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Suppose a ball is attached to a string of length $l$ and the string is attached to the ceiling. Now when I keep the ball close to the point of suspension and release it, it will travel a distance $l$ and then impart an impulsive tension to the string.

Is it possible to calculate this value of the tension developed in the string by knowing the final velocity: $(2gl)^{1/2}$ and mass of the ball $m$.

I thought the impulse is $2mv$ since the ball immediately goes back up with the same speed. But to find the force I also need the time interval for the velocity change. Now how can I find that without physical measurement??

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  • $\begingroup$ Yes you can. If you make a cut around the call, you can very easily find the force in the rope. You can differentiate this to maximise the force. This should be enough to calculate the tension. $\endgroup$ – Finn Eggers Jun 1 at 7:58
  • $\begingroup$ Could you explain in detail as to how I can get the equation of force to maximize $\endgroup$ – Lelouche Lamperouge Jun 1 at 8:24
  • $\begingroup$ I can give a full answer to this in 10 minutes when I am on my pc $\endgroup$ – Finn Eggers Jun 1 at 8:29
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This requires some information about accalerations in circular motions etc but I will show every formula but here it is:

Mass of the ball: $m$

Length of the rope: $l$

Velocity of the ball: $v$

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$$F_n = \dfrac{m}{l}\cdot v^2$$ $$F_t = \text{not needed here}$$ $$F_g = m \cdot g$$

Furthermore, you need to calculate $v$. You can simply use conservation of energy: $$v = \sqrt{2 \cdot g \cdot h}$$ with h being the height difference from where you started.

Assuming that the string starts with 0 velocity at a horizontal state:

$$h = cos(\phi) \cdot l$$

Just do a "simple" force equilibrium in the direction of the force and you get:

$$F = \dfrac{m}{l} * (2 g\cdot cos(\phi) l ) + cos(\phi) \cdot m g$$

$$\Rightarrow F = cos(\phi) \cdot 3 \cdot m \cdot g$$

Hope this helps.

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By making idealizing assumptions, such as considering the string as a spring with constant $k$ and mass $m$. Mechanics is all about modelling, so I would guess this is a valid model. Without extra information, the impulse can also be thought as instantaneous, since rigid bodies are thought as capable of enduring infinite forces, and always try to maintain valid velocity values, which makes the string go straight back up so that the total length can never be longer than $l$. But this case is the limit where $k\to\infty$ so the first model should be a valid generalization for this system.

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  • $\begingroup$ im sorry i didn't understand how this could help me get the value of max tension produced. If you could elaborate on the steps please. $\endgroup$ – Lelouche Lamperouge Jun 1 at 3:59

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