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A ball is tied to a string and whirled in a horizontal circle at a constant 4 revolutions per second. Categorize the kinematic properties according to whether they are constant and zero, constant and nonzero, or not constant in value. A ball attached to a string swings around in a circular path in a horizontal plane in a counterclockwise direction as seen from above.

: linear speed (scalar)
⃗ : linear velocity (vector)
r : radial (centripetal) acceleration (scalar)
⃗ r : radial (centripetal) acceleration (vector)
: angular speed (scalar) ⃗ : angular velocity (vector)
: angular acceleration (scalar)
⃗ : angular acceleration (vector)

Which are constant and zero, constant and nonzero, and not constant?

: linear speed (scalar) - Not constant
⃗ : linear velocity (vector) - Not constant
r : radial (centripetal) acceleration (scalar) - Constant and nonzero
⃗ r : radial (centripetal) acceleration (vector) - Constant and nonzero
: angular speed (scalar) - Constant and nonzero
⃗ : angular velocity (vector) - Constant and nonzero
: angular acceleration (scalar) - Constant and zero
⃗ : angular acceleration (vector) - Constant and zero

User CChi
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1 Answer

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Final answer:

In analyzing a ball in uniform circular motion, linear speed remains constant and nonzero, linear velocity is not constant, radial acceleration (both scalar and vector) is constant and nonzero, angular speed and angular velocity are constant and nonzero, and angular acceleration (both scalar and vector) is constant and zero.

Step-by-step explanation:

In determining the kinematic properties of a ball whirled in a horizontal circle at a constant rate, we analyze both linear and angular characteristics of the motion. For the ball tied to a string and moving in a circular path:

The linear speed (scalar) remains constant and nonzero because the ball travels at a steady rate.

The linear velocity (vector) is not constant because its direction is always changing, even though its magnitude (speed) stays the same.

The radial (centripetal) acceleration (scalar) is constant and nonzero since there's a continuous change in the direction of the velocity vector requiring centripetal force.

The radial (centripetal) acceleration (vector) also remains constant and nonzero as it's always directed towards the center of the circle.

Angular speed (scalar) is constant and nonzero given the ball makes 4 revolutions per second continuously.

The angular velocity (vector) remains constant and nonzero since it does not change in magnitude or direction (oriented perpendicularly to the plane of rotation).

The angular acceleration (scalar) is constant and zero because the angular velocity is uniform, indicating no change in rate of rotation.

Similarly, the angular acceleration (vector) is also constant and zero for the same reason as the scalar quantity.

These categorizations are essential for understanding uniform circular motion and the forces involved in such movements.

User Bloo
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