Why is a differential needed?

To allow the outer wheel, which travels a longer path, to turn faster while sharing torque equally

To make both wheels always turn at the same speed

What do the planet pinions do on a straight road?

They do not spin on their own axes, so the cage and both side gears rotate together

They spin fast on their own axes

Practice · 5 marks

Differential gear action in an automobile

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.

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.

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

   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).

✓ Answer checked by Claude Opus 5.5 (high effort)

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