Question:hard

A load of "W" kN is acting on a tyre having 160 mm nominal width. The effective friction coefficient of tyre and ground interaction is "\(\mu\)" and the kingpin offset is 10 mm. Assuming the tyre impression on ground as circle with diameter equal to nominal tyre width, the kingpin torque of the tyre in N.m will be

Show Hint

The unit conversion factors cancel out beautifully:
\(1\text{ kN} \times 1\text{ mm} = 10^3\text{ N} \times 10^{-3}\text{ m} = 1\text{ N}\cdot\text{m}\).
Therefore, you can compute directly using raw values to get \(W\mu\sqrt{3300}\).
  • \(W\mu\sqrt{3210}\)
  • \(\frac{W\mu\sqrt{3210}}{1000}\)
  • \(W\mu\sqrt{3300}\)
  • \(\frac{W\mu\sqrt{3300}}{1000}\)
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Fix the geometry of the contact patch.
The tyre print is taken as a circle whose diameter equals the nominal tyre width of 160 mm, so the radius of the contact patch is \( r = 80 \text{mm} \). The kingpin axis sits offset from the wheel centre by \( e = 10 \text{mm} \).
Step 2: Find the effective radius for spin resistance over a circular patch.
When a wheel pivots about the kingpin while stationary, every element of the circular contact area resists rotation with a friction force proportional to its distance from the patch centre. The characteristic radius that represents this distributed resistance for a full circle works out to \( r/\sqrt{2} \), so here it is \( 80/\sqrt{2} = 56.57 \text{mm} \), and squared this gives \( r^2/2 = 3200 \ \text{mm}^2 \).
Step 3: Combine the offset and the spin resistance term.
The kingpin offset and the spin resistance act as two independent contributors to the resisting torque, so they combine as \( \sqrt{e^2 + r^2/2} = \sqrt{100 + 3200} = \sqrt{3300} \ \text{mm} \).
Step 4: Attach load and friction, and check units.
The kingpin torque is \( T = W \mu \sqrt{3300} \). Since W is in kN and the length term is in mm, the product kN times mm already equals N.m directly (1 kN x 1 mm = 1000 N x 0.001 m = 1 N.m), so no extra scaling factor of 1000 is needed, which rules out the options carrying a division by 1000.
\[ \boxed{W\mu\sqrt{3300}} \]
Was this answer helpful?
0