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A mass is attached at one end of a uniform bar of mass whose other end is pivote dat point O as shown in the following figure. Determine the natural frequency of vibration of the resulting pendulum for small angular displacements. (10 marks)2.

I_{1 \text { aroundo }}=m_{1} l^{2}

I_{2 \text { coundmiddle point }}=\frac{1}{12} m_{2} I^{2}

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1. Derive the constants of X, and & for an underdamped system, where 3 is the damping ratio, by using initial conditionsof x(0)=x, and *(0)=x, (5 marks)

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7. To familiarize yourself with the nature of the forced response, plot the solution of a forced response of equation mit kx =Fcos at with @=2 rad /s, for a variety of values of the initial conditions and @,, as given in the following chart:

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Consider the system in Fig.3, write the equation of motion,and calculate the response assuming that the system is initially at rest for the values k = 100N / m k, = 500N / m and9.m = 89 kg.

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8. Consider the system in Fig. 2, write the equation of motion,and calculate the response assuming (a) that the system is initially at rest, and (b) that the system has an initial displacement of 0.05 m.

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6. The free response of a 1000-kg automobile with stiffness ofk= 400,000N/m is observed to be of the form give in Figure 6. Modeling the automobile as a single-degree-of-freedom oscillation in the vertical direction, determine the damping coefficient if the displacement at is measured to be 2cm and0.22 cm at

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Repeated problem for the system in Figure 5.5.

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4. Calculate the natural frequency and damping ratio for the100system in Figure 4. It is given the values m =10 kg, c =kg/s, = 4000 N/m, = 200 N/m, = 1000 N/m. Ignore the friction between the cart and the ground. Is the system overdamped, critically damped, or underdamped?

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3. Set up the differential equation of motion for the system shown in Fig.2, Determine the expression for (a) the critical damping coefficient, and (b) the natural frequency of the damped oscillation

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2. The pendulum illustrated in Figure.1 consist of a rigid bar of mass ": and length /, Attached to the end is a lumped mass m. Find the pendulum's natural frequency of oscillation.

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