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Chapter 22: Problem 5

When a \(3-\mathrm{kg}\) block is suspended from a spring, the spring is stretched a distance of \(60 \mathrm{~mm}\). Determine the natural frequency and the period of vibration for a \(0.2-\mathrm{kg}\) block attached to the same spring.

Short Answer

Expert verified
In conclusion, by using Hooke's law to calculate the spring constant, and then applying the formulae for natural frequency and period of vibration, it is possible to find the natural frequency and the period of vibration for a 0.2-kg block attached to the same spring.

Step by step solution

01

Calculation of Spring Constant

Hooke's law states that the force exerted by a spring is proportional to its change in length. So, \(F = k \cdot x\), where \(F\) is the force, \(k\) is the spring constant, and \(x\) is the change in length. In this case, \(F\) is the weight of the 3-kg block (\(F = m \cdot g\), where \(m\) is the mass of the block and \(g\) is the acceleration due to gravity, approximately \(9.81 \, m/s^2\)), and \(x\) is 60 mm (0.06 m). Solving the equation for \(k\), you get \(k = F / x\).
02

Determination of Natural Frequency

The natural frequency (\(f\)) of a spring-mass system is given by \(f= (1/2\pi) \cdot \sqrt{k/m}\), where \(k\) is the spring constant (as calculated in Step 1) and \(m\) is the mass of the block. For the 0.2-kg block, \(m = 0.2\) kg. Substituting the known values, calculate the natural frequency.
03

Determination of Period of Vibration

The period of vibration (\(T\)) is the reciprocal of the frequency. Thus, \(T = 1/f\). Once you have calculated the natural frequency, you can calculate the period of vibration by taking its reciprocal.

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Most popular questions from this chapter

A 6-lb weight is suspended from a spring having a stiffness \(k=3 \mathrm{lb} / \mathrm{in} .\) If the weight is given an upward velocity of \(20 \mathrm{ft} / \mathrm{s}\) when it is \(2 \mathrm{in}\). above its equilibrium position, determine the equation which describes the motion and the maximum upward displacement of the weight, measured from the equilibrium position. Assume positive displacement is downward.

If the lower end of the 6 -kg slender rod is displaced a small amount and released from rest, determine the natural frequency of vibration. Each spring has a stiffness of \(k=200 \mathrm{~N} / \mathrm{m}\) and is unstretched when the rod is hanging vertically.

Find the differential equation for small oscillations in terms of \(\theta\) for the uniform rod of mass \(m\). Also show that if \(c<\sqrt{m k} / 2\), then the system remains underdamped. The rod is in a horizontal position when it is in equilibrium.

A 4-lb weight is attached to a spring having a stiffness \(k=10 \mathrm{lb} / \mathrm{ft}\). The weight is drawn downward a distance of 4 in. and released from rest. If the support moves with a vertical displacement \(\delta=(0.5 \sin 4 t)\) in., where \(t\) is in seconds, determine the equation which describes the position of the weight as a function of time.

A spring has a stiffness of \(800 \mathrm{~N} / \mathrm{m}\). If a \(2-\mathrm{kg}\) block is attached to the spring, pushed \(50 \mathrm{~mm}\) above its equilibrium position, and released from rest, determine the equation that describes the block's motion. Assume that positive displacement is downward.
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