A force F=a+bx acts on a particle in the x-directioin, where a and b are constants. Find the work done by this force during a displacement form `x=0 to x=d`.

Explanation

To find the work done by the variable force \( F = a + bx \) during a displacement from \( x = 0 \) to \( x = d \), we will follow these steps:

### Step 1: Understand the Work Done Formula
The work done \( W \) by a force when it varies with position is given by the integral of the force over the displacement:
\[ W = \int_{x_1}^{x_2} F \, dx \]
where \( x_1 \) and \( x_2 \) are the initial and final positions, respectively.

### Step 2: Substitute the Force Function
In this case, the force \( F \) is given as:
\[ F = a + bx \]
So, we can write the work done as:
\[ W = \int_{0}^{d} (a + bx) \, dx \]

### Step 3: Split the Integral
We can split the integral into two parts:
\[ W = \int_{0}^{d} a \, dx + \int_{0}^{d} bx \, dx \]

### Step 4: Calculate Each Integral
1. **First Integral**:
\[ \int_{0}^{d} a \, dx = a \int_{0}^{d} 1 \, dx = a[x]_{0}^{d} = a(d - 0) = ad \]

2. **Second Integral**:
\[ \int_{0}^{d} bx \, dx = b \int_{0}^{d} x \, dx = b\left[\frac{x^2}{2}\right]_{0}^{d} = b\left(\frac{d^2}{2} - 0\right) = \frac{bd^2}{2} \]

### Step 5: Combine the Results
Now, we combine the results from both integrals:
\[ W = ad + \frac{bd^2}{2} \]

### Final Result
Thus, the work done by the force \( F \) during the displacement from \( x = 0 \) to \( x = d \) is:
\[ W = ad + \frac{bd^2}{2} \]

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