The potential energy of a conservative system is given by U=`ay^(2)`-by,where y represents the position of the particle and a as well b are constants.What is the force acting on the system?

Options

A. `-ay`
B. `-by`
C. `2ay-b`
D. `b-2ay`

Explanation

To find the force acting on the system given the potential energy \( U = ay^2 - by \), we can follow these steps:

### Step 1: Understand the relationship between force and potential energy
In a conservative system, the force \( F \) can be determined from the potential energy \( U \) using the formula:
\[ F = -\frac{dU}{dy} \]
This means that the force is the negative gradient (or derivative) of the potential energy with respect to position.

### Step 2: Differentiate the potential energy with respect to \( y \)
Given the potential energy function:
\[ U = ay^2 - by \]
we need to differentiate \( U \) with respect to \( y \):
\[ \frac{dU}{dy} = \frac{d}{dy}(ay^2 - by) \]

### Step 3: Apply the differentiation
Using the power rule of differentiation:
- The derivative of \( ay^2 \) is \( 2ay \).
- The derivative of \( -by \) is \( -b \).

Thus, we have:
\[ \frac{dU}{dy} = 2ay - b \]

### Step 4: Substitute into the force equation
Now, substitute \( \frac{dU}{dy} \) back into the force equation:
\[ F = -\frac{dU}{dy} = - (2ay - b) \]
This simplifies to:
\[ F = -2ay + b \]

### Step 5: Rearrange the expression for clarity
Rearranging the expression gives:
\[ F = b - 2ay \]

### Final Answer
Thus, the force acting on the system is:
\[ F = b - 2ay \]