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Chapter 5: Problem 25

In emergencies with major blood loss, the doctor will order the patient placed in the Trendelenburg position, in which the foot of the bed is raised to get maximum blood flow to the brain. If the coefficient of static friction between the typical patient and the bed sheets is \(1.20,\) what is the maximum angle at which the bed can be tilted with respect to the floor before the patient begins to slide?

Short Answer

Expert verified
The maximum angle is approximately 50.2 degrees.

Step by step solution

01

Understanding the problem

We need to determine the maximum angle at which a bed can be tilted before a patient begins to slide, given that the coefficient of static friction between the patient and the bed sheets is 1.20. This involves using the relationship between static friction force, gravitational force components, and the angle of incline.
02

Identifying the acting forces

The forces acting on the patient are the component of gravitational force pulling the patient down the slope and the static frictional force resisting this motion. The component of gravitational force along the slope is given by \(mg \sin \theta\) and the static frictional force is given by \(f_s = \mu_s N\), where \(N\) is the normal force.
03

Expressing the normal force

The normal force \(N\) is perpendicular to the inclined plane and is given by the component of gravitational force perpendicular to the slope. Thus, \(N = mg \cos \theta\).
04

Setting up the friction condition

For the patient to just start sliding, the maximum static friction force equals the component of the gravitational force along the incline: \(mg \sin \theta = \mu_s mg \cos \theta\).
05

Simplifying the equation

We can cancel \(mg\) from both sides of the equation, giving us \(\sin \theta = \mu_s \cos \theta\). Simplifying further, we get \( \tan \theta = \mu_s\), hence \(\theta = \arctan(\mu_s)\).
06

Calculating the maximum angle

Substitute the given coefficient of static friction, \(\mu_s = 1.20\), to find \(\theta\).\[\theta = \arctan(1.20)\]Use a calculator to find \(\theta\).
07

Finding the angle

Using a calculator, \(\theta = \arctan(1.20) \approx 50.2^\circ\).

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Key Concepts

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

Static Friction
Static friction is the force that keeps objects at rest when they are in contact with a surface, preventing them from sliding. It acts parallel to the contact surface and is a crucial factor when considering objects on inclined planes.
Its magnitude can vary depending on the forces acting upon the object, but it reaches its maximum just before the object begins to move.
This maximum static friction force is given by the formula:
  • \[ f_s = \mu_s \, N \]
where \( f_s \) is the static friction force, \( \mu_s \) is the coefficient of static friction, and \( N \) is the normal force.
In our problem, the static friction is what prevents the patient from sliding down the inclined bed, and we use this formula to calculate the point where friction no longer holds the patient in place.
Inclined Plane
An inclined plane is a flat surface tilted at an angle to the horizontal. It is a simple machine that reduces the amount of force needed to lift an object by extending the distance over which the force is applied.
In this exercise, the inclined plane is the bed tilted at an angle to help in medical situations.
To analyze problems on an inclined plane, we break down the gravitational force into two components:
  • One parallel to the plane, which tries to pull the object down.
  • One perpendicular to the plane, which contributes to the normal force.
The angle of the incline significantly affects these components, especially when concerning the balance between gravitational force and static friction.
Gravitational Force
Gravitational force is the force that pulls objects toward each other, keeping everything anchored on Earth's surface.
It acts downward due to the object's mass and Earth's gravity:\[ F_g = mg \]
where \( m \) is the mass and \( g \) is the acceleration due to gravity. On an inclined plane, we consider how this force divides into:
  • \( mg \sin \theta \) - the component pulling the patient down the incline.
  • \( mg \cos \theta \) - the component contributing to the normal force.
Managing these components helps us understand how gravity affects movement on an inclined plane.
Normal Force
The normal force is the force exerted by a surface perpendicular to an object resting on it.
On an inclined plane, this force counteracts the component of gravitational force that presses an object into the surface.
  • \[ N = mg \cos \theta \]
This equation describes the normal force on an incline, where \( \theta \) is the angle of the plane.
Since normal force is crucial for calculating the static friction force, understanding it gives insight into how it helps balance forces at work.
In the context of the inclined bed, the normal force is integral to maintaining the patient鈥檚 position as it influences the static friction that resists their sliding.

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Most popular questions from this chapter

The "Giant Swing" at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end (Fig. E5.46). Each arm supports a seat suspended from a cable 5.00 m long, the upper end of the cable being fastened to the arm at a point 3.00 \(\mathrm{m}\) from the central shaft. (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of \(30.0^{\circ}\) with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution?

Two objects with masses 5.00 \(\mathrm{kg}\) and 2.00 \(\mathrm{kg}\) hang 0.600 \(\mathrm{m}\) above the floor from the ends of a cord 6.00 \(\mathrm{m}\) long passing over a frictionless pulley. Both objects start from rest. Find the maximum height reached by the 2.00 -kg object.

A 25.0 -kg box of textbooks rests on a loading ramp that makes an angle \(\alpha\) with the horizontal. The coefficient of kinetic friction is \(0.25,\) and the coefficient of static friction is \(0.35 .\) (a) As the angle \(\alpha\) is increased, find the minimum angle at which the box starts to slip. (b) At this angle, find the acceleration once the box has begun to move. (c) At this angle, how fast will the box be moving after it has slid 5.0 \(\mathrm{m}\) along the loading ramp?

A man pushes on a piano with mass 180 \(\mathrm{kg}\) so that it slides at constant velocity down a ramp that is inclined at \(11.0^{\circ}\) above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes (a) parallel to the incline and (b) parallel to the floor.

An astronaut is inside a \(2.25 \times 10^{6} \mathrm{kg}\) rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound \((331 \mathrm{m} / \mathrm{s})\) as quickly as possible, but you also do not want the astronaut to black out. Medical tests have shown that astronauts are in danger of blacking out at an acceleration greater than 4\(g .\) (a) What is the maximum thrust the engines of the rocket can have to just barely avoid blackout? Start with a free-body diagram of the rocket. (b) What force, in terms of her weight \(w,\) does the rocket exert on the astronatt? Start with a free-body diagram of the astronaut. (c) What is the shortest time it can take the rocket to reach the speed of sound?
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