Question

Q.] The potential energy for a conservative force system is given by U = ax² - bx. Where a and b are constants find out (a) The expression of force (b) Potential energy at equilibrium.
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Answer 1

★ GivEn ThaT :-

• Potential Energy for a conservative force system is given by U = ax² - bx
• Here, a and b are the constants

★ To FinD :-

• (a) The expression of Force
• (b) Potential Energy at Equilibrium

★ As We KnOw ThaT :-

$\sf\red{Force, (F)=− \frac{dU}{dx} }$

$\sf\red{Equilibrium \: position \: is \: where \: F=0}$

★ SoLuTioN :-

$\sf\large{(a) \: The \: Expression \: of \: the \: Force }$

$\sf{= - \frac{d}{dx}(a {x}^{2} - bx)}$

$\sf{= - \frac{d}{dx}(a {x}^{2} )} + \frac{d}{dx} (bx)$

$\sf{= - 2ax + b}$

$\sf\underline\pink{= \: b - 2ax \: }$

(b) Now, at Equilibrium F = 0

$\sf{\therefore \: b - 2ax = 0}$

$\sf{ b = 2ax }$

$\sf{ x = \frac{b}{2a}}$

Then,

$\sf{The \: Equilibrium \: occurs \: at \: x = \frac{b}{2a} }$

$\sf{Potential \: energy \: at \: Equilibrium, U_{eq} = a{( \frac{b}{2a})}^{2} - b( \frac{b}{2a} )}$

$\sf{U_{eq} = ( \frac{ {b}^{2} }{4a}) - ( \frac{ {b}^{2} }{2a} )}$

$\sf{U_{eq} = \frac{ {b}^{2} }{4a} - \frac{ {2b}^{2} }{4a} }$

$\sf{U_{eq} = \frac{ { {b}^{2} - 2b}^{2} }{4a} }$

$\sf\underline\pink{\therefore \: U_{eq} = - \frac{ {b}^{2} }{4a} \: }$

Which are your required answers!!

Answer 2

♠︎★★★I hope it's helpful for you dear

be brainly#