dv• Show that āN the normal and tangential components ofdtRacceleration.

Answered: dv • Show that ā =T+ N the normal and…

International Edition---engineering Mechanics: Statics, 4th Edition

4th Edition

ISBN:9781305501607

Author:Andrew Pytel And Jaan Kiusalaas

Publisher:Andrew Pytel And Jaan Kiusalaas

Chapter9: Moments And Products Of Inertia Of Areas

Section: Chapter Questions

Problem 9.27P

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Question

dv
• Show that ā
N the normal and tangential components of
dt
R
acceleration.

Expert Solution

Step 1

The diagram of the unit vectors of tangential and normal direction is given by,

The expression for unit vector of radial or normal direction is given by,

$\hat{N} = \cos θ \hat{i} + \sin θ \hat{j}$

The expression for unit vector of tangential direction is given by,

$\hat{T} = \sin θ (- \hat{i}) + \cos θ \hat{j}$

The diagram of the particle possess circular motion is given by,

Step 2

The expression for the vector of the radius of the about which particle is rotating is given by,

$\vec{r} = r \hat{N}$

Substitute all the known values in above equation,

$\vec{r} = r (\cos θ \hat{i} + \sin θ \hat{i})$

The expression for the normal velocity of the particle is given by,

$\vec{v} = \frac{d \vec{r}}{d t}$

Substitute all the known values in above equation,

$\vec{v} = \frac{d (r (\cos θ \hat{i} + \sin θ \hat{j}))}{d t} \vec{v} = r (\frac{- \sin θ d θ}{d t} \hat{i} + \frac{\cos θ d θ}{d t} \hat{j}) \vec{v} = r ω (- \sin θ \hat{i} + \cos θ \hat{i})$

Here, ω is the angular velocity of the particle.

Substitute the value of unit vector of tangential direction in above equation,

$\vec{v} = r ω (\hat{T})$

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dv • Show that ā =T+ N the normal and tangential components of dt R acceleration.