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Concept explainers
(a)
The period
(a)
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Answer to Problem 19.51P
The period
Explanation of Solution
Given information:
A thin homogeneous wire is bent into the shape of an isosceles triangle of sides are b, b and 1.6b.
Calculation:
Show the position of centroid and distance as in Figure (1).
Write the equation for mass moment of inertia
Here, r is distance of each particle from the axis of rotation.
Calculate the expression for mass moment of inertia
Here,
Modify the above equation,
Calculate the centroid equation
Calculate the distance equation AG as below
Substitute
The external forces in the system are force due mass of the thin wire and the effective restoring couple is
Take moment about A in the system for external forces.
Substitute
Take moment about A in the system for effective forces.
Substitute
Equate the moment about A in the system for external and effective forces.
Compare the differential Equation (1) with the general differential equation of motion
Calculate the period of small oscillation
Substitute
Therefore, the period
(b)
The period
(b)
![Check Mark](/static/check-mark.png)
Answer to Problem 19.51P
The period
Explanation of Solution
Given information:
A thin homogeneous wire is bent into the shape of an isosceles triangle of sides are b, b and 1.6b.
Calculation:
Calculate the expression for mass moment of inertia
Substitute
Calculate the equation for distance GB by using the Pythagoras theorem:
Substitute
The external forces in the system are force due mass of the thin wire and the effective restoring couple is
Take moment about B in the system for external forces.
Substitute
Take moment about B in the system for effective forces.
Substitute
Equate the moment about B in the system for external and effective forces.
Compare the differential Equation (2) with the general differential equation of motion
Calculate the period of small oscillation
Substitute
Therefore, the period
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Chapter 19 Solutions
VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
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- Two 40-g weights are attached at A and B to the rim of a 1.5-kg uniform disk of radius r = 100 mm. Determine the frequency of small oscillations when β = 60°.arrow_forwardA small collar of mass 1 kg is rigidly attached to a 3-kg uniform rod of length L = 750 mm. Determine (a) the distance d to maximize the frequency of oscillation when the rod is given a small initial displacement, (b) the corresponding period of oscillation.arrow_forwardThe inner rim of an 85-lb flywheel is placed on a knife edge, and the period of its small oscillations is found to be 1.26 s. Determine the centroidal moment of inertia of the flywheel.arrow_forward
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- Figure 3 shows a uniform beam is supported horizontally by two equal springs of the same stiffness. Derive the equation of the two frequencies of small oscillations in the vertical plane. Show that the a two frequencies will be equal only if b = c = √√3 2a b G Fig. 3 Carrow_forwardA thin plate of length l rests on a half cylinder of radius r Derive an expression for the period of small oscillations of the plate.arrow_forwardProblem 38.3 Two uniform rods, each of weight W = 24 lb and length L = 2 ft, are welded together to form the assembly shown. Knowing that the constant of each spring is k = 3 lb/in. and that end A is given a small displacement and released, determine the frequency of the resulting motion. B wwarrow_forward
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