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Chapter 14: Problem 66

A block with mass \(M\) rests on a friction less surface and is connected to a horizontal spring of force constant \(k .\) The other end of the spring is attached to a wall (Fig. \(\mathbf{P 1 4 . 6 6 )} .\) A second block with mass \(m\) rests on top of the first block. The coefficient of static friction between the blocks is \(\mu_{\mathrm{s}}\). Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

Short Answer

Expert verified
The maximum amplitude of oscillation such that the top block will not slip on the bottom block is given by \( A = \frac{\mu_s \times (M + m) \times g}{k} \)

Step by step solution

01

Identify Maximum Static Friction

Calculate the maximum static friction using the formula \( F_s = \mu_s \times m \times g \), where \( \mu_s \) is the coefficient of static friction, \( m \) is the mass of the smaller block and \( g \) is the acceleration due to gravity.
02

Calculate Force by the Oscillating Block

Calculate the force exerted by the bigger block on the smaller one when it's in maximum displacement (amplitude). The maximum acceleration it can have without the smaller block slipping is when the force exerted equals the static friction. The force exerted on the smaller block by the bigger one due to oscillation can be given by \( F = m \times a \) where \( a \) is the acceleration and can be determined using Hooke's law (for spring force) which states that the force exerted by the spring is equal to \( k \times A \) where \( A \) is the amplitude of oscillation.
03

Equate Forces and Solve for Amplitude

Set the two forces equal to each other and solve for the amplitude \( A \). Thus, \( F_s = F \) implies \( \mu_s \times m \times g = m \times a = m \times \frac{k \times A}{M+m} \). Solving for \( A \) will give the maximum amplitude of oscillation without the smaller block slipping.

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Key Concepts

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

Static Friction
Static friction plays a central role in ensuring that objects do not slip past each other. It is the force that keeps the smaller block on top of the larger one without slipping. The maximum static friction is calculated using the formula \( F_s = \mu_s \times m \times g \). Here, \( \mu_s \) is the coefficient of static friction, \( m \) is the mass of the top block, and \( g \) represents the acceleration due to gravity.

Static friction only comes into play up to a certain maximum limit. This maximum is directly proportional to how rough or sticky the surfaces in contact are, indicated by \( \mu_s \), and the weight of the top block (\( m \times g \)). Once the force exceeds \( F_s \), the top block will begin to slide off, which we want to avoid in this setup.
  • The maximum static friction ensures that the top block remains stationary relative to the bottom block.
  • It must counteract any force trying to move the smaller block.
Spring Force
The spring force is responsible for the oscillation of the system. It acts to restore the displacement back to equilibrium. Hook's Law tells us the spring force can be expressed as \( F = k \times x \), where \( k \) is the spring constant and \( x \) is the displacement.

For a mass-spring system, the displacement when the force is maximum is known as the amplitude \( A \). The spring force ensures that the oscillation is continuous as it always pulls back to the equilibrium point. In this problem, the spring force is balanced with static friction to ensure that there is no slipping.
  • Spring force is directly proportional to how far the spring is stretched or compressed from its original length.
  • The spring constant \( k \) determines the stiffness of the spring, affecting how much force is needed for displacement.
Oscillation Amplitude
Oscillation amplitude refers to the maximum extent of the system's movement from its mean position. In this scenario, it is crucial to determine the amplitude because it affects whether the top block will slip.

To find the amplitude \( A \) for which the smaller block does not slip, you need to assess the balance between static friction and spring force. The amplitude is linked with the maximum acceleration that the system can endure without the top block slipping off. By solving the equation \( \mu_s \times m \times g = m \times \frac{k \times A}{M+m} \) for \( A \), you get the safe amplitude range.
  • Amplitude determines the range of motion for the oscillating system.
  • Crucial to maintain amplitude within bounds to prevent slippage.
Mass-Spring System
A mass-spring system is a model used to describe the motion of mass attached to a spring. In this setup, a block (or blocks) is connected to a spring fixed at one end. The entire system can oscillate back and forth when displaced from its equilibrium position.

The dynamics of a mass-spring system depend heavily on the spring constant \( k \) and the mass of the object \( M \) and \( m \). Together, they influence the motion patterns and maximum possible amplitude of the system before the top block slips.
  • A classic example of simple harmonic motion (SHM).
  • Critical understanding of how mass distributions affect oscillation and force balance.
  • Hooke's Law is pivotal in describing how the system behaves dynamically.

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Most popular questions from this chapter

BIO (a) Music. When a person sings, his or her vocal cords vibrate in a repetitive pattern that has the same frequency as the note that is sung. If someone sings the note \(\mathrm{B}\) flat, which has a frequency of \(466 \mathrm{~Hz}\), how much time does it take the person's vocal cords to vibrate through one complete cycle, and what is the angular frequency of the cords? (b) Hearing. When sound waves strike the eardrum, this membrane vibrates with the same frequency as the sound. The highest pitch that young humans can hear has a period of \(50.0 \mu \mathrm{s}\). What are the frequency and angular frequency of the vibrating eardrum for this sound? (c) Vision. When light having vibrations with angular frequency ranging from \(2.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) to \(4.7 \times 10^{15} \mathrm{rad} / \mathrm{s}\) strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light? (d) Ultrasound. High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as x rays do. To detect small objects such as tumors, a frequency of around \(5.0 \mathrm{MHz}\) is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

A \(0.150 \mathrm{~kg}\) toy is undergoing \(\mathrm{SHM}\) on the end of a horizontal spring with force constant \(k=300 \mathrm{~N} / \mathrm{m}\). When the toy is \(0.0120 \mathrm{~m}\) from its equilibrium position, it is observed to have a speed of \(0.400 \mathrm{~m} / \mathrm{s} .\) What are the toy's (a) total energy at any point of its motion; (b) amplitude of motion; (c) maximum speed during its motion?

A \(1.80 \mathrm{~kg}\) monkey wrench is pivoted \(0.250 \mathrm{~m}\) from its center of mass and allowed to swing as a physical pendulum. The period for small-angle oscillations is \(0.940 \mathrm{~s}\). (a) What is the moment of inertia of the wrench about an axis through the pivot? (b) If the wrench is initially displaced 0.400 rad from its equilibrium position, what is the angular speed of the wrench as it passes through the equilibrium position?

A rifle bullet with mass \(8.00 \mathrm{~g}\) and initial horizontal velocity \(280 \mathrm{~m} / \mathrm{s}\) strikes and embeds itself in a block with mass \(0.992 \mathrm{~kg}\) that rests on a friction less surface and is attached to one end of an ideal spring. The other end of the spring is attached to the wall. The impact compresses the spring a maximum distance of \(15.0 \mathrm{~cm} .\) After the impact, the block moves in SHM. Calculate the period of this motion.

An unhappy \(0.300 \mathrm{~kg}\) rodent, moving on the end of a spring with force constant \(k=2.50 \mathrm{~N} / \mathrm{m},\) is acted on by a damping force \(F_{x}=-b v_{x}\). (a) If the constant \(b\) has the value \(0.900 \mathrm{~kg} / \mathrm{s},\) what is the frequency of oscillation of the rodent? (b) For what value of the constant \(b\) will the motion be critically damped?
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