Problem 58 - College Physics

Question

A block with mass \(M\) rests on a frictionless surface and is connected to a horizontal spring of force constant \(k\), the other end of which is attached to a wall (Figure 11.37 ). 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.

Step-by-Step Solution

Answer

The maximum amplitude is \( A = \frac{\mu_s g M}{k} \).

Step 1: Analyze Forces on the Blocks

Identify that the static friction force between the two blocks is what prevents the top block from slipping. This force must at least equal the maximum force exerted by the acceleration of the top block during oscillation.

Step 2: Determine Maximum Acceleration

Use the formula for maximum acceleration in simple harmonic motion, given by the spring's force constant and the amplitude: \( a_{max} = A \omega^2 \), where \( \omega = \sqrt{\frac{k}{M}} \). Hence, \( a_{max} = A \frac{k}{M} \).

Step 3: Apply Static Friction Condition

Recognize that the static friction force \( f_s \) must satisfy \( f_s \leq \mu_s m g \), where \( f_s = m a_{max} \) since it's necessary to prevent slipping. Thus, \( mA \frac{k}{M} \leq \mu_s m g \).

Step 4: Solve for Maximum Amplitude

Simplify the inequality obtained from the static friction condition: \( A \frac{k}{M} \leq \mu_s g \). Hence, the maximum amplitude \( A \) is given by \( A = \frac{\mu_s g M}{k} \).

Key Concepts

Static Friction

In the context of this problem, static friction plays a vital role in ensuring that the block resting on top of the lower block doesn't slip during oscillation. Static friction is the force that prevents two surfaces from sliding past each other. It is a variable force because it adjusts itself to balance the other forces up to a maximum limit. The maximum static friction force can be calculated using the formula:

• \( f_s = \,\mu_s \,\times \, F_N \)

where \( \mu_s \) is the coefficient of static friction and \( F_N \) is the normal force. In this exercise, the normal force is the weight of the top block, which equals \( m \,g \), where \( g \) is the acceleration due to gravity.
This means the maximum force that static friction can exert before slipping occurs is \( \mu_s \, m \, g \). This concept ensures that in oscillations, the top block keeps pace with the lower block without slipping.

Oscillation Amplitude

Oscillation amplitude, represented as \( A \) in simple harmonic motion (SHM), is defined as the maximum displacement from the equilibrium position. It is a crucial factor in determining how far the system, in this case, the mass-spring system, moves on either side of its resting position.In our problem, finding the maximum amplitude is essential because it defines the boundary within which the top block can move without slipping due to excessive acceleration. The maximum amplitude is determined by balancing the maximum static friction with the spring force-induced acceleration.From the solution, we calculate it using:

• \( A = \frac{\mu_s \, g \, M}{k} \)

This formula stems from combining the maximum static frictional force and its limit to withstand the spring’s force.

Force Constant

The force constant, often referred to as the spring constant \( k \), is a measure of a spring's stiffness. In simple harmonic motion, it determines how much force is needed to stretch or compress the spring by a unit length.The larger the value of \( k \), the stiffer the spring and the more force it takes to displace it.This constant is pivotal in determining the frequency of oscillation and the system's maximum acceleration.The spring force can be calculated using Hooke's Law:

• \( F = -k \, x \)

where \( F \) is the force applied, and \( x \) is the spring's displacement from its equilibrium position. For our mass-spring system, the force constant directly influences the system's natural frequency \( \omega = \sqrt{\frac{k}{M}} \) and relates to how the amplitude affects the ability to keep the upper block from slipping.

Mass-Spring System

A mass-spring system is a classic example of simple harmonic motion often studied in physics. In this setup, a block's connection to a spring enables oscillations back and forth about an equilibrium point when displaced. This setup's properties are governed mostly by the mass of the block and the spring constant. In our particular problem, a second block resting on the first introduces complexity, with static friction becoming a critical factor. To analyze such a system, we assume:

• The surface is frictionless, simplifying calculations by removing external damping forces.
• The motion is idealized as undamped simple harmonic motion.
• The primary interactions are between the spring and the masses connected to it.

Understanding these foundational principles helps clarify how this system operates and affects the amplitude of oscillation that is safe to prevent slipping.