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Physics Problem Juan posted on 2017-02-22 14:58:53 |
"A particle 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. " Does someone know how to solve? |
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Replied by Amanda on 2017-02-23 14:46:52 |
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. |