/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 68 A block with mass \(M\) rests on... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 68

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 the spring is attached to a wall (Fig. 13.36\() .\) A second block with mass \(m\) rests on top of the first block. The coefficient of static friction between the blocks is \(\mu_{s}\) 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 is \( A_{max} = \frac{\mu_s \cdot m \cdot g}{k} \).

Step by step solution

01

Understand the System

We have a system of two blocks: a block of mass \(M\) on a frictionless surface connected with a spring of force constant \(k\), and a second block of mass \(m\) on top of the first. The upper block should not slide over the lower block when the system oscillates. The static friction between the blocks has a coefficient \(\mu_s\).
02

Apply Static Friction Condition

To ensure that the top block doesn't slip, the maximum static friction force \(f_s = \mu_s \cdot m \cdot g\) should be equal to or greater than the maximum horizontal force due to the spring oscillation exerted on the mass \(m\). Here, \(g\) stands for gravitational acceleration.
03

Hooke's Law and Maximum Force Condition

The spring force applied to the upper block can be expressed using Hooke's Law: \(F = k \cdot A\), where \(A\) is the amplitude of oscillation. For the top block not to slip, we require \(k \cdot A \leq \mu_s \cdot m \cdot g\).
04

Solve for Maximum Amplitude

Rearrange the inequality to find the maximum possible amplitude \(A_{max}\) that satisfies the no-slip condition: \[ A \leq \frac{\mu_s \cdot m \cdot g}{k} \]. This gives us the maximum amplitude at which the top block will not slip.

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Key Concepts

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

Static Friction
Static friction is the force that keeps an object at rest when a force is applied to it. Imagine trying to slide a book across a table, but it doesn鈥檛 move. That's static friction at work! It acts between two surfaces that are not moving relative to each other. In our exercise, the top block and the bottom block have a static frictional force between them. This force helps the top block stay in place and not slide off.
  • Role of static friction: It counteracts the applied forces to prevent slipping.
  • Determining static friction: The maximum static friction can be calculated using the formula: \( f_s = \mu_s \cdot m \cdot g \), where \( \mu_s \) is the static friction coefficient, \( m \) is the mass of the top block, and \( g \) is the gravitational acceleration.
Static friction is essential in determining the maximum amplitude of oscillation in this system. If the force due to the oscillation exceeds this static friction, the top block will slip.
Hooke's Law
Hooke's Law describes how springs exert force. When a spring is either compressed or stretched, it applies a force to return to its original shape. This is why if you pull a spring, it pulls back!In our scenario, as the spring stretches or compresses, it exerts a force on the connected blocks. This force is described mathematically by Hooke's Law:- The formula is: \( F = k \cdot A \) - \( F \) represents the force applied by the spring. - \( k \) is the spring constant, a measure of the stiffness. - \( A \) is the amplitude of the oscillation, or how far the spring is stretched or compressed from its rest position.By knowing how to apply Hooke's Law, you can predict the behavior of blocks in oscillation and calculate the conditions that prevent slipping. Simply put, it's understanding the push and pull of spring systems.
Force Constant k
The force constant, often represented by the letter \( k \), is a critical factor in understanding spring behavior. Do you remember testing the `stiffness` of a spring with your hands? The stiffer the spring, the higher the force constant.Here鈥檚 what makes \( k \) important:- **Definition:** It quantifies the spring鈥檚 stiffness. Larger \( k \) means a stiffer spring.- **Role in oscillation:** Determines how much force the spring exerts for a given displacement: \( F = k \cdot A \).- **Connection to amplitude:** A strong spring (large \( k \)) exerts more force for the same amplitude.Understanding \( k \) allows us to comprehend how much force the spring can exert, which is vital for calculating how much the spring can oscillate without causing the top block to slip off.
Frictionless Surface
A frictionless surface sounds like something out of science fiction, but it's a concept often used in physics problems to simplify situations. Imagine an ice rink so smooth, nothing slows you down! In our exercise, the surface the bottom block rests on is frictionless, meaning: - What it means: There is no friction resisting the motion of the block along the surface. - Simplification: Helps focus on the forces at play, like the spring force and static friction between blocks, without surface friction complicating things. - Consequences: The only horizontal force acting is from the spring, allowing simple calculation of oscillation effects. This setup allows us to easily study the effect of the spring on the block system by eliminating additional frictional forces. It helps in focusing purely on the interaction between the spring and the blocks.

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Most popular questions from this chapter

Inside a NASA test vehicle, a \(3.50-\mathrm{kg}\) ball is pulled along by a horizontal ideal spring fixed to a friction-free table. The force constant of the spring is 225 \(\mathrm{N} / \mathrm{m}\) . The vehicle has a steady acceleration of \(5.00 \mathrm{m} / \mathrm{s}^{2},\) and the ball is not oscillating. Suddenly, when the vehicle's speed has reached 45.0 \(\mathrm{m} / \mathrm{s}\) , its engines turn off, thus eliminating its acceleration but not its velocity. Find (a) the amplitude and (b) the frequency of the resulting oscillations of the ball. (c) What will be the ball's maximum speed relative to the vehicle?

A uniform, \(1.00-m\) stick hangs from a horizontal axis at one end and oscillates as a physical pendulum. An object of small dimensions and with mass equal to that of the meter stick can be clamped to the stick at a distance \(y\) below the axis. Let \(T\) represent the period of the system with the body attached and \(T_{0}\) the period of the meter stick alone. (a) Find the ratio \(T / T_{0}\) . Evaluate your expression for \(y\) ranging from 0 to 1.0 \(\mathrm{m}\) in steps of \(0.1 \mathrm{m},\) and \(\operatorname{graph} T / T_{0}\) versus \(y .\) (b) Is there any value of \(y,\) other than \(y=0\), for which \(T=T_{0} ?\) If so, find it and explain why the period is unchanged when \(y\) has this value.

A 175 -g glider ona horizontal, frictionless air trackis attached to a fixed ideal spring with force constant 155 \(\mathrm{N} / \mathrm{m}\) . At the instant you make measurements on the glider, it is moving at 0.815 \(\mathrm{m} / \mathrm{s}\) and is 3.00 \(\mathrm{cm}\) from its equilibrium point. Use energy conservation to find (a) the amplitude of the motion and (b) the maximum speed of the glider. (c) What is the angular frequency of the oscillations?

A uniform, solid metal disk of mass 6.50 \(\mathrm{kg}\) and diameter 24.0 \(\mathrm{cm}\) hangs in a horizontal plane, supported at its center by a vertical metal wire. You find that it requires a horizontal force of 4.23 \(\mathrm{N}\) tangent to the rim of the disk to turn it by \(3.34^{\circ},\) thus twisting the wire. You now remove this force and release the disk from rest. (a) What is the torsion constant for the metal wire? (b) What are the frequency and period of the torsional oscillations of the disk? (c) Write the equation of motion for \(\theta(t)\) for the disk.

When a body 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 Ha. Find (a) the period of the motion; \((b)\) the angular frequency; (c) the mass of the body.
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