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Chapter 9: Problem 16

In this project we will study a phenomenon called stickslip friction. If you pull a block along a flat table with a soft spring, you will find that the block does not move continuously with a constant velocity, instead it moves in small jumps. This intermittent motion is called stick-slip friction, and it is the origin of the high-frequency vibrating tone you often hear from wheels that are not well lubricated. It is also one of the basic mechanisms leading to the wide distribution of earthquake sizes. Here, we will introduce and study a model for stick-slip friction for a block pulled by a spring sliding over a flat, horizontal surface, as illustrated in Fig. \(9.18\). The block has mass \(m\). A massless spring (with spring constant \(k\) and equilibrium length \(b\) ) is attached to the block at the point \(x\). The free end (the right-hand end in Fig. 9.18) of the spring is at the point \(x_{b}\). We move \(x_{b}\), the free end of the spring, with a constant velocity \(u\). The static and dynamic coefficients of friction for the contact between the block and the bottom surface are \(\mu_{s}\) and \(\mu_{d}\) respectively. The acceleration of gravity is \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\). The block starts at the position \(x\left(t_{0}\right)=0\) at the time \(t_{0}=0\). The position \(x_{b}\) of the free end of the spring is \(x_{b}\left(t_{0}\right)=x\left(t_{0}\right)+b\) at \(t_{0}\). (a) Draw a free-body diagram for the block. (b) Find the position of the spring attachment point \(x_{b}(t)\) as a function of time. (c) Show that the force, \(\mathbf{F}\) on the block from the spring is \(\mathbf{F}=k\left(x_{b}-x-b\right) \mathbf{i}\). Stationary state: First, let us characterize the stationary state, where the block is moving at a constant velocity. (d) Identify the forces acting on the block and draw a free-body diagram for the block in the stationary state. (e) Introduce force models for all the forces acting on the block. Find the normal force, \(N\), on the block. (f) Find the acceleration of the block in the stationary state. (g) Find the elongation \(\Delta L\) of the spring in the stationary state. (h) Find the position \(x(t)\) of the block as a function of time in the stationary state. Starting from rest: Let us now address the situation where the block starts from rest. That is, we assume that the block starts at \(x\left(t_{0}\right)=0 \mathrm{~m}\) with \(v\left(t_{0}\right)=0 \mathrm{~m} / \mathrm{s}\) at the time \(t_{0}=0 \mathrm{~s}\). (i) Identify the force acting on the block and draw a free-body diagram of the block before the block starts moving. Introduce force models for all the forces. (j) Find the elongation \(\Delta L\) of the spring at the instant the block starts moving. (k) Assume that the block starts at rest. Find the friction force on the block as a function of time in the period before the block starts moving. Sketch the friction force as a function of time until some time after the block has started moving. (1) Show that the acceleration of the block immediately after it starts moving is \(a=(k / m)\left(x_{b}-x-b\right)-\mu_{d} g\). Explain why you cannot use this relation for the acceleration to determine the subsequent motion of the block. General motion: Now, we will develop a general method to find the motion, \(x(t)\), of the block. First, we study the case when \(u=0 \mathrm{~m} / \mathrm{s}\) and the coefficients of friction are zero, \(\mu_{s}=\mu_{d}=0\). (m) Find an expression for the horizontal acceleration of the block. Show that \(x(t)=\) \(\left(v_{0} / \omega\right) \sin \omega t\), where \(\omega=(k / m)^{1 / 2}\), when \(v(0)=v_{0} .\) (n) Write a numerical algorithm to find the position and velocity of the block at a time \(t_{i}+\Delta t, x\left(t_{i}+\Delta t\right)\) and \(v\left(t_{i}+\Delta t\right)\), given the position and velocity of the block at a time \(t_{i}, x\left(t_{i}\right)\) and \(v\left(t_{i}\right)\). (o) Implement the numerical algorithm in a program to find the position of the block as a function of time for \(m=0.1 \mathrm{~kg}, k=100 \mathrm{~N} / \mathrm{m}, b=0.1 \mathrm{~m}\) and \(v_{0}=0.1 \mathrm{~m} / \mathrm{s}\). Plot the behavior for a simulation of \(2 \mathrm{~s}\), and compare the result of your program with exact solution. Ensure that you choose a time-step \(\Delta t\) the reproduces the exact solution with sufficient accuracy. What happens if you choose a too large time-step \(\Delta t ?\) Let us now address the situation when the block is pulled at a finite velocity, \(u\). (p) Modify your program to find the position of the block when \(u=0.1 \mathrm{~m} / \mathrm{s}\) and the block starts at rest. In this case, the exact solution is: $$ x(t)=u t-\frac{u}{\omega} \sin \omega t . $$ Compare your result with the exact solution by plotting both the simulated \(x\) and the exact \(x\) in the same plot. General motion with friction: Finally, we address the full complexity of the situation, and introduce non-zero friction forces. (q) Modify your program to include friction using \(\mu_{s}=0.6, \mu_{d}=0.3\). Show a plot of \(x(t)\) for \(m=0.1 \mathrm{~kg}\) and for \(m=1.0 \mathrm{~kg}\). (r) Plot the spring force \(F\) on the block as a function of time for both values of \(m\) and explain the differences. (s) What happens if you instead decrease \(k\) to \(k=10 \mathrm{~N}\) for \(m=0.1 \mathrm{~kg}\). Can you explain the behavior?

