Aparticle is moving around in a circle and its position is given in polar coordinates as x = rcosθ, and y = rsinθ, where r is the radius of the circle, and θ is in radians. from these equations derive the equation for centripetal acceleration.

“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

answer; ///(74) degrees is your answer..