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Chapter 2: Problem 27

A \(12 \mathrm{~kg}\) spool that is \(1 \mathrm{~m}\) in radius is pinned to a viscoelastic rod of negligible mass with effective properties \(k=10 \mathrm{~N} / \mathrm{m}\) and \(c=8 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). The end of the rod is attached to a rigid support as shown. Determine the natural frequency of the system if the spool rolls without slipping.

Short Answer

Expert verified
Answer: The natural frequency of the system with the spool rolling without slipping is \(\sqrt{\frac{5}{6}} \mathrm{~rad/s}\).

Step by step solution

01

Determine the moment of inertia of the spool

To determine the moment of inertia, we will use the formula for a cylinder, which is \(I = \frac{1}{2}mR^2\), where \(I\) is the moment of inertia, \(m\) is the mass, and \(R\) is the radius. In this case, we have \(m = 12 \mathrm{~kg}\) and \(R = 1 \mathrm{~m}\): \(I = \frac{1}{2}(12 \mathrm{~kg})(1 \mathrm{~m})^2 = 6 \mathrm{~kg⋅m^2}\)
02

Apply the equation of motion

Considering the forces acting on the spool - gravitational force and the forces exerted by the rod, we will apply Newton's second law of motion. For rolling without slipping, the translational and rotational acceleration are related as \(\alpha R = a\), where \(\alpha\) is the angular acceleration and \(a\) is the linear acceleration. The forces exerted by the rod can be calculated as: Spring force: \(F_s = kx\) Damping force: \(F_d = cv\) Total force: \(F_T = F_s + F_d = kx + cv\) For translational motion: \(F_T = ma\) For rotational motion: \(F_T R = I\alpha\) Substituting the rotational equation with the translational one, we get: \(kx + cv = I\frac{a}{R}\)
03

Find the natural frequency

To find the natural frequency, we use the formula \(\omega_n=\sqrt{\frac{k}{m}}\), where \(\omega_n\) is the natural frequency, \(k\) is the stiffness, and \(m\) is the mass. \(\omega_n=\sqrt{\frac{10 \mathrm{~N}/\mathrm{m}}{12 \mathrm{~kg}}} = \sqrt{\frac{5}{6}} \mathrm{~rad/s}\) The natural frequency of the system with the spool rolling without slipping is \(\sqrt{\frac{5}{6}} \mathrm{~rad/s}\).

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Most popular questions from this chapter

Determine the natural period of a typical ice cube floating in water. Measure the dimensions of a typical cube from your refrigerator and calculate its natural frequency in water. (The dimensions may vary depending on your particular ice tray.) Confirm your "experiment." Place an ice cube in water, displace it slightly and release it. Make an approximate measure of the period of an oscillation with your wrist watch, or a stop watch if available. Repeat this operation several times and compare the average measured

A screen door of mass \(m\), height \(L\) and width \(\ell\) is attached to a door frame as indicated. A torsional spring of stiffness \(k_{T}\) is attached as a closer at the top of the door as indicated, and a damper is to be installed near the bottom of the door to keep the door from slamming. Determine the limiting value of the damping coefficient so that the door closes gently, if the damper is to be attached a distance a from the hinge.

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\) and a viscous damper whose coefficient is \(2 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). (a) Determine the response of the horizontally configured system if the mass is displaced 1 meter to the right and released from rest. Plot and label the response history of the system. (b) Determine the response and plot its history if the damping coefficient is \(8 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\). If the coefficients of static and kinetic friction between the block and the surface it moves on are respectively \(\mu_{s}=\) \(0.12\) and \(\mu_{k}=0.10\), determine the drop in amplitude between successive periods during free vibration. What is the frequency of the oscillations?

A single degree of freedom system is represented as a \(4 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(6 \mathrm{~N} / \mathrm{m}\). If the coefficients of static and kinetic friction between the mass and the surface it moves on are \(\mu_{s}=\mu_{k}=0.1\), and the mass is displaced 2 meters to the right and released with a velocity of \(4 \mathrm{~m} / \mathrm{sec}\), determine the time after release at which the mass sticks and the corresponding displacement of the mass.
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