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Chapter 9: Problem 1

Walking on ice is usually more tiring than walking on a dry road. Why?

Short Answer

Expert verified
Walking on ice is more tiring because there is less friction, which makes it harder to maintain balance and traction.

Step by step solution

01

- Understand the Concept of Friction

Friction is the force that resists the motion of one surface relative to another with which it is in contact. It is essential in helping us move, as it provides the grip needed to push against the ground and propel ourselves forward.
02

- Compare Friction on Ice and Dry Road

Ice has significantly less friction compared to a dry road due to its smooth and slippery surface. A dry road, on the other hand, has much more friction, which helps in better traction and easier movement.
03

- Analyze Movement on Ice

When walking on ice, the reduced friction makes it harder to get a firm footing, causing muscles to work harder to maintain balance and move forward. This extra effort makes walking on ice more tiring.
04

- Conclusion

The primary reason that walking on ice is more tiring than walking on a dry road is the lack of friction, which requires more energy to maintain balance and traction.

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Key Concepts

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

Friction
Friction is a crucial force in our daily lives. It is the resistance that one surface or object encounters when moving over another. This force allows us to walk, drive, write, and perform many other activities. Without friction, surfaces would slide past each other uncontrollably. Friction occurs because surfaces are not perfectly smooth; even the smoothest surfaces have tiny bumps and grooves. These irregularities catch on each other, resisting movement. There are two main types of friction: static and kinetic. Static friction acts on objects that are not moving, helping them stay in place, while kinetic friction acts on moving objects, slowing them down. The amount of friction depends on the materials involved and how much they are pressed together. More friction means more grip and control, helping us move efficiently and safely.
Surface Traction
Surface traction is the grip or hold that a surface provides to prevent slipping and sliding. It is a direct result of friction. High traction surfaces, like rubbery floors or asphalt roads, offer strong resistance against slipping. This is because they provide a lot of friction, making it easy to maintain control while walking or driving. On the other hand, low traction surfaces, such as ice or wet tiles, provide very little resistance. This lack of friction makes it challenging to maintain balance and control. For example, many sports shoes have specialized treads to enhance traction on specific surfaces, helping athletes perform better. Traction is essential for safety in everyday activities. Proper traction can prevent falls and accidents, especially in slippery conditions. Walking on ice is difficult because of the low traction, requiring more effort to stay balanced and move.
Energy Expenditure
Energy expenditure refers to the amount of energy a person uses to perform activities. The body generates this energy through metabolic processes, fueling muscles to produce movement. Different activities require different amounts of energy. Walking on a dry, high-traction road is relatively easy and smooth, requiring less energy to maintain a steady pace. However, walking on ice is more tiring because of the lack of friction and traction. The muscles have to work harder to keep balance and prevent slipping. This increased effort means the body consumes more energy. Everyday tasks can also vary in energy expenditure based on the conditions. For example:
  • Walking uphill requires more energy than walking on a flat surface.
  • Carrying a heavy load increases energy expenditure compared to walking empty-handed.
Understanding how different factors affect energy expenditure can help in planning and performing activities more efficiently, reducing fatigue and improving performance.

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Most popular questions from this chapter

