A particle is moving around in a circleand its position is given in po

1.	A particle is moving around in a circleand its position is given in polarcoordinates as x-RC059, andy Rsinwhere R is the radius of the circle, andθ is in radians. From these equationsderive the equation for centripetalacceleration.
Answer» 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.

A particle is moving around in a circleand its position is given in polarcoordinates as x-RC059, andy Rsinwhere R is the radius of the circle, andθ is in radians. From these equationsderive the equation for centripetalacceleration.

Answer»

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.

A particle is moving around in a circleand its position is given in polarcoordinates as x-RC059, andy Rsinwhere R is the radius of the circle, andθ is in radians. From these equationsderive the equation for centripetalacceleration.

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