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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

Two uniform solid spheres, each with mass \(M=0.800 \mathrm{~kg}\) and radius \(R=0.0800 \mathrm{~m},\) are connected by a short, light rod that is along a diameter of each sphere and are at rest on a horizontal tabletop. A spring with force constant \(k=160 \mathrm{~N} / \mathrm{m}\) has one end attached to the wall and the other end attached to a friction less ring that passes over the rod at the center of mass of the spheres, which is midway between the centers of the two spheres. The spheres are each pulled the same distance from the wall, stretching the spring, and released. There is sufficient friction between the tabletop and the spheres for the spheres to roll without slipping as they move back and forth on the end of the spring. Show that the motion of the center of mass of the spheres is simple harmonic and calculate the period.

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 \(1.80 \mathrm{~kg}\) 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 harmonic oscillator has angular frequency \(\omega\) and amplitude \(A\). (a) What are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (Assume that \(U=0\) at equilibrium.) (b) How often does this occur in each cycle? What is the time between occurrences? (c) At an instant when the displacement is equal to \(A / 2,\) what fraction of the total energy of the system is kinetic and what fraction is potential?

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?
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