A pendulum consists of a stick of length L and mass M that is pivoted about one

Question

A pendulum consists of a stick of length L and mass M that is pivoted about one end, and attached to a spring of stiffness k a distance l from the pivot, as shown in Fig. 1.

(a) Find the natural angular frequency of this pendulum’s small oscillation.

(b) Compute from part (a) if l = (2/3)L, M = 1 kg, and k = 300 N/m.

A pendulum consists of a stick of length L and mass M that is pivoted about one end, and attached to a spring of stiffness k a distance l from the pivot, as shown in Fig. 1. (a) Find the natural angular frequency ? of this pendulum?s small oscillation. (b) Compute ? from part (a) if l = (2/3)L, M = 1 kg, and k = 300 N/m.

Explanation / Answer

Let the length of the stick = L The mass of bob = M The spring constant = k For small angled () oscillations we have ML(d2/dt2) = -Mg - kL = -(Mg + kL) = -2              From the above we have                          2 = (Mg + kL)                           = sqrt(Mg + kL) So the natural angular frequency, = sqrt(Mg + kL) b) The length,l = (2/3)L Mass, M = 1 kg Spring constant, k = 300 N/m The angular frequency, = sqrt(1*9.8 + 300(2/3)L)                                      = sqrt(9.8 + 200L)                           = sqrt(Mg + kL) So the natural angular frequency, = sqrt(Mg + kL) b) The length,l = (2/3)L Mass, M = 1 kg Spring constant, k = 300 N/m The angular frequency, = sqrt(1*9.8 + 300(2/3)L)                                      = sqrt(9.8 + 200L)

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