Why does taking a long step increase the chance of slipping? Me and my friend were walking and it was raining. He didn't have any grip on the slippers so he took smaller steps to avoid slipping. We both were wondering why does taking a long step increase the chance of slipping on the slippers and not on a short step?
 A: You probably intuitively know the answer but just don't have the math, but it's because the angle relative to vertical your leg makes is smaller with small steps than with large steps. As a result, deviations in the angle your leg makes with the ground (ie, error in foot placement) correspond to changes in torque your leg needs to keep stable which are proportional to $\mbox{Sin}(\theta)$, where $\theta$ is the angle your leg makes with the normal to the ground (assuming flat ground). For small angles, this counterbalancing your brain and muscles need to make is proportional to the angle $\theta$. Thus placing your foot down with small steps make it easier to maintain balance. 
How does surface slipperiness come in to play? Consider what happens when you aren't able to provide that compensating torque. Your foot will have a force component which points horizontally (you will try to slide forward). If this force component is above the static friction threshold your foot makes with the ground, you slip. When the ground is icy, this static friction threshold is small, and so when you make a small mistake in foot placement and your foot begins to push horizontally, you slip. Note that the horizontal force component that a vector with an angle $\theta$ relative to surface normal is proportional to $\mbox{Sin}(\theta)$. Thus large steps both increase the likelihood that you will make a mistake, and that the mistake will result in slippage.
