fixed distance from the pivot, with the proviso that it cannot provide a push, only a pull. The tension in the string and the component of the weight of the object along the string together provide a net inward force (the centripetal force) to keep the object moving in a circle :

$\text{inward component of weight + tension = centripetal force}$.

When the string or rod is below the horizontal, the component of weight is outward - ie it is -ve in this equation. If the object is still moving, centripetal force is greater than zero, so tension is always +ve in this situation.

When the string or rod is above the horizontal the component of the weight of the object is inwards, in the same direction as the tension force. As the object rises the inward component of weight increases. The required centripetal force decreases because the object loses speed as it rises. So the tension must also decrease. If the object is not moving fast enough then at some point the component of weight could exceed the required centripetal force. Then the tension becomes less than zero - ie a push is required from the string instead of a pull. The string cannot supply a push, but the rod can.

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