A steel spur pinion has a module (m) of 1.25 mm, 20 teeth and 20° pressure angle. The pinion rotates at 1200 rpm and transmits power to a 60 teeth gear. The face width (F) is 50 mm, Lewis form factor Y = 0.322 and a dynamic factor K_v = 1.26. The bending stress (σ) induced in a tooth can be calculated by using the Lewis formula given below.

If the maximum bending stress experienced by the pinion is 400 MPa, the power transmitted is ______ kW (round off to one decimal place).

Lewis formula : $\sigma =\frac{K_v{W}^t}{FmY},$ where W_t is the tangential load acting on the pinion.

The correct answer is 10.

Here is the step-by-step derivation and logical explanation of the solution:

1. Identify the given parameters:
- Module of the pinion, $m=1.25\text{mm}$
- Number of teeth on the pinion, $Z_p=20$
- Rotational speed of the pinion, $N=1200\text{rpm}$
- Face width, $F=50\text{mm}$
- Lewis form factor, $Y=0.322$
- Dynamic factor (or velocity factor), $K_v=1.26$
- Maximum bending stress induced, $\sigma =400\text{MPa}=400{\text{N/mm}}^2$

2. Calculate the Tangential Load ($W_t$):
We are given the Lewis formula:
$$\sigma =\frac{K_v{W}_t}{FmY}$$
Rearranging the formula to solve for the tangential load ($W_t$):
$$W_t=\frac{\sigma FmY}{K_v}$$
Substituting the given values into this equation:
$$W_t=\frac{400\times 50\times 1.25\times 0.322}{1.26}$$
Evaluating the numerator:
$$400\times 50\times 1.25\times 0.322=8050$$
Thus, the tangential force is:
$$W_t=\frac{8050}{1.26}\approx 6388.89\text{N}$$

3. Calculate the pitch circle velocity ($v$) of the pinion:
First, find the pitch circle diameter of the pinion ($D_p$):
$$D_p=m\times Z_p=1.25\text{mm}\times 20=25\text{mm}=0.025\text{m}$$
The linear velocity ($v$) is given by:
$$v=\frac{\pi D_{p}N}{60}$$
Substituting $D_p=0.025\text{m}$ and $N=1200\text{rpm}$:
$$v=\frac{\pi \times 0.025\times 1200}{60}=0.5\pi \approx 1.5708\text{m/s}$$

4. Calculate the Power Transmitted ($P$):
The power transmitted can be calculated by:
$$P=W_t\times v$$
Substituting our calculated values:
$$P=6388.89\text{N}\times 1.5708\text{m/s}\approx 10035.7\text{W}$$
Converting the power into kilowatts (kW):
$$P=\frac{10035.7}{1000}\approx 10.0\text{kW}$$
Rounding off to one decimal place, we obtain 10 kW.