A crane lifts a 2500-kg beam using a steel cable whose mass per unit length is 0.650 kg/m.What is the speed of transverse waves on this cable if the upward acceleration of the beam is a steady 3.00 m/s²?

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Quizplus · Accepted Answer

The speed of transverse waves on the cable can be calculated using the formula $v = \sqrt{\frac{T}{\mu}}$, where $v$ is the speed of the wave, $T$ is the tension in the cable, and $\mu$ is the mass per unit length of the cable. The tension $T$ in the cable is due to the weight of the beam and the additional force from its acceleration, so $T = mg + ma = m(g + a)$, where $m$ is the mass of the beam, $g$ is the acceleration due to gravity (approximately $9.81 m/s^2$), and $a$ is the upward acceleration of the beam. Substituting the given values, $m = 2500 kg$, $g = 9.81 m/s^2$, $a = 3.00 m/s^2$, and $\mu = 0.650 kg/m$, we get:$$T = 2500 kg \times (9.81 m/s^2 + 3.00 m/s^2) = 2500 kg \times 12.81 m/s^2 = 32025 N$$Then, the speed of the wave is:$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{32025 N}{0.650 kg/m}} = \sqrt{49269.23 m^2/s^2} \approx 222 m/s$$

A Crane Lifts a 2500-Kg Beam Using a Steel Cable