The potential energy of a body mass m is `U=ax+by` the magnitude of acceleration of the body will be-

Options

A. `(ab)/(m)`
B. `((a+b)/(m))`
C. `(sqrt(a^(2)+b^(2)))/(m)`
D. `(a^(2)+b^(2))/(m)`

Answer

To find the magnitude of the acceleration of a body with a given potential energy \( U = ax + by \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Potential Energy Function**:
The potential energy \( U \) is given as:
\[ U = ax + by \]
where \( a \) and \( b \) are constants, and \( x \) and \( y \) are the coordinates of the body.

2. **Calculate the Force from Potential Energy**:
The force \( \vec{F} \) in a conservative field can be derived from the potential energy using the relation:
\[ \vec{F} = -\nabla U \]
where \( \nabla U \) is the gradient of \( U \).

3. **Compute the Gradient**:
The gradient in two dimensions is given by:
\[ \nabla U = \left( \frac{\partial U}{\partial x}, \frac{\partial U}{\partial y} \right) \]
We calculate the partial derivatives:
- For \( x \):
\[ \frac{\partial U}{\partial x} = a \]
- For \( y \):
\[ \frac{\partial U}{\partial y} = b \]

4. **Express the Force Vector**:
Therefore, the force vector \( \vec{F} \) becomes:
\[ \vec{F} = -\left( a \hat{i} + b \hat{j} \right) = -a \hat{i} - b \hat{j} \]

5. **Calculate the Magnitude of the Force**:
The magnitude of the force \( F \) is given by:
\[ F = \sqrt{(-a)^2 + (-b)^2} = \sqrt{a^2 + b^2} \]

6. **Apply Newton's Second Law**:
According to Newton's second law, the acceleration \( \vec{a} \) is given by:
\[ \vec{a} = \frac{\vec{F}}{m} \]
Therefore, the magnitude of the acceleration \( a \) is:
\[ a = \frac{F}{m} = \frac{\sqrt{a^2 + b^2}}{m} \]

### Final Result:
The magnitude of the acceleration of the body is:
\[ a = \frac{\sqrt{a^2 + b^2}}{m} \]