Question

The speed of a particle moving in a circle 2m in a radius increases at the constant rate of 3m/s^2 At some instant, the magnitude of the total acceleration is 5m/s^2 At this this instant. Find (1) the centripetal acceleration of the particle and (2) its speed.

Answer 1

$\huge\mathbb{\purple{QUESTION-}}$

The speed of a particle moving in a circle 2m in a radius increases at the constant rate of 3m/s² At some instant, the magnitude of the total acceleration is 5m/s² At this this instant. Find

(1) the centripetal acceleration of the particle and

(2) its speed.

$\huge\mathbb{\purple{SOLUTION-}}$

$\large\underline{\underline{\sf Given:}}$

• Radius of circle (r) = 2m
• Tangential Acceleration ${\sf a_t}$=3m/s²
• Net Acceleration ${\sf a_{net}}$ = 5m/s²

$\large\underline{\underline{\sf To\:Find:}}$

• Centripetal Acceleration ${\sf (a_c)}$
• Speed (v)

✪ Centripetal Acceleration ➝

✪$\large{\boxed{\bf \blue{a_{net}=\sqrt{a_t^2+a_c^2}} }}$

$\implies{\sf a_c^2=a_{net}^2-a_t^2}$

$\implies{\sf 5^2-3^2 }$

$\implies{\sf 25-9}$

$\implies{\sf 16 }$

$\implies{\bf \red{a_c=16\:m/s^2} }$

✪ Speed (v) ➝

✪$\large{\boxed{\bf \blue{ a_c=\dfrac{v^2}{r} }}}$

$\implies{\sf v^2=4×2 }$

$\implies{\sf v^2=8 }$

$\implies{\bf \red{v=2.82\:m/s }}$

$\huge\mathbb{\purple{ANSWER-}}$

Centripetal Acceleration ${\bf \red{4\:m/s^2}}$

Speed ${\bf \red{2.82\:m/s}}$