In this project we address the motion of an object sliding on a slippery surface-such as a ski sliding in a snowy track. You will learn how to find the equation of motion for sliding systems both analytically and numerically, and to interpret the results. We start by studying a simplified situation called frictional motion: A block is sliding on a surface. moving with a velocity \(v\) in the positive \(x\)-direction. The forces from the interactions with the surface results in an acceleration: $$ a=\left\\{\begin{array}{cc} -\mu(|v|) g & v>0 \\ 0 & v=0 \\ \mu(|v|) g & v<0 \end{array}\right. $$ where \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) is the acceleration of gravity. Let us first assume that \(\mu(v)=\) \(\mu=0.1\) for the surface. That is, we assume that the coefficient of friction does not depend on the velocity of the block. We give the block a push and release it with a velocity of \(5 \mathrm{~m} / \mathrm{s}\). (a) Find the velocity, \(v(t)\), of the block. (b) How long time does it take until the block stops? (c) Write a program where you find \(v(t)\) numerically using Euler's or Euler- Cromer's method. (Hint: You can find a program example in the textbook.) Use the program to plot \(v(t)\) and compare with your analytical solution. Use a timestep of \(\Delta t=0.01 .\) The description of friction provided above is too simplified. The coefficient of friction is generally not independent of velocity. For dry friction, the coefficient of friction can in some cases be approximated by the following formula: $$ \mu(v)=\mu_{d}+\frac{\mu_{s}-\mu_{d}}{1+v / v^{*}} $$ where \(\mu_{d}=0.1\) often is called the dynamic coefficient of friction, \(\mu_{s}=0.2\) is called the static coefficient of friction, and \(v^{*}=0.5 \mathrm{~m} / \mathrm{s}\) is a characteristic velocity for the contact between the block and the surface. (d) Show that the acceleration of the block is: $$ a(v)=-\mu_{d g}-g \frac{\mu_{s}-\mu_{d}}{1+v / v^{*}} $$ for \(v>0\). (e) Use your program to find \(v(t)\) for the more realistic model, with the same starting velocity, and compare with your previous results. Are your results reasonable? Explain. The model we have presented so far is only relevant at small velocities. At higher velocities the snow or ice melts, and the coefficient of friction displays a different dependency on velocity: $$ \mu(v)=\mu_{m}\left(\frac{v}{v_{m}}\right)^{-\frac{1}{2}} \text { when } v>v_{m} $$ where \(v_{m}\) is the velocity where melting becomes important. For lower velocities the model presented above with static and dynamic friction is still valid. (f) Show that $$ \mu_{m}=\mu_{d}+\frac{\mu_{s}-\mu_{d}}{1+v_{m} / v^{*}}, $$ in order for the coefficient of friction to be continuous at \(v=v_{m}\). (g) Modify your program to find the time development of \(v\) for the block when \(v_{m}=1.5 \mathrm{~m} / \mathrm{s}\). Compare with the two other models above: The model without velocity dependence and the model for dry friction. Comment on the results. (h) The process may be clearer if you plot the acceleration for all the three models in the same plot. Modify your program to plot \(a(t)\), plot the results, and comment on the results. What would happen if the initial velocity was much higher or much lower than \(5 \mathrm{~m} / \mathrm{s}\) ?

Step by step solution

01

- Understanding the Problem

The problem involves a block sliding on a surface with given initial velocity. The first task is to find the velocity of the block over time considering constant friction coefficient.

02

- Define Parameters and Initial Conditions

Given parameters are: \( \mu = 0.1 \), \( g = 9.8 \mathrm{~m/s}^2 \), and \( v(0) = 5 \mathrm{~m/s} \).

03

- Finding Analytical Solution for Velocity

Use the given acceleration formula, \[ a = -\mu g \text{ for } v > 0 \]. Since acceleration \( a = \frac{dv}{dt} \), integrate to find velocity: \[ \frac{dv}{dt} = -0.1 \cdot 9.8 \]. Integrating both sides, we get: \[ v(t) = 5 - 0.98t \].

04

- Determine Time When Block Stops

Set \( v(t) = 0 \) and solve for \( t \): \[ 0 = 5 - 0.98t \]. Solving for \( t \), we get: \[ t = \frac{5}{0.98} \approx 5.102 \,\text{s} \].

05

- Numerical Solution Using Euler's Method

For numerical solution, use Euler's method: \[ v(t + \Delta t) = v(t) + a \Delta t \]. Set \( \Delta t = 0.01 \). Initialize \( v = 5 \) and iterate until \( v \leq 0 \).

06

- More Realistic Model

Using the given formula for \( \mu(v) \), \( \mu(v) = \mu_d + \frac{\mu_s - \mu_d}{1 + v/v^*} \), calculate acceleration: \[ a(v) = -g (\mu_d + \frac{\mu_s - \mu_d}{1 + v/v^*}) \].

