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A rod of length L lies along the x-axis from x = 0 to x = L. Its linear mass density varies with distance x from the origin as rho = a + bx. (a) State the SI units of a and b. (b) Express the total mass of the rod in terms of a, b and L.

1. Units: a in kg/m, b in kg/m²; Mass = aL + bL²/2
2. Units: a in kg/m, b in kg/m²; Mass = aL + bL²
3. Units: a in kg, b in kg/m; Mass = aL/2 + bL²
4. Units: a in kg/m², b in kg/m; Mass = aL + bL²/2

Correct answer: Units: a in kg/m, b in kg/m²; Mass = aL + bL²/2

Solution

Since rho is in kg/m, a is kg/m and bx must be kg/m so b is kg/m². Mass = integral of (a+bx) dx = aL + bL²/2.

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