By changing the length of your grip, you change the moment of inertia of the badminton racket.
The Wikipedia article on MOI says: The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis.
Your hand position is the (approximate) axis of rotation. When you use a shorter grip, you are repositioning the axis of rotation. Using a shorter grip reduces the moment of inertia.
A lower MOI means less difficulty in changing the angular motion. In other words, a shorter grip lets you get power from a short swing: it takes less effort to get the racket moving
. Because it’s easier to get the racket moving, you also experience a sense of increased control: there is less drag
from the racket head.
This does not imply that short grip = more power
. Angular momentum of the racket is converted into linear momentum of the shuttlecock: it’s not only about how fast the racket is rotating, but also about how long the lever is.
When two racket swings have the same angular velocity, the one with the longer grip has more angular momentum. Consequently, the one with the longer grip will make the shuttlecock fly faster.
Here’s another way to think about it: in order to achieve the same shuttlecock speed, you need to make a faster swing with a short grip. A long grip can achieve the same result with a slower swing.
Biomechanical limits
The problem with short grips is that you cannot keep making the swing faster. As well as the physical properties of the racket, we have to consider the physical capabilities of the human body.
There is a limit to how fast your arm can swing. This means that you cannot fully compensate for using a short grip: you will not be able to reach the required arm speed to match the power of a long grip.
It’s a compromise: short grips give you more immediate power, but long grips give you greater maximum power. A similar compromise exists when comparing rackets with different weight distributions: a heady-heavy racket can potentially offer more power, but it doesn’t offer as much immediate power as an evenly balanced racket.