Problem 13 - College Physics

Question

Effect on blood of walking. While a person is walking, his arms swing through approximately a \(45^{\circ}\) angle in \(\frac{1}{2}\) s. As a reasonable approximation, we can assume that the arm moves with constant speed during each swing. A typical arm is 70.0 \(\mathrm{cm}\) long, measured from the shoulder joint. (a) What is the acceleration of a 1.0 gram drop of blood in the fingertips at the bottom of the swing? (b) Make a free-body diagram of the drop of blood in part (a).(c) Find the force that the blood vessel must exert on the drop of blood in part (b). Which way does this force point? (d) What force would the blood vessel exert if the arm were not swinging?

Step-by-Step Solution

Answer

(a) 1.73 m/s², (c) 0.00173 N upward, (d) 0.00981 N downward.

Step 1: Calculate the Distance Traveled by the Arm in One Swing

The arm swings through a angle of \(45^\circ\). First, we convert this angle to radians: \(45^\circ = \frac{\pi}{4}\) radians. The length of the arm acts as the radius of the swing, which is \(70.0\, \text{cm} = 0.7\, \text{m}\). The arc length \(s\) that the arm travels can be calculated using the formula for arc length: \[ s = r \theta = 0.7 \times \frac{\pi}{4} \. \approx 0.55 \, \text{m}. \]

Step 2: Calculate the Velocity of the Arm

The arm completes each swing in \(\frac{1}{2}\) seconds. Assuming constant speed, calculate the velocity \(v\) using the formula: \[ v = \frac{s}{t} = \frac{0.55}{0.5} = 1.1\, \text{m/s}. \]

Step 3: Calculate the Centripetal Acceleration of the Blood Drop

The centripetal acceleration \(a_c\) is calculated using the formula for centripetal acceleration: \[ a_c = \frac{v^2}{r} = \frac{(1.1)^2}{0.7} \. \approx 1.73 \, \text{m/s}^2. \] This is the acceleration felt by the drop of blood at the fingertips.

Step 4: Draw the Free-Body Diagram

In the free-body diagram for the blood drop, represent the forces: 1. The force of gravity \(F_g\), acting downwards. 2. The centripetal force \(F_c\), required for the circular motion, directed towards the center of the swing (upwards). Label these forces clearly in your sketch.

Step 5: Calculate the Centripetal Force on the Blood Drop

The centripetal force \(F_c\) can be calculated using the formula: \[ F_c = m \cdot a_c, \]where \(m = 0.001\, \text{kg}\), the mass of the blood drop, and \(a_c = 1.73\, \text{m/s}^2\). Substituting these values, \[ F_c = 0.001 \times 1.73 = 0.00173 \, \text{N}. \] This force points towards the center of the arm swing.

Step 6: Calculate the Force if the Arm is Not Swinging

If the arm were not swinging, only the force of gravity would act on the drop of blood. Calculate this force using: \[ F_g = m \cdot g = 0.001 \times 9.81 = 0.00981 \, \text{N}, \] where \(g = 9.81\, \text{m/s}^2\) is the acceleration due to gravity. This force points downward.

Key Concepts

Blood Circulation

Blood circulation is an essential process in the human body, responsible for transporting nutrients and oxygen to cells while removing waste products. The heart acts as a pump, circulating blood through a network of arteries, veins, and capillaries. This vital system ensures that all parts of the body are nourished and maintained.

The curious aspect of blood circulation within the context of the given exercise is the effect of movement on blood, particularly when walking. When arms swing during walking, blood experiences artificial forces due to angular motion. These forces result in additional accelerations acting on blood compared to when the body is static.

• Blood vessels must adapt by exerting forces to counterbalance angular motion forces.
• In a swinging arm, blood in the fingertips is subjected to centripetal acceleration towards the shoulder joint along with gravity pulling it downwards.

Understanding these concepts highlights the body's complex biomechanical adaptations to regular activities like walking.

Free-Body Diagram

A free-body diagram is a powerful tool in physics. It helps visualize forces acting on an object, making it easier to understand and solve problems involving motion. In the exercise discussed, the object in focus is a drop of blood at the tip of a swinging arm.

Free-body diagrams involve isolating the object and illustrating all forces acting upon it. Here's how to create one for this specific scenario:

• First, acknowledge the drop of blood as the object.
• Identify the forces: gravity pulling downward and the centripetal force pulling upward towards the shoulder.
• Represent these forces with arrows: the direction and length of the arrows indicate the direction and magnitude of forces.

In our case, the diagram would show a downward arrow for gravity and an upward arrow for centripetal force. This visual representation simplifies understanding the balance of forces involved when an arm swings.

Forces in Physics

Forces are fundamental in physics, governing how objects interact and move. They come in different forms - gravitational, centripetal, frictional, and more. Understanding forces provides insights into behaviors experienced in everyday activities.

In this exercise, two main forces act on the drop of blood:

• Gravity, always acting downward, with a magnitude determined by the mass of the object and the acceleration due to gravity.
• Centripetal force, which in circular motion acts towards the center of the circle traced by the motion's arc. It depends on the object's speed and distance from the center.

The net effect of these forces results in changes in motion that must be counteracted by internal forces within the body, such as those exerted by blood vessels. Analyzing forces allows us to predict how moving bodies behave and respond to external stimuli.

Angular Motion

Angular motion refers to the movement of an object along a circular path. It involves understanding not just the speed but also the rotation of objects about a fixed point or axis. In the context of a swinging arm, the shoulder acts as a pivot, and the arm describes an arc-shaped path during each swing.

Key components of angular motion relevant here include:

• Angular displacement - the angle through which the arm moves, here, \(45^\circ \) or \(\pi/4 \) radians.
• Angular velocity - the rate of change of angular displacement, which in this scenario is the consistent swing speed.
• Centripetal acceleration - direction towards the center of the swing's circular path, it changes the velocity’s direction but not its magnitude.

Understanding these elements helps explain why blood in fingertips needs to cope with changing forces, as the arm swings through its circular path, adding complexity to bodily functions that maintain equilibrium.