Short Answer

Expert verified
Calculate forces, derive motion equations, numerically simulate for given parameters. Compare with exact solutions and examine effects of varying mass and spring constant.

Step by step solution

01

- Free-body Diagram for the Block

Draw a block on a horizontal surface. Indicate the gravitational force (mg) acting downward, the normal force (N) acting upward, the frictional force (F_f) acting opposite to the direction of motion, and the spring force (F_s) acting towards the point of attachment of the spring.
02

- Position of Spring Attachment Point

The position of the spring attachment point x_b(t) can be found using the initial conditions given. As the free end of the spring moves with constant velocity u, this can be expressed as:x_b(t) = x_b(t_0) + ut
03

- Force on the Block from the Spring

The force on the block from the spring can be derived using Hooke's Law. The displacement of the spring from its equilibrium length is given by (x_b - x - b). Thus, the spring force is:\(\mathbf{F} = k(x_b - x - b) \mathbf{i} \)
04

- Stationary State Free-body Diagram

In the stationary state, when the block is moving at a constant velocity, draw a free-body diagram indicating the forces: gravitational force, normal force, friction force, and spring force all balanced.
05

- Force Models and Normal Force in Stationary State

Introduce force models for all the forces: friction force F_f = μ_d N, spring force F_s = k(x_b - x - b). The normal force can be calculated using the gravitational force: N = mg.
06

- Acceleration in Stationary State

In the stationary state, the block's velocity and thus acceleration are zero. Therefore, a = 0 m/s².
07

- Elongation of Spring in Stationary State

The elongation of the spring ΔL in the stationary state can be found by solving the equation F = ma = 0 for the displacement: ΔL = x_b - x - b.
08

- Position of Block as a Function of Time

Since the block is moving at constant velocity in the stationary state, its position can be expressed as x(t) = ut.
09

- Free-body Diagram before Block Starts Moving

Draw the forces acting on the block when at rest: gravitational force, normal force, static friction force (equal and opposite to the spring force), and the spring force F_s.
10

- Elongation at Instant Block Starts Moving

At the instant the block starts moving, elongation ΔL can be found through static friction threshold: kΔL = μ_s mg; solve for ΔL.
11

- Friction Force before Movement Starts

The friction force before the block starts moving is static friction: F_f(t) = μ_s N when the block is not moving. Sketch this as a horizontal line on a force vs. time graph until friction force becomes dynamic.
12

- Acceleration Immediately After Block Starts Moving

Show that the acceleration a is given by:a = (k/m)(x_b - x - b) - μ_d gExplain this involves dynamic friction immediately after starting to move, thus larger friction force initially transitioning to steady acceleration.
13

- Horizontal Acceleration with Zero Velocity and Friction

With zero velocity u and no friction coefficients, horizontal acceleration simplifies to F/m = k(x_b - x - b)/m. Show this results in harmonic motion: x(t) = (v_0/ω) sin(ωt), with ω = (k/m)^(1/2).
14

- Numerical Algorithm for Position and Velocity

Write a numerical algorithm to update the block's position and velocity using:x(t + Δt) = x(t) + v(t)Δtv(t + Δt) = v(t) + a(t)Δtwhere a(t) is instantaneous acceleration.
15

- Implementing Numerical Algorithm in Program

Implement the algorithm in a program setting m = 0.1 kg, k = 100 N/m, b = 0.1 m, v_0 = 0.1 m/s. Simulate position for 2 seconds and compare with exact solution x(t) = (v_0/ω) sin(ωt). Discuss accuracy with different Δt.
16

- Modifying Program for Non-zero Velocity

Modify program to move free end with u = 0.1 m/s, compare simulated x(t) with exact x(t) = ut - (u/ω) sin(ωt). Plot both in the same graph.
17

- Modifying Program to Include Non-zero Friction

Include friction with μ_s = 0.6, μ_d = 0.3. Plot x(t) for m = 0.1 kg and m = 1.0 kg, show differences due to mass.
18

- Spring Force as Function of Time

Plot the spring force F as a function of time for both masses m = 0.1 kg and m = 1.0 kg. Discuss differences due to varying inertia and spring force required.
19

- Effect of Decreasing Spring Constant

Decrease k to 10 N/m for m = 0.1 kg, discuss the effect on block motion. Explain behavior changes due to lower restoring spring force.

