Does turning a spoon in water raise the temperature? I read about Joule's experiment proving the transformation of mechanical work into heat. But say I have a bowl with some water, and I start turning a spoon in it very fast, thus doing work — the water won't get hotter! What am I missing?
I think maybe the work I put is simply kinetic, and won't turn into heat. But then how do you explain Joule's experiment?
 A: One of the reasons that makes you not to believe that one cannot heat up water by stirring it, might be that we usually experience the opposite effect. Namely, one usually stirs a hot tea or soup to cool it down. Why a cup of hot tea or a bowl of warm soup cools down when one stirs it? The reason is that the liquid/air interface where the heat exchange occurs is increased by stirring and therefore the warm liquid cools faster. The other contribution comes from the spoon which is usually a metal and sucks the heat fast from the liquid and dissipates it to air or to one's fingers.
The other reason that makes you not to believe might be that you have never experienced a glass of water warmed up because of stirring. There are couple of reasons why we normally do not experience that. The most obvious one is that the increase in temperature that one can induce is way too lower than our senses can detect. The other reason might be that if one does such an experiment in an ordinary glass the added heat dissipates so fast that one never observes the increase in temperature.
I do not know where do you live but never you rubbed your hands to each other to heat them up in a cold winter day?
A: Do the maths and calculate how much energy is needed to raise the water's temperature by 1K. If you have a fast moving stirrer, you should be able to measure the increase in temperature of a liquid in an isolated pot. 
By the way: the microwaves in your microwave oven turn around the water molecules very fast and heat up your food this way.
A: Well first you have the energy in the form of kinetic energy of the spinning water. Once you let that water settle, it DOES get hotter. 
The only problem is that water has a high specific heat (it takes a LOT of energy to heat up water), so you don't notice the water getting hotter since the amount it's heating up is not very noticeable. Coincidentally, it is this property of water that makes the earth a habitable planet--we have moderate temperatures compared to other planets because our oceans, bays, and lakes can absorb or release large amounts of heat to moderate the atmospheric temperatures.
If you want a more observable experiment, try taking a piece of metal (maybe a paper clip?) and bending it back and forth a lot of times. Although it'll eventually break, you should be able to notice it getting hotter
A: While not exactly the same as your example using water, try this.
Put a few scoops of ice cream in a bowl (preferably a plastic bowl). If you use a glass or ceramic or metal bowl that was sitting at room temperature, you will see the outside edges of the ice cream begin to melt quite quickly. Using a plastic bowl will significantly lessen that effect.
Now, if you did nothing else, the ice cream would eventually melt on its own. Well, of course, not actually "on its own", but rather from the surrounding air that is at a higher temperature. But, that would take a while.
As a "control", before you start stirring the ice cream, put about the same amount of ice cream in a separate plastic bowl that you will just leave undisturbed.
Now, take the first bowl of ice cream and a spoon, and break up the ice cream a bit then begin to stir it rapidly. You will notice that it will fairly quickly soften considerably, and within a couple of minutes, you will see that a significant amount of the ice cream will be melted.
Compare it to the "undisturbed" ice cream to see the effect.
I don't know, and I'm sure others will correct me if I'm wrong, but I assume that the specific heat of the fats, sugars, and other solids in the ice cream is lower than the specific heat of water, which causes the the temperature of the ice cream to increase quicker.
