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In physics, angular velocity is a measure of how quickly an object rotates or revolves around a central point. It鈥檚 often symbolized by the Greek letter \( \omega \). Angular velocity is expressed in radians per second, which aligns with the calculations you would typically perform in physics problems.
To calculate angular velocity, you need to know how many radians an object covers in a given amount of time. Here's the simple formula used:

• \( \omega = \frac{\text{angle in radians}}{\text{time in seconds}} \).

For example, in our exercise, the arm swings through a 45 degree angle, which was converted into \( \frac{\pi}{4} \) radians. If this swing happens over 0.5 seconds, the angular velocity is:

• \( \omega = \frac{\pi/4}{0.5} = \frac{\pi}{2} \) rad/s

This gives the measure of how quickly the arm swings.鈥淎ngular velocity鈥� is crucial for computing centripetal forces in rotational motion.

Centripetal acceleration describes the inward acceleration of an object moving in a circular path. This acceleration is always directed towards the center of the circle. It's absolutely essential when considering objects that move in curved, rather than straight, paths.
Centripetal acceleration \( a_c \) is calculated using the square of the angular velocity \( \omega \) and the radius \( r \). Here鈥檚 the formula:

• \( a_c = r \omega^2 \)

In our swing problem, the fingertips travel at a constant radius of 0.7 meters from the shoulder. With an angular velocity of \( \frac{\pi}{2} \) rad/s, the centripetal acceleration becomes:

• \( a_c = 0.7 \times \left(\frac{\pi}{2}\right)^2 = 1.73 \text{ m/s}^2 \)

This indicates how quickly the blood in the fingertips is being pulled inward during the swing, showing how centripetal forces keep the blood moving in a curved path, following the swing of the arm.

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. This is expressed in the equation \( F = ma \). It鈥檚 a central concept in understanding dynamics of motion and forces.
In our exercise, this law helps us determine the force that the blood vessel must exert to maintain the movement of the blood within the swinging arm. With a mass \( m \) of 1.0 gram (converted to 0.001 kg) and a centripetal acceleration \( a \) of 1.73 m/s\(^2\), the force is:

• \( F = 0.001 \times 1.73 = 0.00173 \text{ N} \)

This computed force tells us how much effort the blood vessel needs to exert in order to counter this circular inward pull. When the arm is stationary, the only force it needs to exert is due to gravity, calculated as \( F = 0.0098 \text{ N} \) pulling downward. Newton's Second Law thus gives us the framework to quantify these essential forces.