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Key Concepts

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

elastic mechanics
Elastic mechanics is the branch of physics that studies the behavior of elastic bodies under the action of forces. In this problem, the spring attached to the block is a perfect example of an elastic object. A key property of elastic materials is their ability to return to their original shape when the force deforming them is removed. The potential energy stored in the compressed or stretched spring follows Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from equilibrium.

This is given by the formula:
\(\mathbf{F} = -k \Delta x\).
Here, ‘k’ is the spring constant, and \(\Delta x\) is the displacement from equilibrium.

In our exercise, the restoring force of the spring is what causes the block to move back and forth. As the block moves, it stretches or compresses the spring, which then exerts a force back on the block. This interaction is foundational to understanding the stick-slip phenomenon.
numerical methods in physics
Numerical methods are techniques used to approximate solutions for mathematical problems that cannot be solved analytically. In studying the motion of our block-spring system, we turn to numerical methods to simulate the system's behavior over time.

One common approach is to use a simple numerical integration method like Euler's Method. For small time steps \(\Delta t\), we can approximate the position and velocity of the block at a new time \(t_{i+1}\) based on their values at time \(t_i\).

The basic process can be summarized as:
  • Update the position: \[x(t_{i+1}) = x(t_i) + v(t_i) \Delta t\]
  • Update the velocity: \[v(t_{i+1}) = v(t_i) + a(t_i) \Delta t\]
These approximations help simulate complex dynamical systems efficiently.

For higher precision, more advanced methods such as the Runge-Kutta methods may be employed. The choice of the time step \(\Delta t\) is crucial. Too large a time step may lead to inaccurate results, while too small a step increases the computational load.
dynamical systems
Dynamical systems are systems that evolve over time according to a set of rules. The system we are analyzing – a block attached to a spring – is a classic example of a simple harmonic oscillator, which is a type of dynamical system.

In dynamical systems, we are interested in the state of the system as it changes over time. The state can be described by variables such as position and velocity. For our block-spring system, the equations of motion are derived from Newton's second law, which is given by: \[F = ma\]
or, substituting the spring force \[k(x_b - x - b) - \mu_d mg = ma \].

To analyze and predict the motion, we solve these equations to get functions for position and velocity as a function of time. However, due to the complexity and non-linearity introduced by friction forces, numerical methods are required for practical solutions.

Understanding the nature of the stick-slip motion involves looking at how the forces interact to create periods of 'stick' where static friction holds the block in place, and 'slip' where dynamic friction allows the block to move.
force models
Force models are mathematical representations of the forces acting on an object. In the stick-slip friction problem, several forces are at play: gravitational force, normal force, frictional forces, and spring force.

Each force can be modeled as follows:
  • Gravitational force: \(F_g = mg\), acting downward.
  • Normal force: \(N\), equal to the gravitational force in magnitude but acting upward on the block balanced by the gravitational pull. N = mg
  • Frictional force: Can be static or dynamic. Static friction force prevents motion: \(F_f = \mu_s N\). When the block starts moving, dynamic friction takes over: \(F_f = \mu_d N\).
  • Spring force: Modeled by Hooke’s Law, \(F_s = k(x_b - x - b)\), acting in the direction to restore the spring to equilibrium.
Understanding these forces and how they interact are key to analyzing the block's motion under stick-slip conditions.
harmonic motion
Harmonic motion refers to a type of periodic motion where the restoring force is directly proportional to displacement. In the absence of friction, the block-spring system exhibits simple harmonic motion (SHM).

The mathematical form of SHM can be described by: \[x(t) = (v_0/\omega) \sin(\omega t)\].
Here, \(\omega = (k/m)^{1/2}\) is the angular frequency of oscillation.

The block oscillates back and forth with a frequency determined by the mass of the block and the spring constant.

However, when friction is introduced, the motion becomes more complex. The block undergoes damped harmonic motion where the amplitude of oscillation decreases over time due to energy loss from friction until it reaches a stationary state. In this state, the block eventually sticks and slips intermittently, demonstrating a more complicated and realistic behavior often seen in real-world systems.

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Most popular questions from this chapter

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