Question

The potential energy of a body is given by \(\mathrm{U}=\mathrm{A}-\mathrm{Bx}^{2}\) (where \(\mathrm{x}\) is displacement). The magnitude of force acting on the particle is (A) constant (B) proportional to \(\mathrm{x}\) (C) proportional to \(\mathrm{x}^{2}\) (D) Inversely proportional to \(\mathrm{x}\)

Short answer

The magnitude of the force acting on the particle is proportional to x (Option B), as the force acting on the particle is given by \(F = 2Bx\), where F is the force, B is a constant, and x is the displacement.

Step-by-step solution

Write down the potential energy function

The given potential energy function is \(U = A - Bx^2\), where U is the potential energy, A and B are constants, and x is the displacement.

Differentiate the potential energy function with respect to x

We need to find the negative gradient of the potential energy function with respect to x. This is the same as finding the first derivative of the given function with respect to x and multiplying it by -1. Differentiate the function with respect to x: \( \frac{dU}{dx} = \frac{d(A - Bx^2)}{dx} \) Using the power rule, we get: \( \frac{dU}{dx} = -2Bx \) Now, multiply by -1: \( F = -\frac{dU}{dx} = 2Bx \)

Analyze the relationship between force and displacement

Now, we have the force acting on the particle: \(F = 2Bx\) Comparing this with the given options, we can eliminate options (A) and (C) since the force is neither constant nor proportional to x^2. Since the force is directly proportional to x, we can eliminate option (D) as well. Therefore, the correct answer is: (B) proportional to x

Key concepts

Differentiation

Differentiation is a core concept in calculus that helps us understand how functions change. It involves finding the derivative of a function, which gives us the rate at which the function's value changes with respect to a change in one of its variables. In the context of potential energy, differentiation allows us to find how the energy changes with displacement.
In the exercise, we are given a potential energy function: \[U = A - Bx^2\]where \(U\) is the potential energy, \(x\) is the displacement, and \(A\) and \(B\) are constants.
To find the force derived from this potential energy, we need the derivative of \(U\) with respect to \(x\). Using the power rule of differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\), we can find:

• The derivative of a constant is zero.
• The derivative of \(-Bx^2\) is \(-2Bx\).

Force-Displacement Relationship

The force-displacement relationship describes how the force acting on an object is related to its displacement. In many physics problems, especially those involving potential energy, understanding this relationship helps us predict how objects will move.
When dealing with potential energy functions, the force \(F\) acting on a particle is derived by finding the negative of the derivative of potential energy \(U\) with respect to displacement \(x\). For the given potential energy function:\[F = -\frac{dU}{dx}\]Substituting the derivative we found earlier, \(-2Bx\), we have:\[F = 2Bx\]This equation shows that the force is directly proportional to displacement \(x\), indicating that as the displacement increases, the force increases proportionately as well.
This is an important detail that allows us to conclude that option (B), proportional to \(x\), is the correct choice in the exercise.

Negative Gradient

The negative gradient is a concept that is essential when interpreting force from a potential energy function. In the physical world, forces often arise from the tendency of systems to move towards lower potential energy.
The gradient of a function provides the direction and rate of the steepest increase of that function. For potential energy, we often desire to know in which direction and how steep the slope is to understand how forces will act.

• The force is derived as the negative of the gradient (derivative) of potential energy.
• A negative gradient indicates that the force acts in the direction that reduces potential energy the fastest.

For the potential energy function in the exercise, the calculation:\[F = -\left(-2Bx\right) = 2Bx\]reveals that the negative gradient leads to a positive force, which aligns with the tendency to lower potential energy through movement. Thus, the concept of the negative gradient helps to understand how forces drive motion relative to energy landscapes.