Question

Derive expression for linear acceleration in uniform circular
motion​

Answer 1

Answer:-

As you know,

$ds = d \ \beta \times r$

(Beta is angular displacement).

Differentiating w.r.t time

$\frac{ds}{dt} = \frac{d \ \beta }{dt } \times r$

$v = w \times r$

This the relation between Linear speed and angular speed.

Now,

again Differentiating w.r.t time

$\frac{dv}{dt } = \frac{dw}{dt} \times r$ + w × dr/dt

(following the chain rule in differentiating)

[$a = \alpha \times r$ + w × v]

Expression for Linear acceleration in uniform circular motion.

Note:-

$a \: represents \: linear \: acceleration$

$w \: reprents \: angular \: velocity$

$\alpha \: represents \: angular \: acceleration$

$v \: represents \: linear \: velocity$

$dt \: represents \: small \: time \: interval$

$\beta \: represent \: angular \: displacement$