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

in simple terms, the principle states that the buoyancy force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational acceleration, g. this is also known as upthrust.

no

sodium is solid at rt