07

- Analytical Acceleration for Realistic Model

From previous step, substitute \( \mu_d = 0.1, \mu_s = 0.2, \text{ and } v^* = 0.5 \): \[ a(v) = -9.8 (0.1 + \frac{0.2-0.1}{1 + v/0.5}) \].

08

- Modify Program for Realistic Model

Implement the numerical solution using the updated acceleration formula in the code. Compare results with analytical and first numerical solution.

09

- Higher Velocity Model

Use \( v_m = 1.5 \mathrm{~m/s} \) and the given formula for \( \mu(v) \) when \( v > v_m \): \( \mu(v) = \mu_m \left( \frac{v}{v_m} \right)^{-\frac{1}{2}} \).

10

- Continuous Friction at \( v = v_m \)

To show \( \mu_m \) for continuity, set: \[ \mu_m = \mu_d + \frac{\mu_s - \mu_d}{1 + v_m / v^*}, \].

11

- Modify Program for Higher Velocity Model

Update the program to integrate over intervals considering \( \mu(v) \) dependency when \( v > v_m \). Compare results with previous models.

12

- Plot and Analyze Accelerations

Modify program to plot \( a(t) \) for all three models. Analyze and comment on differences and what would happen at different initial velocities.

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Key Concepts

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

Equation of Motion

The equation of motion is fundamental in kinematics and helps describe how an object's velocity and position change over time. In this problem, we focus on a block sliding on a surface with friction. The forces at play generate an acceleration that is dependent on the velocity. The derived acceleration function is:

\[a = \begin{cases}-\mu g & \,v > 0\ 0 & \,v = 0\ \mu g & \,v < 0\end{cases}\]

With this piecewise function, where \(\mu\) is a constant friction coefficient and \(g\) is the gravitational acceleration \(9.8 \, \mathrm{m/s}^2\), we can derive the velocity function over time. Knowing the initial conditions, such as the initial velocity \(v(0) = 5 \, \mathrm{m/s}\), allows us to find the analytical solution for velocity using standard integration techniques. The derived equation for velocity over time is:

\[v(t) = 5 - 0.98t\]

This simple linear equation helps predict when the block will come to a stop by setting \(v(t) = 0\) and solving for \(t\). This leads to the block stopping at around \(t \approx 5.102 \, \text{seconds}\).

Analytical and Numerical Techniques

Analytical techniques involve solving equations algebraically to obtain exact solutions. In this exercise, we started with the analytical solution for velocity. However, real-world problems frequently require numerical approaches for practical or complex systems.

Numerical techniques approximate solutions using algorithms and iteration. Euler's method is a simple numerical technique that we apply here. It uses the following update step:

\[v(t + \Delta t) = v(t) + a \Delta t\]

Where \(\Delta t\) is a small time step (e.g., \(0.01 \, \text{seconds}\)). By iteratively updating the velocity and position, we can simulate the block's motion. This method, while approximate, is powerful for solving problems that are analytically intractable or highly non-linear.

Euler's Method

Euler's method is a straightforward approach for numerical integration. Here’s how it works:

• Step 1: Initialize with known values, such as \(v(0) = 5 \, \mathrm{m/s}\).
• Step 2: Calculate acceleration \(a\) at the current time step.
• Step 3: Update the velocity using \(v(t + \Delta t) = v(t) + a \Delta t\).
• Step 4: Repeat the process for subsequent time steps until the stopping condition (e.g., \(v \leq 0\)) is met.

For the given problem, Euler's method helps us numerically determine the velocity over time, considering both constant and velocity-dependent friction. By comparing the numerical results with the analytical solution, we validate the accuracy and reliability of Euler's method for predicting the motion of the block.

Velocity-Dependent Friction

Friction in real-life situations is often velocity-dependent. For dry friction, the coefficient of friction can be modeled as:

\[ \mu(v) = \mu_d + \frac{\mu_s - \mu_d}{1 + v / v^*} \]

Where \(\mu_d\) is the dynamic coefficient, \(\mu_s\) is the static coefficient, and \(v^*\) is a characteristic velocity. This relationship adjusts the friction depending on the block's speed. For the acceleration, the formula becomes:

\[ a(v) = -g \left( \mu_d + \frac{\mu_s - \mu_d}{1 + v / v^*} \right) \]

This velocity-dependent friction is crucial for realistic simulations. It accounts for changes in friction as the object's velocity varies. Implementing this in the numerical model modifies our previous Euler's method approach. Programs that include such dependencies offer more accurate predictions and insights, especially when properties like snow or ice melt at higher speeds.

Understanding and modeling these dependencies enable us to predict and compare results for different friction models—static, dynamic, and those influenced by velocity.