My instructor makes a lot of use of the terms cadence and tempo in the spinning class, but I have not been able to get a satisfactory answer on what the difference between the two is. What is the difference between the two words?
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You would have to ask your instructor to clarify for specifics.
Cadence is used to count the number of times one foot rotates a complete circle around the crankset. So if you are counting the right foot, to go from top to bottom to top is one. Count for a minute, and you get your cadence.
Tempo is very often used as a measure of intensity, especially when combined with devices that measure power, or when compared to a race pace. Something that is done "at tempo" is near to the same pace you would use when racing a particular distance.
However, your instructor may be using them interchangeably or with different meanings, best is to ask him/her for an explanation.
Note: This is also the same terminology used in running, you will hear references to running cadence and tempo workouts.
I agree with JohnP's answer. I am also a fitness instructor, and I've been teaching indoor cycling classes for years. For me, when I say tempo (which isn't often), I tend to use these term interchangeably in my class, but that's because I always explicitly let them know what zone or intensity level I want them to target, and refer to that intensity in another manner.
You really need to ask the instructor to clarify his/her meaning - if they can't, that might be a sign this instructor really needs to do some more research into what he/she is talking about. Tempo is commonly used as another term for pace, whether it be running or cycling, but there tends to be an associated intensity with it (e.g. a run at 10k tempo pace or a 2-hour ride at an endurance tempo).
If your instructor is a cyclist, I have a possible explanation:
In cycling, "tempo" is a pace where you are working hard but not so hard that you will be forced to stop soon. It's roughly correlated to zone 3 and pretty close to your lactate threshold.