In this project we will create a model for an atom with mass \(m\) in the vicinity of two large molecules. We will assume that the molecules are massive and do not move. We will first look at a one-dimensional movement along the \(\mathrm{x}\)-axis, and later expand to higher dimensions. The interactions between the atom and the molecules can be described by the potential $$ U(x)=U_{0}\left(\left[\left(\frac{x}{d}\right)^{2}-1\right]^{2}\right)=\frac{U_{0}}{d^{4}}\left(x^{2}-d^{2}\right)^{2}=\frac{U_{0}}{d^{4}}\left(x^{4}-2 x^{2} d^{2}+d^{4}\right) \quad(9.80) $$ where \(U_{0}\) is a known constant measured in Joules, \(x\) is the position of the atom, and \(d\) is a known length. We will assume all other forces on the atom are small in comparison, and neglect them in our model. (a) Make an energy diagram. Find the equilibrium points, mark these on the diagram and characterize their stability. (b) Choose two different energies that give two distinct types of motions, draw them into the diagram, and describe the motion in each case. (c) If the atom starts at rest at \(x=2 d\), what is the velocity of the atom at the point \(x=d ?\) (d) If the atom starts at rest at \(x=d\) with the velocity \(v_{0}\), how large must \(v_{0}\) be for the atom to reach the point \(x=-d\) ? (e) Show that the acceleration of the atom is $$ a=-\frac{4 U_{0}}{m d^{4}}\left(x^{3}-x d^{2}\right) $$ (f) Write a numerical algorithm to find the position and velocity of the atom at a time \(t+\Delta t\), given the position and velocity of the atom at a time \(t\). (g) Implement the numerical algorithm in a program to find the position of the atom as a function of time from \(t=0 \mathrm{~ns}\) to \(t=10 \mathrm{~ns}\) with a timestep of \(\Delta t=0.01 \mathrm{~ns}\) for \(m=1, d=0.1 \mathrm{~nm}\) and \(U_{0}=1 \mathrm{~nJ}\). (h) Make a plot of two distinct \(\mathrm{x}(\mathrm{t})\) from \(t=0 \mathrm{~ns}\) to \(t=10 \mathrm{~ns}\), the first from a running of the program with the initial conditions \(x_{0}=d\) and \(v_{0}=0.5 \mathrm{~m} / \mathrm{s}\), and the other from a running of the program with the initial conditions \(x_{0}=d\) and \(v_{0}=1.5 \mathrm{~m} / \mathrm{s}\). Make another plot of \(v(t)\) for the same initial conditions. (i) Describe the behavior of the atom in both simulations and sketch the motion in an energy-diagram. We will now look at the same system, but in two dimensions. The Atom interacts with a surface in such a way that the potential of the atom is given as $$ U(r)=U_{0}\left(\left[\left(\frac{r}{d}\right)^{2}-1\right]^{2}\right)=\frac{U_{0}}{d^{4}}\left(r^{2}-d^{2}\right)^{2}=\frac{U_{0}}{d^{4}}\left(r^{4}-2 r^{2} d^{2}+d^{4}\right)(9.82) $$ where \(r=\sqrt{x^{2}+y^{2}}\) is the distance to the origin. (We can no longer interpret this interaction as an effect from two molecules. Instead we interpret it as an approximation of a more complicated interaction from many surrounding atoms.) (j) Show that the acceleration on the atom can be written $$ \mathbf{a}=-\frac{4 U_{0}}{m d^{4}}\left(r^{3}-r d^{2}\right) \frac{\mathbf{r}}{r} $$ (k) Rewrite your program to find the velocity and position of the atom using the new expression for the force \(F\). Use vectorized expressions in your code. (I) Plot the motion of an atom starting in \(\mathbf{r}_{0}=(d, 0)\) from \(t=0 \mathrm{~ns}\) to \(t=20 \mathrm{~ns}\) for the initial velocities \(\mathbf{v}=(0,0.5 \mathrm{~m} / \mathrm{s}), \mathbf{v}=(0,1 \mathrm{~m} / \mathrm{s})\) and \(\mathbf{v}=(0,1.5 \mathrm{~m} / \mathrm{s})\). (m) Can you choose initial conditions \(\mathbf{r}_{0}\) and \(\mathbf{v}_{0}\) in such a manner that the atom moves in a circular orbit with a constant radius? If so, what initial conditions are those? Plot the motion for these conditions.

A fireman of mass \(m\) is in hurry, and jumps onto a vertical pole leading into the garage. He holds the pole so tight that he slides downwards with constant velocity. (a) Draw a free-body diagram of the fireman on the pole. (b) Determine the friction force on the fireman. (c) The dynamic coefficient of friction between the fireman and the pole is \(\mu_{d}\). Find the normal force from the fireman on the pole.

A homogeneous rope with mass \(m\) hangs between two equally high poles. The angle between the rope and the horizontal at each of the attachment points is \(\alpha\). (a) Find the tension at each end of the rope. (b) Find the tension at the lowest point of the rope. (c) Is it possible to tighten the rope so much that \(\alpha=0\) ?

In this project you will learn to use Newton's laws and the force model for air resistance in a wind field to address the motion of a light object in strong winds. We start from a simple model without wind and gradually add complexity to the model, until we finally address the motion in a tornado. Motion without wind: First, we address the motion of the feather without wind. (a) Identify the forces acting on a feather while it is falling and draw a free-body diagram for the feather. (b) Introduce quantitative force models for the forces, and find an expression for the acceleration of the feather. You may assume a quadratic law for air resistance. (c) If you release the feather from rest, its velocity will tend asymptotically toward the terminal velocity, \(v_{T}\). Show that the terminal velocity is \(v_{T}=-(m g / D)^{1 / 2}\), where \(D\) is the constant in the air resistance model. (d) We release the feather from a distance \(h\) above the floor and measure the time \(t\) until the feather hits the floor. You may assume that the feather falls with a constant velocity equal to the terminal velocity. Show how you can determine \(D / m g\) by measuring the time \(t\). Estimate \(D / m g\) when you release the feather from a height of \(2.4 \mathrm{~m}\) above the floor and it takes \(4.8 \mathrm{~s}\) until it hits the floor. (e) We will now develop a more precise model where we do not assume that the velocity is constant. You release the feather from the height \(h\) at the time \(t=0 \mathrm{~s}\). Find the equation you have to solve to find the position of the feather as a function of time. What are the initial conditions? (f) Write a program that solves this equation to find the velocity and position as a function of time \(t\). Use the parameters you determined above, and test the program by ensuring that it produces the correct terminal velocity. (g) Fig. \(9.19\) shows the position and velocity calculated with the program using the parameters found above. Was the approximation in part (d) reasonable? Explain your answer. Model with wind: We have now found a model that can be used to find the motion of the feather. We will now find the motion of the feather in three dimensions while it is blowing. The velocity of the wind varies in space, so that the wind velocity \(\mathbf{w}\) is a function of the position \(\mathbf{r}\). We write this as \(\mathbf{w}=\mathbf{w}(\mathbf{r})\). (h) Find an expression for the acceleration of the feather. The expression may include the wind velocity \(\mathbf{w}(\mathbf{r})\). Let the z-axis correspond to the vertical direction. (i) Assume that the feather is moving in an approximately horizontal plane- that is you may assume that the vertical acceleration is negligible. How does the wind have

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?
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