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Chapter 14: Problem 25

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. The amplitude of the motion is \(0.165 \mathrm{~m}\). The maximum speed of the block is \(3.90 \mathrm{~m} / \mathrm{s}\). What is the maximum magnitude of the acceleration of the block?

Short Answer

Expert verified
The maximum acceleration is approximately 91.52 m/s虏

Step by step solution

01

Identifying the givens

From the problem, the maximum speed of the block, \(v_{max}\), is given as 3.90 m/s, and the amplitude, A, is given as 0.165 m.
02

Apply the formula for max acceleration

We will use the formula which relates maximum acceleration, maximum speed and amplitude in a simple harmonic motion, which is \(a_{max} = \frac{{v_{max}}^2}{A}\)
03

Set up the equation

By substituting the given values into the formula, we get \(a_{max} = \frac{{(3.90 \, \text{m/s})^2}}{0.165 \, \text{m}}\)
04

Calculate the acceleration

Finally, calculate the maximum acceleration by squaring the maximum speed and dividing by the amplitude.

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Key Concepts

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

Maximum Acceleration in Simple Harmonic Motion
In simple harmonic motion, acceleration is not a constant value. Instead, it varies depending on the object's position. The maximum acceleration occurs when the object is at its maximum displacement from the equilibrium position, also known as the amplitude. At this point, the restoring force, which follows Hooke's law, is at its greatest. This maximum acceleration can be determined using the formula:
  • \( a_{max} = \frac{{v_{max}}^2}{A} \)
Here, \( v_{max} \) is the maximum speed, and \( A \) is the amplitude of motion.
By substituting the known values into the formula, you can find out how much the acceleration reaches at its peak. For example, in our scenario, the calculation involves squaring the maximum speed and dividing it by the amplitude, giving the maximum acceleration in the system.
The Role of Amplitude in Simple Harmonic Motion
Amplitude is a crucial component of simple harmonic motion (SHM). It symbolizes the greatest distance that the oscillating object travels from its equilibrium or central position. The amplitude directly affects the dynamics of SHM, influencing both the maximum speed and maximum acceleration.
In mathematical terms, amplitude \( A \) is constant for a particular motion and is one of the factors that determine other quantities. For instance, 鈥� As seen in the formula \( a_{max} = \frac{{v_{max}}^2}{A} \), a lower amplitude can lead to higher acceleration for the same maximum speed.
  • The constant amplitude ensures that the energy within the system remains consistent.
  • It helps in calculating other parameters like period and frequency as well.
Understanding the amplitude's role makes it easier to predict and calculate changes in motion as the block moves to extreme positions.
Maximum Speed in Simple Harmonic Motion
The maximum speed in simple harmonic motion occurs as the object passes through the equilibrium position. It is at this point that kinetic energy is at its peak, and potential energy is zero.
  • This speed is dependent on both the spring force and the amplitude.
  • The relationship is captured by the formula \( v_{max} = \omega A \) where \( \omega \) is the angular frequency.
Reaching maximum speed tells us when the kinetic energy is fully expressed, as the object rapidly passes through the equilibrium with all the energy that was stored as potential energy at the amplitudes.
Knowing how to compute maximum speed is vital for solving problems involving motion dynamics because it is integral in understanding energy conversions in SHM.

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Most popular questions from this chapter

BIO Weighing a Virus. In February 2004, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just \(30 \mathrm{nm}\) long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached \(\left(f_{\mathrm{S}+\mathrm{V}}\right)\) to the frequency without the virus \(\left(f_{\mathrm{S}}\right)\) is given by \(f_{\mathrm{S}+\mathrm{V}} / f_{\mathrm{S}}=1 / \sqrt{1+\left(m_{\mathrm{V}} / m_{\mathrm{S}}\right)},\) where \(m_{\mathrm{V}}\) is the mass of the virus and \(m_{\mathrm{S}}\) is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of \(2.10 \times 10^{-16} \mathrm{~g}\) and a frequency of \(2.00 \times 10^{15} \mathrm{~Hz}\) without the virus and \(2.87 \times 10^{14} \mathrm{~Hz}\) with the virus. What is the mass of the virus, in grams and in femtograms?

\(\mathrm{A} 50.0 \mathrm{~g}\) hard-boiled egg moves on the end of a spring with force constant \(k=25.0 \mathrm{~N} / \mathrm{m} .\) Its initial displacement is \(0.300 \mathrm{~m} . \mathrm{A}\) damping force \(F_{x}=-b v_{x}\) acts on the egg, and the amplitude of the motion decreases to \(0.100 \mathrm{~m}\) in \(5.00 \mathrm{~s}\). Calculate the magnitude of the damping constant \(b\).

When an object of unknown mass is attached to an ideal spring with force constant \(120 \mathrm{~N} / \mathrm{m},\) it is found to vibrate with a frequency of \(6.00 \mathrm{~Hz}\). Find (a) the period of the motion; (b) the angular frequency; (c) the mass of the object.

Consider the system of two blocks and a spring shown in Fig. \(\mathrm{P} 14.66 .\) The horizontal surface is friction less, but there is static friction between the two blocks. The spring has force constant \(k=150 \mathrm{~N} / \mathrm{m} .\) The masses of the two blocks are \(m=0.500 \mathrm{~kg}\) and \(M=4.00 \mathrm{~kg} .\) You set the blocks into motion by releasing block \(M\) with the spring stretched a distance \(d\) from equilibrium. You start with small values of \(d,\) and then repeat with successively larger values. For small values of \(d,\) the blocks move together in SHM. But for larger values of \(d\) the top block slips relative to the bottom block when the bottom block is released. (a) What is the period of the motion of the two blocks when \(d\) is small enough to have no slipping? (b) The largest value \(d\) can have and there be no slipping is \(d=8.8 \mathrm{~cm} .\) What is the coefficient of static friction \(\mu_{\mathrm{s}}\) between the surfaces of the two blocks?

\(\mathrm{A}\) mass is oscillating with amplitude \(A\) at the end of a spring. How far (in terms of \(A\) ) is this mass from the equilibrium position of the spring when the elastic potential energy equals the kinetic energy?
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