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Chapter 5: Problem 10

Verify that all three expressions for radial acceleration $\left(v \omega, v^{2} / r, \text { and } \omega^{2} r\right)$ have the correct dimensions for an acceleration.

Short Answer

Expert verified
Answer: Yes, all three expressions vω, v²/r, and ω²r have the correct dimensions for an acceleration, which is LT⁻².

Step by step solution

01

a) \(v\omega\) (Velocity x Angular velocity)

: To find the dimensions of \(v\omega\), we simply multiply the dimensions of velocity and angular velocity: Dimension of \(v\omega = [V][ω] = (LT^{-1})(T^{-1}) = LT^{-2}\)
02

b) \(v^2 /r\) (Velocity^2 / Radius)

: To find the dimensions of \(v^2 /r\), we square the dimensions of velocity and divide it by the dimensions of radius: Dimension of \(v^2 /r = [V]^2 /[r] = (LT^{-1})^2 / L = L^2T^{-2} / L = LT^{-2}\)
03

c) \(\omega^2r\) (Angular velocity^2 x Radius)

: To find the dimensions of \(\omega^2r\), we square the dimensions of angular velocity and multiply it with the dimensions of radius: Dimension of \(\omega^2r = [ω]^2[r] = (T^{-1})^2 L = T^{-2}L = LT^{-2}\) As we can see, all three expressions \(v\omega,\) \(v^2 /r,\) and \(\omega^2r\) have the dimensions \(LT^{-2}\), which is the same as the dimension of acceleration. Therefore, we can verify that all three expressions for radial acceleration have the correct dimensions for an acceleration.

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