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Chapter 6: Problem 21

A worker wishes to pile a cone of sand onto a circular area in his yard. The radius of the circle is \(R\), and no sand is to spill onto the surrounding area (Fig. 6-46). If \(\mu^{\text {stat }}\) is the static coefficient of friction between each layer of sand along the slope and the sand beneath it (along which is might slip), show that the greatest volume of sand that can be stored in this manner is \(\pi \mu^{\text {stat }} R^{3} / 3\). (The volume of a cone is \(A h / 3\), where \(A\) is the base area and \(h\) is the cone's height.)

Short Answer

Expert verified
\(V = \frac{\pi \mu^{\text{stat}} R^3}{3}\)

Step by step solution

01

Understand the problem

Identify the given parameters: the radius of the circular base of the cone, which is denoted by \(R\), and the static coefficient of friction between layers of sand, denoted by \(\mu^{\text{stat}}\). The goal is to find the greatest volume of sand that can be stored in the shape of a cone without spilling.
02

Determine the relationship between height and radius

For a cone, the height \(h\) and the radius \(R\) are related by the static coefficient of friction. The maximum height \(h\) of the cone is when the slope angle \(\theta\) equals the angle of repose, which can be expressed as \(\tan \theta = \mu^{\text{stat}}\). Therefore, \(h = R \mu^{\text{stat}}\).
03

Calculate the base area of the cone

The base area \(A\) of the cone is the area of the circle at the bottom, which is given by \(A = \pi R^2\).
04

Use the volume formula for a cone

The volume \(V\) of a cone is given by the formula \(V = \frac{1}{3} A h\), where \(A\) is the base area and \(h\) is the height. Substitute the values of \(A\) and \(h\) obtained from the previous steps.
05

Perform the substitution

Substitute \(A = \pi R^2\) and \(h = R \mu^{\text{stat}}\) into the volume formula: \[V = \frac{1}{3} \pi R^2 (R \mu^{\text{stat}})\].
06

Simplify the expression

Simplify the expression to find the maximum volume: \[V = \frac{1}{3} \pi R^2 R \mu^{\text{stat}} = \frac{1}{3} \pi R^3 \mu^{\text{stat}}\]

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Key Concepts

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

Static Coefficient of Friction
The static coefficient of friction (often denoted as \( \mu^{\text{stat}} \) or just \( \mu \)) is a measure of the resistance to the initiation of sliding motion between two surfaces. This coefficient depends on the materials in contact and is a dimensionless number. For instance, rubber on concrete might have a higher static coefficient of friction compared to ice on steel. In the context of the cone of sand problem, \( \mu^{\text{stat}} \) is crucial because it determines the steepest angle at which the sand can pile up without beginning to slide down. This relationship can be expressed as \[ \tan(\theta) = \mu^{\text{stat}} \]. This equation ties the static coefficient of friction to the angle of repose, or the steepest angle of a stable pile of sand.
Volume of a Cone
The volume of a cone is a fundamental geometric concept. For a cone with a circular base, the volume \( V \) is given by the formula \[ V = \frac{1}{3} \pi R^2 h \]. Here, \( R \) is the radius of the circular base and \( h \) is the height of the cone. This formula can be understood as a result of how a cone is essentially a pyramid with a circular base, and its volume is one-third the product of the base area and the height. In the cone of sand problem, knowing this volume formula helps us determine how much sand can be piled up within the given dimensions without exceeding the circle's boundary. Specifically, to maximize this volume without spilling, we use the relationship between \( h \) and \( R \) provided by the angle of repose influenced by the static coefficient of friction.
Angle of Repose
The angle of repose is the steepest angle at which a pile of granular material (like sand) remains stable without sliding. It can be visualized by imagining a sandpile; the sides of the pile form a slope at this maximum angle. The angle of repose is influenced by the material's properties and the static coefficient of friction. For sand, a common range for the angle of repose is between 30 and 34 degrees, but it can vary depending on the moisture content and grain size. Mathematically, the angle of repose \(\theta\) is related to the static coefficient of friction by the equation \[ \tan(\theta) = \mu^{\text{stat}} \]. Understanding this connection helps us realize why the maximum height of the cone in the problem can be expressed as \ h = R \mu^{\text{stat}} \, depicting a direct relationship between the base radius of the cone and its height through the frictional properties of sand.

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Most popular questions from this chapter

A motorcycle and \(60.0 \mathrm{~kg}\) rider accelerate at \(3.0 \mathrm{~m} / \mathrm{s}^{2}\) up a ramp inclined \(10^{\circ}\) above the horizontal. (a) What is the magnitude of the net force acting on the rider? (b) What is the magnitude of the force on the rider from the motorcycle?

We know that as an object passes through the air, the air exerts a resistive force on it. Suppose we have a spherical object of radius \(R\) and mass \(m\). What might the force plausibly depend on? \- It might depend on the properties of the object. The only ones that seem relevant are \(m\) and \(R\). \- It might depend on the object's coordinate and its derivatives: \(\vec{r}, \vec{v}, \vec{a}, \ldots\) \- It might depend on the properties of the air, such as the density, \(\rho\). (a) Explain why it is plausible that the force the air exerts on a sphere depends on \(R\) but implausible that it depends on \(m\). (b) Explain why it is plausible that the force the air exerts depends on the object's speed through it, \(|\vec{v}|\), but not on its position, \(\vec{r}\), or acceleration, \(\vec{a}\). (c) Dimensional analysis is the use of units (e.g., meters, seconds, or newtons) associated with quantities to reason about the relationship between the quantities. Using dimensional analysis, construct a plausible form for the force that air exerts on a spherical body moving through it.

Suppose the coefficient of static friction between the road and the tires on a Formula One car is \(0.6\) during a Grand Prix auto race. What speed will put the car on the verge of sliding as it rounds a level curve of \(30.5 \mathrm{~m}\) radius?

An amusement park ride consists of a car moving in a vertical circle on the end of a rigid boom of negligible mass. The combined weight of the car and riders is \(5.0 \mathrm{kN}\), and the radius of the circle is \(10 \mathrm{~m}\). What are the magnitude and direction of the force of the boom on the car at the top of the circle if the car's speed there is (a) \(5.0 \mathrm{~m} / \mathrm{s}\) and (b) \(12 \mathrm{~m} / \mathrm{s}\) ?

For sport, a \(12 \mathrm{~kg}\) armadillo runs onto a large pond of level, frictionless ice with an initial velocity of \(5.0 \mathrm{~m} / \mathrm{s}\) along the positive direction of an \(x\) axis. Take its initial position on the ice as being the origin. It slips over the ice while being pushed by a wind with a force of \(17 \mathrm{~N}\) in the positive direction of the \(y\) axis. In unit-vector notation, what are the animal's (a) velocity and (b) position vector when it has slid for \(3.0 \mathrm{~s}\) ?
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