In Circular Motion, Assuming V = W * R , Obtain an Expression for the Resultant Acceleration of a Particle in Terms of Tangential and Radial Component. - Physics

In circular motion, assuming bar v = bar w xx bar r , obtain an expression for the resultant acceleration of a particle in terms of tangential and radial component.

Solution

Acceleration of a particle,

a=lim_(deltat->0) ((delta v)/(delta t)) ... delta t ->0; delta t ne 0

therefore a=(dv)/(dt)

But, v = r omega

therefore a=d/dt (r omega)

=r (d omega)/dt + omega (dr)/(dt)

because "r is constant"

(dr)/(dt) = 0

therefore a=r (d omega)/(dt)

because (d omega)/(dt) = alpha

alpha = r alpha

Given that:

bar "v"= bar omega xx bar "r"

Differentiating w.r.t. time

(d bar v)/(dt)=d/dt (bar omega xx bar "r")

(d barv)/(dt) = (d baromega)/(dt) xx barr + baromega xx (d barr)/(dt)

(d barv)/(dt) = baralpha xx barr + baromega xx barv

therefore bara = bara_T + bara_r

Where,

a = Linear acceleration

aT = Tangential component of linear acceleration

ar = Radial component of linear acceleration

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