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The potential energy for a conservative ...

The potential energy for a conservative force system is given by `U=ax^(2)-bx`. Where a and b are constants find out (a) The expression of force (b) Potential energy at equilibrium.

A

F = constant

B

F = bx - 2a

C

F = b - 2ax

D

F = 2 ax

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will find the expression for force and then determine the potential energy at equilibrium. ### Step 1: Find the expression for force The potential energy \( U \) is given by the equation: \[ U = ax^2 - bx \] To find the force \( F \), we use the relationship between force and potential energy: \[ F = -\frac{dU}{dx} \] Now, we differentiate \( U \) with respect to \( x \): \[ \frac{dU}{dx} = \frac{d}{dx}(ax^2 - bx) = 2ax - b \] Thus, the expression for force becomes: \[ F = -\left(2ax - b\right) = b - 2ax \] ### Step 2: Find the equilibrium position At equilibrium, the force \( F \) is equal to zero: \[ b - 2ax = 0 \] Solving for \( x \): \[ 2ax = b \implies x = \frac{b}{2a} \] ### Step 3: Calculate potential energy at equilibrium Now, we need to find the potential energy \( U \) at the equilibrium position \( x = \frac{b}{2a} \): \[ U\left(\frac{b}{2a}\right) = a\left(\frac{b}{2a}\right)^2 - b\left(\frac{b}{2a}\right) \] Calculating \( U \): \[ U\left(\frac{b}{2a}\right) = a\left(\frac{b^2}{4a^2}\right) - \frac{b^2}{2a} \] This simplifies to: \[ U\left(\frac{b}{2a}\right) = \frac{ab^2}{4a^2} - \frac{b^2}{2a} \] Now, we can combine the terms: \[ U\left(\frac{b}{2a}\right) = \frac{b^2}{4a} - \frac{2b^2}{4a} = -\frac{b^2}{4a} \] ### Final Answers (a) The expression for force is: \[ F = b - 2ax \] (b) The potential energy at equilibrium is: \[ U = -\frac{b^2}{4a} \]
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