--- title: "Differential gear action in an automobile" subject: "Machine Design II" chapter: 5 key: "semester_7/machine_design_ii:automobile-differential-gear-action" source: https://www.bctengineer.com/q/semester_7/machine_design_ii/automobile-differential-gear-action site: BCT Engineering --- # Differential gear action in an automobile Machine Design II · Chapter 5: System Design - Asked in: year not given, 5 marks - Video: [Differential explained - How differential works open, limited slip](https://www.youtube.com/watch?v=VN4_asSBEKY) — The Engineering Mindset - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/machine_design_ii/automobile-differential-gear-action - Whole chapter: https://www.bctengineer.com/md/semester_7/machine_design_ii/chapter-5 ## Question Explain the need, construction and working of the differential of an automobile. Derive the relation between the speeds of the two axle shafts and the crown wheel. In a differential, the pinion has 10 teeth and the crown wheel 40 teeth. The pinion runs at 1800 rpm and the left wheel runs at 400 rpm while turning a curve; find the speed of the right wheel. ## Short answer The differential permits different wheel speeds when turning to avoid tyre scrub. It has a crown wheel carrying a cage with spider pinions meshed with two side gears on the axles. Straight ahead, all rotate together; in a turn, the pinions spin, giving N_L + N_R = 2N_c. With pinion 10T, crown 40T, pinion 1800 rpm, cage speed = 450 rpm, right wheel speed = 500 rpm. ## Answer #### Need When a vehicle turns, the outer wheel travels a longer path than the inner wheel, so it must rotate faster. If both rear wheels were rigidly fixed to one shaft, one would slip and wear the tyres. A **differential** lets the wheels turn at different speeds while still sharing the driving torque equally. #### Construction ```text Prop shaft ---> Pinion | Crown wheel (carries the cage) | Left axle - Side gear \ / Side gear - Right axle Spider (planet) pinions ``` The drive pinion meshes with the crown wheel. The crown wheel is bolted to the differential cage, which carries the spider shaft and two or four **planet pinions**. The planets mesh with two **side (sun) gears** splined to the left and right axle shafts. #### Working - Straight road: both wheels meet equal resistance, the planets do not spin on their own axes and the cage and both side gears rotate together. - Turning: the inner wheel resists more, so the planets rotate about the spider axis, speeding up one side gear and slowing the other by the same amount. - Torque is equal on both axles in a bevel-gear differential (a drawback on slippery surfaces, which limited-slip or locking differentials overcome). #### Speed relation Cage speed $N_c$, side gears $N_L$ and $N_R$. Relative to the cage the two side gears turn in opposite directions with equal ratio (planets are idlers): $N_L - N_c = -(N_R - N_c)$, therefore $$ N_L + N_R = 2N_c $$ #### Numerical $N_c = 1800\times\dfrac{10}{40} = 450$ rpm. $$ N_R = 2\times450 - 400 = 500\ \text{rpm} $$ **Answer:** right wheel = 500 rpm (the outer wheel in the turn). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is a differential needed? A. To make both wheels always turn at the same speed B. To allow the outer wheel, which travels a longer path, to turn faster while sharing torque equally C. To reduce engine speed only D. To lock the wheels in a turn Answer: B. Without it one wheel would slip and wear the tyre on a curve. (Key point: Need for a differential) 2. What do the planet pinions do on a straight road? A. They spin fast on their own axes B. They stop the cage C. They do not spin on their own axes, so the cage and both side gears rotate together D. They disconnect one wheel Answer: C. Equal resistance on both wheels means no relative rotation, so planets only revolve with the cage. (Key point: Straight road action) 3. Which relation holds between the axle speeds and the cage (crown wheel) speed? A. N_L = N_R = N_c B. N_L x N_R = N_c C. N_L - N_R = 2 N_c D. N_L + N_R = 2 N_c Answer: D. Relative to the cage the side gears turn equally in opposite directions, so N_L + N_R = 2 N_c. (Key point: Speed relation) 4. The pinion has 10 teeth at 1800 rpm and the crown wheel has 40 teeth. If the left wheel runs at 400 rpm, what is the right wheel speed? A. 500 rpm B. 50 rpm C. 1300 rpm D. 400 rpm Answer: A. N_c = 1800 x 10/40 = 450 rpm, and N_R = 2 x 450 - 400 = 500 rpm. (Key point: Numerical) ## Related in Machine Design II - [Gear ratios of a four-speed car](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Aautomobile-gear-ratio-selection) — Machine Design II, chapter 5 - [Gearbox shaft design with spur gear](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Agearbox-shaft-design-asme) — Machine Design II, chapter 5 - [Six-speed machine tool gearbox design](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Amachine-tool-speed-ladder-gear) — Machine Design II, chapter 5 ## Related in other subjects - [Compound gear train speeds, torques and tooth forces](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Acompound-gear-train-forces) — Machine Design I, chapter 9 - [Tooth forces on a spur gear pinion](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aspur-gear-tooth-forces-numerical) — Theory of Mechanism and Machine I, chapter 8 - [Straight bevel gear geometry and formative teeth](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Astraight-bevel-gear-geometry-formative-teeth) — Theory of Mechanism and Machine I, chapter 5 - [Engine test: four-cylinder at 1800 rpm](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aengine-test-1800-rpm) — Power Plant Equipment, chapter 5, asked 5×