A spur gear with 20° full depth teeth is transmitting 20 kW at 200 rad/s. The pitch circle diameter of the gear is 100 mm. The magnitude of the force applied on the gear in the radial direction is

To find the magnitude of the force applied on the gear in the radial direction, we can break down the calculation into three main steps: finding the torque transmitted, determining the tangential force on the pitch circle, and then calculating the radial force using the pressure angle.

Step 1: Calculate the transmitted torque ($T$)
The relation between power ($P$), torque ($T$), and angular velocity ($\omega$) is given by:
$P=T·\omega$
Given:
Power, $P=20\text{kW}=20000\text{W}$
Angular velocity, $\omega =200\text{rad/s}$
Substituting these values:
$T=\frac{P}{\omega }=\frac{20000}{200}=100\text{N·m}$

Step 2: Calculate the tangential force ($F_t$)
The tangential force acting on the gear teeth at the pitch circle radius is related to torque by:
$T=F_t·r$
where $r$ is the pitch circle radius.
Given the pitch circle diameter $d=100\text{mm}=0.1\text{m}$, the radius is:
$r=\frac{d}{2}=50\text{mm}=0.05\text{m}$
Now, solve for tangential force ($F_t$):
$F_t=\frac{T}{r}=\frac{100}{0.05}=2000\text{N}=2\text{kN}$

Step 3: Calculate the radial force ($F_r$)
The radial force is related to the tangential force by the pressure angle ($\varphi$):
$F_r=F_t·tan\left(\varphi \right)$
Given the pressure angle for full depth teeth $\varphi =20°$:
$F_r=2\text{kN}·tan(20°)$
Using $tan(20°)\approx 0.364$:
$F_r\approx 2\text{kN}·0.364=0.728\text{kN}\approx 0.73\text{kN}$

Therefore, the magnitude of the force applied on the gear in the radial direction is 0.73 kN.