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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

A thin metal disk with mass \(2.00 \times 10^{-3} \mathrm{~kg}\) and radius \(2.20 \mathrm{~cm}\) is attached at its center to a long fiber (Fig. \(\mathbf{E 1 4 . 4 0}\) ). The disk, when twisted and released, oscillates with a period of \(1.00 \mathrm{~s}\). Find the torsion constant of the fiber.

Object \(A\) has mass \(m_{A}\) and is in \(\mathrm{SHM}\) on the end of a spring with force constant \(k_{A} .\) Object \(B\) has mass \(m_{B}\) and is in \(\mathrm{SHM}\) on the end of a spring with force constant \(k_{B}\). The amplitude \(A_{A}\) for object \(A\) is twice the amplitude \(A_{B}\) for the motion of object \(B\). Also, \(m_{B}=4 m_{A}\) and \(k_{A}=9 k_{B}\). (a) What is the ratio of the maximum speeds of the two objects, \(v_{\max , A} / v_{\max , B} ?\) (b) What is the ratio of their maximum accelerations, \(a_{\max , A} / a_{\max , B} ?\)

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is \(0.090 \mathrm{~m},\) it takes the block \(2.70 \mathrm{~s}\) to travel from \(x=0.090 \mathrm{~m}\) to \(x=-0.090 \mathrm{~m} .\) If the amplitude is doubled, to \(0.180 \mathrm{~m},\) how long does it take the block to travel (a) from \(x=0.180 \mathrm{~m}\) to \(x=-0.180 \mathrm{~m}\) and (b) from \(x=0.090 \mathrm{~m}\) to \(x=-0.090 \mathrm{~m} ?\)

A \(40.0 \mathrm{~N}\) force stretches a vertical spring \(0.250 \mathrm{~m}\). (a) What mass must be suspended from the spring so that the system will oscillate with a period of \(1.00 \mathrm{~s} ?\) (b) If the amplitude of the motion is \(0.050 \mathrm{~m}\) and the period is that specified in part (a), where is the object and in what direction is it moving \(0.35 \mathrm{~s}\) after it has passed the equilibrium position, moving downward? (c) What force (magnitude and direction) does the spring exert on the object when it is \(0.030 \mathrm{~m}\) below the equilibrium position, moving upward?

A \(10.0 \mathrm{~kg}\) mass is traveling to the right with a speed of \(2.00 \mathrm{~m} / \mathrm{s}\) on a smooth horizontal surface when it collides with and sticks to a second \(10.0 \mathrm{~kg}\) mass that is initially at rest but is attached to one end of a light, horizontal spring with force constant \(170.0 \mathrm{~N} / \mathrm{m}\). The other end of the spring is fixed to a wall to the right of the second mass. (a) Find the frequency, amplitude, and period of the subsequent oscillations. (b) How long does it take the system to return the first time to the position it had immediately after the collision?
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