Question

Two mating spur gears have 36 and 108 teeth, respectively. The pinion rotates at 1200 rpm and transmits a torque of 20 Nm. The torque transmitted by the gear is

• A. 10 Nm
• B. 20 Nm
• C. 40 Nm
• D. 60 Nm

Hint:

The torque transmitted by mating gears is proportional to the gear ratio, which is the ratio of the number of teeth.

Answer 1

The Correct Option is D

Step 1: Understand the gear system.
Two mating spur gears have 36 teeth (pinion) and 108 teeth (gear). The pinion rotates at 1200 rpm and transmits a torque of 20 Nm. We need to find the torque transmitted by the gear. Step 2: Determine the gear ratio.
The gear ratio is the ratio of the number of teeth on the gear to the number of teeth on the pinion: \[ \text{Gear ratio} = \frac{\text{Number of teeth on gear}}{\text{Number of teeth on pinion}} = \frac{108}{36} = 3. \] Step 3: Relate the torque using the gear ratio.
In a gear system, the power transmitted is the same (assuming no losses): \[ \text{Power} = \text{Torque} \times \text{Angular velocity}. \] Torque on pinion \( T_p = 20 \, \text{Nm} \),
Speed of pinion \( N_p = 1200 \, \text{rpm} \).
The angular velocity \( \omega \) is: \[ \omega_p = \frac{2\pi N_p}{60} = \frac{2\pi \times 1200}{60} = 40\pi \, \text{rad/s}. \] Power transmitted by the pinion: \[ P = T_p \omega_p = 20 \times 40\pi = 800\pi \, \text{W}. \] The gear ratio also gives the speed ratio (inverse of gear ratio): \[ \frac{N_p}{N_g} = \frac{T_g}{T_p} = 3, \] \[ N_g = \frac{N_p}{3} = \frac{1200}{3} = 400 \, \text{rpm}, \] \[ \omega_g = \frac{2\pi N_g}{60} = \frac{2\pi \times 400}{60} = \frac{40\pi}{3} \, \text{rad/s}. \] Power transmitted by the gear: \[ P = T_g \omega_g, \] \[ 800\pi = T_g \times \frac{40\pi}{3}, \] \[ T_g = \frac{800\pi}{\frac{40\pi}{3}} = 800 \times \frac{3}{40} = 60 \, \text{Nm}. \] Alternatively, use the torque ratio directly: \[ \frac{T_g}{T_p} = \text{Gear ratio}, \] \[ T_g = T_p \times \text{Gear ratio} = 20 \times 3 = 60 \, \text{Nm}. \] Step 4: Select the correct answer.
The torque transmitted by the gear is 60 Nm, matching option (4).