# Calculating the Necessary Stiffness of a Tripod

Stiffer tripods tend to be larger, heavier and more expensive than their smaller, lighter, and more affordable less stiff counterparts.  We therefore don’t want a high end systematic tripod just to support a mirror-less camera and wide angle lens on a hiking trip.  Neither though do we want the photos from that trip to come back blurry because our tripod was not stiff enough for the job.  Or perhaps you are just starting out photographing wildlife.  You need a tripod that is stable enough for your new long focal length lens, but won’t break the bank.  So how stiff of a tripod do you need?

To answer that question, we are going to be working in the torsion spring framework.  Briefly, this means that the tripod behaves like a rotating spring, returning the camera to where it was originally pointing when subjected to some external forces, or more precisely, torques.  For critical sharpness, lets say that the image cannot move by more than one pixel width while the exposure is taken.  One pixel’s width worth of movement still causes a detectable loss of sharpness, but its a good practical line at which there becomes minimal loss of sharpness to the human eye.  Using the small angle approximation, one pixel width ($d$) is equivalent to an angle in radians ($\theta$) given by:

$\theta^c = d/f$

where $f$ is the focal length of the lens being used.  $\theta^c$ is the critical angle for adequate image sharpness.

When the tripod or attached camera is subjected to some torque $\tau$, the pointing of the camera will move by an angle given by:

$\kappa\theta = \tau$

where $\kappa$ is the stiffness (or spring) constant of the tripod in question.  Note here that I am making an important assumption, that the torque applied is relatively slow and constant.  Given that the natural oscillation frequency of most camera/tripod setups will be in the tens of Hz, and most external torques such as from a hand or the wind will be much slower, this is reasonable.  For other very fast excitations, such as from the camera’s shutter or mirror, this is not a reasonable assumption to make.  Combining these two expressions, we get:

$\kappa^c = \tau f/d$

where $\kappa^c$ is the amount of stiffness necessary to obtain critical sharpness.  Intuitively this all makes perfect sense.  More torque, longer focal length, and smaller pixels all necessitate having a stiffer tripod.  This expression is relatively simple to compute, with the very notable exception of the amount of torque that the tripod is expected to receive.  This torque will be the subject of future posts as I take measurements of how much torque various events exert.