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Chapter 2: Problem 7

A \(30 \mathrm{~cm}\) aluminum rod possessing a circular cross section of \(1.25 \mathrm{~cm}\) radius is inserted into a testing machine where it is fixed at one end and attached to a load cell at the other end. At some point during a torsion test the clamp at the load cell slips, releasing that end of the rod. If the \(20 \mathrm{~kg}\) clamp remains attached to the end of the rod, determine the frequency of the oscillations of the rod-clamp system. The radius of gyration of the clamp is \(5 \mathrm{~cm}\). Fig. P2.7 Fig. P2.7

Short Answer

Expert verified
Answer: To determine the frequency of oscillations (f), follow the steps below: Step 1: Calculate the moment of inertia of the rod and clamp using the formulas \(I_{rod} = \frac{1}{4} \pi r_{rod}^4 \rho_{Al} L\) and \(I_{clamp} = M_{clamp} k^2\). Step 2: Calculate the torsional stiffness of the rod using the formula \(K = \frac{GJ}{L}\), where \(J = \pi r_{rod}^4 / 2\). Step 3: Determine the frequency of oscillations using the formula \(f = \frac{1}{2\pi} \sqrt{\frac{K}{I_{total}}}\), where \(I_{total} = I_{rod} + I_{clamp}\). Plug in the given values and solve for the frequency, \(f\).

Step by step solution

01

Calculate moment of inertia of the rod and the clamp

To calculate the moment of inertia of the rod and the clamp, we need to use the formulas: \(I_{rod} = \frac{1}{4} \pi r_{rod}^4 \rho_{Al} L\) for the rod and \(I_{clamp} = M_{clamp} k^2\) for the clamp, where \(I_{rod}\) and \(I_{clamp}\) are the moment of inertia of the rod and clamp, respectively, \(r_{rod}\) is the radius of the rod, \(\rho_{Al}\) is the density of aluminum, \(L\) is the length of the rod, \(M_{clamp}\) is the mass of the clamp, and \(k\) is the radius of gyration. Plug in the given values for the variables.
02

Calculate torsional stiffness of the rod

Next, we need to find the torsional stiffness of the rod, which can be calculated using the formula \(K = \frac{GJ}{L}\), where \(K\) is the torsional stiffness, \(G\) is the shear modulus of aluminum, \(J\) is the polar moment of inertia of the rod (can be calculated as \(J=\pi r_{rod}^4 / 2\)), and \(L\) is the length of the rod. Plug in the given values for the variables.
03

Determine the frequency of oscillations

Now, we have all the information needed to calculate the frequency of oscillations. The formula for the frequency of oscillations is \(f = \frac{1}{2\pi} \sqrt{\frac{K}{I_{total}}}\), where \(f\) is the frequency of oscillations, \(K\) is the torsional stiffness, and \(I_{total} = I_{rod} + I_{clamp}\) is the total moment of inertia of the rod-clamp system. Plug in the values obtained in Steps 1 and 2 and solve for the frequency, \(f\). Following these steps will help you determine the frequency of oscillations of the rod-clamp system after the clamp slips during a torsion test.

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