When one exposes theirs body parts to cold environment (especially foot and hands), they may be numb, with temporarily blocked both muscles.
What is typical body temperature below which human muscles don't work properly?
Moreover, what is the mechanism behind it?
EDIT:
Interested in case when muscles almost cease to work (e.g one is not able to grasp a cup), not just in small perturbations to strength.
If you dissect striated muscle out of most-any organism, the actual contractile apparatus works over a wide range of temperatures. So that's at the single-muscle-fiber scale. The muscle itself continues to work at all (thawed) temperatures below body temperature -- the problem comes with its regulation.
The shivering response -- a centrally controlled involuntary contractile pattern -- overwhelms voluntary muscle control. So the mechanism behind loss of muscle coordination in hypothermia is the same as the shivering mechanism. When the core temperature drops, the midbrain starts over-riding voluntary control of the muscles. When the core temperature drops far enough; around 32C, the shivering often slows down or stops. Voluntary movement becomes compromised probably because the brain simply isn't working; neuron firing rates are so slow that sensation, processing, and motor responses are all critically impaired.
The feeling of numbness does not actually directly accompany a loss of muscle contractility. You can walk pretty much indefinitely on frozen feet if you can keep your balance (and you keep your core temperature up.) Lots of people survive severe frostbite of their feet (their feet do not often survive, however.) The reason why it seems your hands don't work when they get cold is that you can't feel what you're doing (note; your hands can be much colder than your core body temperature.) But the muscles themselves work right up until they freeze solid.
UPDATE: Here's a paper that directly addresses the scenario posed by OP -- the decrease in grip strength with temperature. Figure 1 of that paper illustrates their experimental setup; they measure the contractile strength of the index finger while manipulating the temperature of the rest of the hand. They show that contractile function is impaired with temperature and look at temperatures as low as 12C.
They measure as much as a 50% impairment on twitch tension upon cooling to 12C. It's interesting that they review results suggesting that some of this effect is intrinsic to the muscle fiber (not neurological), showing that I should refine what is meant by "continuing to work" in my opening paragraph. (I meant having an ability to generate contractile force when equilibrated in solution containing sufficient ATP and Ca$^{2+}$, not the ability to contract optimally.) For fun, I linearly extrapolated the final arm of Figure 5 and found that the 'voluntary tension' approached 25% at 5C. This suggests that total failure of the voluntary contraction happens somewhere below the freezing point of water (muscle would freeze at a temperature lower than 0C because of colligative effects.)
This seems complicated, but I found a reference that might answer these questions (if its right).
Sports physiology common wisdom is that human muscles regularly work in the range of 37C to 40C which is basically body temperature. Since the muscles generate quite a bit of heath when they function, they are usually not functioning in colder temperatures and hyperthermia starts setting in.
Thomas et al. put this to the test by cooling down some muscles in volunteers! They say that the core body temperature is more important than the peripheral temperature in restoring muscle strength (torque produced). Hypothermia makes the victim weaker and weaker apparently and it sets in below 37C pretty quickly apparently. They speculate that the cooling might actually impair the nervous system from activating the muscles, which might be an answer to your question.
This is probably not the final word, but its as far as I got...
