Problem 36 - College Physics

Question

In emergencies involving major blood loss, the doctor will order the patient placed in the Trendelberg position, which is to raise the foot of the bed to get maximum blood flow to the brain. If the coefficient of static friction between the typical patient and the bedsheets is \(1.2,\) what is the maximum angle at which the bed can be tilted with respect to the floor before the patient begins to slide?

Step-by-Step Solution

Answer

The maximum angle is approximately 50.19 degrees.

Step 1: Understanding the Problem

The problem involves determining the maximum angle at which a bed can be tilted before a patient starts sliding due to static friction. We need to find the angle at which the gravitational force component parallel to the bed exceeds the force of static friction.

Step 2: Identify the Forces at Play

When the bed is tilted, the gravitational force acting on the patient has two components: one parallel to the bed and one perpendicular to the bed. The static friction force must counteract the parallel component to prevent sliding.

Step 3: Write the Static Friction Equation

The force of static friction is given by the equation \[ F_{friction} = ext{friction coefficient} imes F_{normal} = ext{friction coefficient} imes mg ext{cos}(\theta), \] where \( F_{normal} \) is the normal force, \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of tilt.

Step 4: Gravitational Force's Parallel Component

The component of gravitational force trying to slide the patient down is \[ F_{parallel} = mg ext{sin}(\theta). \] To prevent sliding, this force must not exceed the static friction force.

Step 5: Set Forces Equal and Solve for Angle

To find the maximum angle before sliding occurs, set the static friction force equal to the parallel gravitational force:\[ mg ext{sin}(\theta) = 1.2 imes mg ext{cos}(\theta). \] Divide both sides by \( mg \) (assuming it is not zero), and simplify to get:\[ ext{tan}(\theta) = 1.2. \] Solve for \( \theta \) using inverse tangent:\[ \theta = ext{tan}^{-1}(1.2). \]

Step 6: Calculate the Maximum Angle

Using a calculator to find the angle, \[ \theta = ext{tan}^{-1}(1.2) \approx 50.19^\circ. \] This is the maximum angle at which the bed can be tilted before the patient starts to slide.

Key Concepts

Newton's Laws of Motion

Newton's Laws of Motion provide the foundation for understanding how objects behave when forces are applied. The first law, often called the law of inertia, tells us that an object will remain at rest or in uniform motion unless acted upon by a force. In this exercise, the patient initially remains stationary due to static friction unless the bed is tilted too much.
The second law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (\( F = ma \)). In our scenario, the gravitational force causes acceleration downhill when the bed is tilted. When the force due to gravity's tilt component exceeds static friction, the patient will slide.
Finally, the third law introduces action and reaction; this is somewhat subtler in this exercise but underlies interactions like the contact between patient and bed. Here, the force exerted by the patient on the bed due to gravity meets an equal and opposite force, mainly static friction, endeavoring to keep them stationary.

Angle of Inclination

The angle of inclination is the angle at which a surface is tilted from the horizontal. It significantly affects how forces act on an object placed on that surface. In the problem at hand, this angle determines when the static friction will fail to hold the patient still.
As the angle increases, the gravitational force's component parallel to the plane also increases—meaning more force tries to slide the patient downward. We calculate the maximum angle using the tangent function, a trigonometric ratio expressing the relationship between the opposite and adjacent sides of a right triangle. For this case:\[\tan(\theta) = 1.2\]Solving this provides:\[\theta = \tan^{-1}(1.2) \approx 50.19^\circ.\]This angle is the threshold where the incline causes sufficient force to overcome static friction, leading to sliding.

Gravitational Force

Gravitational force is the force of attraction between two masses—primarily here between the Earth and the patient. It acts towards the Earth's center and affects how the patient interacts with the tilted bed.
When the bed is inclined, gravity's force is divided into two components: parallel and perpendicular to the bed's surface. The parallel component is crucial because it's responsible for any downward slide; it is calculated by:\[F_{parallel} = mg \sin(\theta)\]where \( m \)is mass and \( g \)is gravitational acceleration (approximately \( 9.8 \ m/s^2 \)on Earth's surface).
The perpendicular component, on the other hand, contributes to the normal force, which static friction depends upon. The interaction between these components and friction allows computation of the critical angle preventing movement. Understanding these forces helps in various applications, including safely using inclined surfaces.