“Without loss of generality, we only need to look at the equation for the x-position, since we know that centripetal acceleration points towards the center of the circle. Thus, when θ = 0, the second derivative of x with respect to time must be the centripetal acceleration.
The first derivative of x with respect to time t is:
dx/dt = —Rsinθ(dθ/dt)
The second derivative of x with respect to time t is:
d2x/dt2 = —Rcosθ(dθ/dt)2—Rsinθ(d2θ/dt2)
In both of the above equations the chain rule of Calculus is used and by assumption θ is a function of time. Therefore, θ can be differentiated with respect to time.
Now, evaluate the second derivative at θ = 0.
We have,
d2x/dt2 = —R(dθ/dt)2
The term dθ/dt is usually called the angular velocity, which is the rate of change of the angle θ. It has units of radians/second.
For convenience we can set w ≡ dθ/dt.
Therefore,
d2x/dt2 = —Rw2
This is the well-known form for the centripetal acceleration equation.”
-Leah
