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Chapter 11: Problem 52

An electric motor of mass \(40 \mathrm{~kg}\) is mounted on four vertical springs each having spring constant of 4000 \(\mathrm{N} / \mathrm{m}\). The period with which the motor vibrates vertically is (a) \(0.314 \mathrm{~s}\) (b) \(3.14 \mathrm{~s}\) (c) \(0.628 \mathrm{~s}\) (d) \(0.157 \mathrm{~s}\)

Short Answer

Expert verified
The period of the motor's vibration is \( 0.314 \mathrm{~s} \).

Step by step solution

01

Identify Given Values

The mass of the motor is given as \( m = 40 \mathrm{~kg} \) and the spring constant for each spring is \( k = 4000 \mathrm{~N/m} \). Since there are four springs, the equivalent spring constant is \( k_{eq} = 4k \).
02

Determine Equivalent Spring Constant

Calculate the equivalent spring constant by multiplying the spring constant of one spring by the number of springs: \[ k_{eq} = 4 \times 4000 \mathrm{~N/m} = 16000 \mathrm{~N/m} \]
03

Use the Formula for Period of Vibration

The formula for the period of vertical vibrations for a mass-spring system is given by: \[ T = 2\pi \sqrt{\frac{m}{k_{eq}}} \] Substitute the known values into this formula.
04

Perform Calculations

Substitute \( m = 40 \mathrm{~kg} \) and \( k_{eq} = 16000 \mathrm{~N/m} \) into the formula: \[ T = 2 \pi \sqrt{\frac{40}{16000}} \] This simplifies to: \[ T = 2 \pi \sqrt{0.0025} \] Calculate further to get: \[ T = 2\pi \times 0.05 = 0.314 \mathrm{~s} \]
05

Select Correct Answer

The period of vibration we calculated is \( 0.314 \mathrm{~s} \), which matches option (a).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Period of Vibration
The period of vibration, often denoted by the symbol \( T \), is the time it takes for one complete cycle of oscillation. In the context of a mass-spring system, this is the duration of time required for the mass to complete one bounce back and forth. Understanding the period is crucial because it helps us to predict how quickly or slowly an object will oscillate.
  • For a mass-spring system, the formula used to calculate the period \( T \) is: \[ T = 2\pi \sqrt{\frac{m}{k_{eq}}} \]
  • The term \( 2\pi \) comes from the circular nature of harmonic motion and represents one complete rotation in radians.
  • The square root component \( \sqrt{\frac{m}{k_{eq}}} \) shows that the period is determined by both the mass \( m \) of the object and the effective spring constant \( k_{eq} \).
This formula implies that a larger mass will increase the period, making vibrations slower, while a stiffer spring (larger spring constant) will decrease the period, leading to faster vibrations.
Understanding this concept is key to solving problems involving oscillatory motion.
Mass-Spring System
A mass-spring system is a common physics model used to study oscillatory motion. It consists of a mass attached to a spring that can stretch and compress. This system makes an excellent example of simple harmonic motion (SHM). Simply put, SHM is characterized by oscillations that repeat over time and are driven by a restoring force proportional to the displacement.
  • The basic setup includes a mass that pulls on a spring, stretching or compressing it from its equilibrium (rest) position.
  • When the system is set into motion, it exhibits periodic motion, meaning the system will continue vibrating back and forth until other forces, such as friction, dampen the motion.
  • The frequency of these vibrations depends on the mass of the object and the spring constant.
A mass-spring system can be parallel or series, impacting the way spring constants combine. In parallel like the example here, the constant is the sum of the individual ones. By understanding how this system works, you can predict and calculate important aspects like the period of vibration.
Spring Constant
The spring constant, denoted by \( k \), measures the stiffness of a spring. It is a fundamental property in physics that indicates how much force is needed to stretch or compress the spring by a certain distance. Simply put, a larger spring constant means a stiffer spring.
  • The stronger the spring, the higher the force needed to stretch or compress it, indicating a larger \( k \) value.
  • In mathematical terms, Hooke鈥檚 Law describes the relationship: \[ F = k \cdot x \] where \( F \) is force, and \( x \) is the displacement from equilibrium.
  • In our given scenario, the spring constant of one spring is \( 4000 \text{ N/m} \).
When springs work together in a system, like the four parallel springs attached to the motor, the total effective spring constant \( k_{eq} \) is the sum of individual spring constants (\( k_{eq} = 4 \times 4000 \text{ N/m} = 16000 \text{ N/m} \)). This makes the system much stiffer collectively, altering the period of vibration.

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Most popular questions from this chapter

In a plane progressive harmonic wave particle speed is always less than the wave speed if (a) amplitude of wave \(<\lambda / 2 \pi\) (b) amplitude of wave \(>\lambda / 2 \pi\) (c) amplitude of wave \(<\lambda\) (d) amplitude of wave \(>\lambda \pi\)

A travelling wave in a gas along the positive \(X\) -direction has an amplitude of \(2 \mathrm{~cm}\), velocity \(45 \mathrm{~m} / \mathrm{s}\) and frequency \(75 \mathrm{~Hz}\). Particle acceleration after an interval of 3 seconds at a distance of \(135 \mathrm{~cm}\) from the origin is (a) \(0.44 \times 10^{2} \mathrm{~cm} / \mathrm{s}^{2}\) (b) \(4.4 \times 10^{5} \mathrm{~cm} / \mathrm{s}^{2}\) (c) \(4.4 \times 10^{3} \mathrm{~cm} / \mathrm{s}^{2}\) (d) \(44 \times 10^{5} \mathrm{~cm} / \mathrm{s}^{2}\)

The equation of motion of a particle executing simple harmonic motion is \(a+16 \pi^{2} x=0 .\) In this equation, \(a\) is the linear acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) of the particle at a displacement \(x\) in metre. The time period in simple harmonic motion is (a) \(\frac{1}{4}\) second (b) \(\frac{1}{2}\) second (c) 1 second (d) 2 seconds

Which of the following statements is correct? (a) Both sound and light waves in air are longitudinal. (b) Both sound and light waves in air are transverse. (c) Sound waves in air are transverse while light longitudinal. (d) Sound waves in air are longitudinal while light waves transverse.

A simple harmonic wave is represented by the relation \(y(x, t)=a_{o} \sin 2 x\left(v t-\frac{x}{\lambda}\right)\) If the maximum particle velocity is three times the wave velocity, the wavelength \(\lambda\) of the wave is (a) \(\pi a_{a} / 3\) (b) \(2 \pi a_{o} / 3\) (c) \(\pi a_{o}\) (d) \(\pi a_{e} / 2\)
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