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for a project I need to create a launching mechanism which operates based on gravity. After some initial designing I have come up with a design where a counterweight (250g) is dropped about 85cm, which is attached to a set of gears. These gears then spin and cause a block to strike a small ball to be launched at a 45 degree angle. The specifics of the design aren't that important, the basic principle is that a projectile is launched by means of a block colliding with it, which is connected to some gears. This brings me to the question:

Is there a specific Torque/RPM ratio that is preferable here? As in, would it be feasible to simply max out RPM to move the block as fast as possible? Or is a balance necessary.

Maxing out the RPMs is the idea that came to me initially, but I'm not sure if its the best move, which is why I'm asking here. Please do not judge me on this question, I simply want to learn :).

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  • $\begingroup$ Balance is always necessary $\endgroup$
    – DKNguyen
    Commented Dec 6, 2021 at 17:46
  • $\begingroup$ Could you specify how I could find a balance / "Sweet Spot" that is specific to this scenario? I've tried online research but I can only seem to find information relevant to cars. $\endgroup$
    – Pranav
    Commented Dec 6, 2021 at 17:52
  • $\begingroup$ Off the top of my head, calculate the energy in the dropping weight. Then transfer that energy to the projectile using 1/2mv^2 to find out what the projectile v should be which could give you the speed that your hammer needs to be. However, this doesn't take into account things like impulse, contact time, and acceleration. $\endgroup$
    – DKNguyen
    Commented Dec 6, 2021 at 17:58

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First off, because the energy is transferred with an impact, which happens at near zero time, the torque applied to the gears does not affect the outcome of the impact. Only the kinetic energy of the device at the moment of impact gives us the "potential" of the device.

The kinetic energy stored at the moment if impact is calculated with $$ KE = \tfrac{1}{2} \left( I_{\rm gear} + m_{\rm block} r^2\right) \omega^2 $$

Where $m_{\rm block}$ is the mass of the block, $I_{\rm gear}$ is the mass moment of inertia of the geartrain at the output axle, $r$ is the radius the block is located, and $\omega$ is the spin rate of the gears.

In more detail though, the ideal device would transfer the entirety of the kinetic energy in the gears into the payload and the gear would stop after the impact.

If only the block carried kinetic energy then this would happen only if $$ m_{\rm block} = m_{\rm payload} $$

But because the block is part of a spinning system you need to calculate the effective mass of the geartrain at the block to dynamically balance the system.

This is done with the following calculation of the effective mass of the block + gear

$$ \frac{1}{ \frac{1}{m_{\rm block}} + \frac{r^2}{I_{\rm gear}} } = m_{\rm payload} $$

This results in the ideal placement of the block, the distance $r$

$$ r = \sqrt{ \left( I_{\rm gear} \left( \tfrac{1}{m_{\rm payload}} - \tfrac{1}{m_{\rm block}} \right) \right)} $$

This creates the strict constraint that $m_{\rm payload} < m_{\rm block}$ so that $r$ is not an imaginary number.

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