While a person is walking, his arms swing through approximately a 45° angle in 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 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?
To solve this problem, we can use the equations of motion for rotational motion. The swinging motion of the arm can be considered as an oscillating pendulum.
(a) Finding the acceleration of the drop of blood at the bottom of the swing:
We know that the arm swings through approximately a 45° angle in 1/2 s. Firstly, let's convert the angle to radians:
45° * (π/180°) = 0.7854 radians.
Next, let's determine the angular velocity (ω), which is the rate at which the angle changes with time. The formula for angular velocity is ω = Δθ / Δt, where Δθ is the change in angle and Δt is the change in time.
Therefore, ω = (0.7854 radians) / (1/2 s) = 1.5708 rad/s.
The next step is to find the linear speed (v) at the bottom of the swing. The linear speed of the swinging arm is given by the formula v = r * ω, where r is the length of the arm measured from the shoulder joint.
So v = (70.0 cm) * (1.5708 rad/s) = 109.96 cm/s.
To find the acceleration (a) at the bottom of the swing, we can use the equation a = v^2 / r. Therefore, a = (109.96 cm/s)^2 / (70.0 cm) = 172.8576 cm/s^2.
However, the given mass of the blood drop is in grams, and since acceleration has SI units of m/s^2, we need to convert the units. 1 cm = 0.01 m, so 1 cm/s^2 = 0.01 m/s^2.
Therefore, the acceleration of the blood drop at the bottom of the swing is 172.8576 cm/s^2 * 0.01 m/s^2 / 1 cm = 1.7286 m/s^2.
(b) Making a free-body diagram of the drop of blood at the bottom of the swing:
The free-body diagram would include the following forces acting on the drop of blood:
1. The gravitational force pulling it downward.
2. The normal force exerted by the blood vessel supporting the drop.
3. The tension force in the blood vessel that is responsible for keeping the blood drop on the swing path.
(c) Finding the force that the blood vessel must exert on the drop of blood:
The force exerted by the blood vessel is the centripetal force required to keep the blood drop moving in the circular path during the swing motion. This force is given by the equation F = m * a, where F is the force, m is the mass, and a is the acceleration.
Since the mass of the blood drop is given as 1.0 gram, we need to convert it to kilograms: 1 gram = 0.001 kg.
Therefore, the force exerted by the blood vessel is F = (0.001 kg) * (1.7286 m/s^2) = 0.0017286 N.
The force points toward the center of the circular path.
(d) Finding the force the blood vessel would exert if the arm were not swinging:
If the arm were not swinging, the blood drop would be at rest, and the force required to support it would be equal to its weight. The weight can be calculated using the formula W = m * g, where W is the weight, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Using the mass of the blood drop as 0.001 kg, the weight would be W = (0.001 kg) * (9.8 m/s^2) = 0.0098 N.
Therefore, if the arm were not swinging, the blood vessel would exert a force of 0.0098 N, which is equal to the weight of the blood drop, and it would point vertically downward.
To find the answers to these questions, we can use the following steps:
Step 1: Determine the angular velocity:
In order to find the acceleration, we need to first find the angular velocity of the swinging arm. We can use the formula: ω = θ / t, where ω represents the angular velocity, θ represents the angle, and t represents the time.
Given that the arm swings through a 45° angle in 1/2 s, we can substitute these values into the formula:
ω = (45°) / (1/2 s) = 90°/s
Step 2: Convert angular velocity to radian per second:
Since most equations in physics use radian per second as the unit of angular velocity, we need to convert our result from step 1. We know that 1 radian = 180°/π, so we can convert as follows:
ω = (90°/s) * (π/180°) = π/2 rad/s
Step 3: Calculate linear velocity:
To find the linear velocity of the fingertips, we need to use the formula: v = r * ω, where v represents linear velocity, r represents the length of the arm, and ω represents the angular velocity.
Given that the length of the arm is 70.0 cm, we can substitute these values into the formula:
v = (70.0 cm) * (π/2 rad/s) = 35π cm/s
Step 4: Convert linear velocity to meters per second:
Since the SI unit for velocity is meters per second, we need to convert our result from step 3. We know that 1 meter = 100 cm, so we can convert as follows:
v = (35π cm/s) * (1 m/100 cm) = (35π/100) m/s = (7π/20) m/s
(a) Find the acceleration of the 1.0 gram drop of blood at the bottom of the swing:
The acceleration of the drop of blood at the bottom of the swing is equal to the centripetal acceleration. We can use the formula for centripetal acceleration: a = v^2 / r, where a represents the acceleration, v represents the linear velocity, and r represents the length of the arm.
Given that the arm length is 70.0 cm, we can substitute these values into the formula:
a = ((7π/20) m/s)^2 / (0.70 m) ≈ 0.65 m/s^2
(b) Make a free-body diagram of the drop of blood:
The free-body diagram of the drop of blood would include the gravitational force (mg) acting downwards and the force exerted by the blood vessel (F) acting upwards.
(c) Find the force that the blood vessel must exert on the drop of blood:
The force that the blood vessel must exert on the drop of blood is equal to the centripetal force. We can use the formula for centripetal force: F = ma, where F represents the force, m represents the mass of the drop of blood, and a represents the acceleration.
Given that the mass of the drop of blood is 1.0 gram, we need to convert this to kilograms: 1 gram = 0.001 kg.
Substituting the values into the formula:
F = (0.001 kg) * (0.65 m/s^2) = 0.00065 N
The force exerted by the blood vessel points upwards to provide the centripetal force.
(d) Find the force that the blood vessel would exert if the arm were not swinging:
If the arm were not swinging, there would be no centripetal force required. In this case, the only force acting on the drop of blood would be its weight (mg). The force exerted by the blood vessel would be zero.