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Chapter 6: Problem 76

A house is built on the top of a hill with a nearby slope at angle \(\theta=45^{\circ}\) (Fig. 6-55). An engineering study indicates that the slope angle should be reduced because the top layers of soil along the slope might slip past the lower layers. If the coefficient of static friction between two such layers is \(0.5\), what is the least angle \(\phi\) through which the present slope should be reduced to prevent slippage?

Short Answer

Expert verified
Reduce the slope by approximately \(18.44^{\circ}\) to prevent slippage.

Step by step solution

01

Identify the Critical Angle of Friction

The critical angle at which slippage occurs is equal to the angle whose tangent is the coefficient of friction. This angle is given by \( \theta_c = \arctan(\mu) \), where \( \mu = 0.5 \) is the coefficient of static friction.
02

Calculate the Critical Angle of Friction

Using \( \mu = 0.5 \), calculate the critical angle: \( \theta_c = \arctan(0.5) \). The calculated angle \( \theta_c \) is approximately \( 26.565^{\circ} \).
03

Determine the Necessary Reduction in Angle

Since the current slope angle is \( \theta = 45^{\circ} \) and the soil is stable only at angles less than \( \theta_c \), we need to reduce the slope angle by \( \phi = \theta - \theta_c = 45^{\circ} - 26.565^{\circ} \).
04

Calculate the Reduction Angle

The reduction angle \( \phi \) is calculated as \( 45^{\circ} - 26.565^{\circ} = 18.435^{\circ} \). This is the least angle by which the slope should be reduced to prevent slippage.

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Key Concepts

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

Critical Angle of Friction
Whenever we deal with the potential for slippage on a slope, understanding the critical angle of friction is essential. The critical angle of friction refers to the specific angle where a surface will begin to slip over another due to insufficient frictional force to hold it in place. In this particular problem, our goal was to determine this angle to make sure that the hill slope remains stable.
The formula we use to find this angle is \( \theta_c = \arctan(\mu) \), where \( \mu \) is the coefficient of static friction. The tangential function is used here because it relates the angle of inclination to the forces acting parallel and perpendicular to the surface.
For instance, with a coefficient of static friction of 0.5, the critical angle of friction \( \theta_c \) is calculated as \( \arctan(0.5) \). This translates to approximately 26.565 degrees, a crucial piece of information for engineers looking to ensure slope stability.
Slope Stability
Slope stability is a key concern in the construction and maintenance of structures such as houses, especially when they are built on or near sloped landscapes. In general, a slope becomes unstable when the gravitational force acting downward along the slope exceeds the frictional force that holds the material in place. When this happens, the material on the slope is likely to start sliding downwards.
Factors affecting slope stability include:
  • Soil type and cohesion
  • Water content
  • Angle of the slope
  • Weight of any structures built on the slope
Reducing the slope angle is a common approach to increasing stability. By ensuring the angle is kept lower than the critical angle of friction, which in this case is calculated to be about 26.565 degrees, we can significantly lower the risk of slippage. This ensures the house and surrounding environment remain safe and secure.
Coefficient of Friction
The coefficient of friction is a measure that represents the degree of interaction between surfaces in contact. The static coefficient of friction, in particular, indicates how much resistance there is to movement when an object is placed on a surface without slipping.
It is essential to understand the coefficient of friction in the context of slope stability because it helps determine how easily one layer of soil might slide over another. It is determined by the nature of the materials in contact; rougher surfaces generally have higher coefficients of friction.
In this exercise, the static coefficient of friction between the soil layers is given as 0.5. This value is used to calculate the critical angle. A higher coefficient would mean greater resistance to slippage, implying more stable slopes, while a lower coefficient might indicate a higher risk. Understanding these concepts allows engineers to better design and modify landscapes to prevent dangerous slippages.

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Most popular questions from this chapter

ILW What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is \(29 \mathrm{~km} / \mathrm{h}\) and the \(\mu_{s}\) between tires and track is \(0.32 ?\)

A worker pushes horizontally on a \(35 \mathrm{~kg}\) crate with a force of magnitude \(110 \mathrm{~N}\). The coefficient of static friction between the crate and the floor is 0.37. (a) What is the value of \(f_{s, \max }\) under the circumstances? (b) Does the crate move? (c) What is the frictional force on the crate from the floor? (d) Suppose, next, that a second worker pulls directly upward on the crate to help out. What is the least vertical pull that will allow the Fig. \(6-2\) first worker's \(110 \mathrm{~N}\) push to move the crate? (e) If, Problem instead, the second worker pulls horizontally to help out, what is the least pull that will get the crate moving?

SSM WWW A bedroom bureau with a mass of \(45 \mathrm{~kg}\), including drawers and clothing, rests on the floor. (a) If the coefficient of static friction between the bureau and the floor is \(0.45\), what is the magnitude of the minimum horizontal force that a person must apply to start the bureau moving? (b) If the drawers and clothing, with \(17 \mathrm{~kg}\) mass, are removed before the bureau is pushed, what is the new minimum magnitude?

minz A sling-thrower puts a stone \((0.250 \mathrm{~kg})\) in the sling's pouch \((0.010 \mathrm{~kg})\) and then begins to make the stone and pouch move in a vertical circle of radius \(0.650 \mathrm{~m}\). The cord between the pouch and the person's hand has negligible mass and will break when the tension in the cord is \(33.0 \mathrm{~N}\) or more. Suppose the slingthrower could gradually increase the speed of the stone. (a) Will the breaking occur at the lowest point of the circle or at the highest point? (b) At what speed of the stone will that breaking occur?

In the early afternoon, a car is parked on a street that runs down a steep hill, at an angle of \(35.0^{\circ}\) relative to the horizontal. Just then the coefficient of static friction between the tires and the street surface is \(0.725 .\) Later, after nightfall, a sleet storm hits the area, and the coefficient decreases due to both the ice and a chemical change in the road surface because of the temperature decrease. By what percentage must the coefficient decrease if the car is to be in danger of sliding down the street?
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