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A thin uniform metallic rod of length L = 0.5 m and radius r = 0.1 m rotates with angular velocity omega = 400 rad/s in a horizontal plane about a vertical axis through one end. The density of the rod is rho = 10⁴ kg/m³ and Young's modulus Y = 2 * 10¹¹ N/m². Find: (a) the maximum tension in the rod (at the fixed end), and (b) the total elongation of the rod.
- T = 2*pi * 10⁶ N, delta = 1/3000 m
- T = pi * 10⁶ N, delta = 1/6000 m
- T = 4*pi * 10⁶ N, delta = 1/1500 m
- T = 2*pi * 10⁶ N, delta = 1/2000 m
Correct answer: T = 2*pi * 10⁶ N, delta = 1/3000 m
Solution
Tension is maximum at the fixed end: T_max = rho*A*omega²*L²/2 = 10⁴ * pi*(0.1)² * (400)² * (0.5)² / 2 = 2*pi * 10⁶ N. Elongation = rho*omega²*L³/(3Y) = 10⁴*(400)²*(0.5)³/(3*2*10¹¹) = 1/3000 m.
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