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Angular velocity \( \omega \) is a measure of how fast an object rotates about an axis. It's similar to the linear velocity, but for rotational motion. An important characteristic of angular velocity is that it's usually measured in rotations per second (rev/s) or radians per second (rad/s). Angular velocity can be affected by angular acceleration \( \alpha \), which is the rate of change of angular velocity over time.

In the example of the electric turntable, the initial angular velocity \( \omega_0 \) was 0.250 rev/s. With a constant angular acceleration of 0.900 rev/s^2, the angular velocity after 0.200 seconds can be found using the equation:

\[ \omega = \omega_0 + \alpha t \]

where
\( \omega_0 = 0.250 \, \text{rev/s} \),
\( \alpha = 0.900 \, \text{rev/s}^2 \), and
\( t = 0.200 \, \text{s} \).

Plug in these values to find:
\[ \omega = 0.250 + 0.900 \times 0.200 = 0.430 \, \text{rev/s} \]

This new value represents how quickly the turntable spins after 0.200 seconds.

Angular acceleration \( \alpha \) determines how quickly the angular velocity changes with time. It's like how acceleration works in linear motion but in the world of rotations.

However, we often need to convert between different units, such as revolutions per second squared (rev/s²) and radians per second squared (rad/s²), because many formulas require angular measurements in radians. This conversion uses \( 2\pi \) since one full revolution equals \( 2\pi \) radians.

To illustrate this, with our electric turntable example, we have:
\( \alpha = 0.900 \, \text{rev/s}^2 \) .

Converting to rad/s² involves:
\[ \alpha = 0.900 \times 2\pi = 5.654 \, \text{rad/s}^2 \]

Knowing angular acceleration helps us understand how rapidly the turntable speeds up or slows down.

Tangential speed \( v \) refers to the linear speed of a point located on the edge of a rotating object. It's a direct function of both the object's angular velocity and its radius. Essentially, it tells how fast a point on the edge is moving in a straight line, usually measured in meters per second (m/s).

If we consider the turntable, the radius \( r \) is half its diameter. Here,
\( r = 0.375 \, \text{m} \).

To find the tangential speed, the angular velocity \( \omega \) must be in rad/s. This conversion is:

\[ \omega = 0.430 \times 2\pi = 2.701 \, \text{rad/s} \]

The formula for tangential speed is:
\[ v = r \times \omega = 0.375 \times 2.701 \approx 1.01 \, \text{m/s} \]

This speed shows how fast a point on the turntable's rim is moving linearly.

In rotational motion, the resultant acceleration of a point on a rotating object includes both tangential and radial components.

Tangential acceleration \( a_t \) tells how fast the tangential speed changes. It's calculated by multiplying angular acceleration by radius:
\[ a_t = r \times \alpha = 0.375 \times 5.654 \approx 2.12 \, \text{m/s}^2 \]

Radial or centripetal acceleration \( a_r \) results from change in direction of the velocity vector and is calculated as follows:
\[ a_r = \omega^2 \times r = (2.701)^2 \times 0.375 \approx 7.26 \, \text{m/s}^2 \]

To find the magnitude of the resultant acceleration \( a \), combine these components using the Pythagorean theorem:
\[ a = \sqrt{a_t^2 + a_r^2} = \sqrt{2.12^2 + 7.26^2} \approx 7.57 \, \text{m/s}^2 \]

This value indicates how the point on the rim of the turntable experiences acceleration in total.