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Ch 09: Rotation of Rigid Bodies
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 09: Rotation of Rigid BodiesProblema 24d
Capitolo 9, Problema 24d

An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s2. What is the magnitude of the resultant acceleration of a point on the rim at t = 0.200 s?

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Step 1: Convert the given angular velocity and angular acceleration from revolutions per second (rev/s) and revolutions per second squared (rev/s²) to radians per second (rad/s) and radians per second squared (rad/s²). Use the conversion factor: 1 revolution = 2π radians.
Step 2: Calculate the angular velocity at time t = 0.200 s using the formula \( \omega = \omega_0 + \alpha t \), where \( \omega_0 \) is the initial angular velocity, \( \alpha \) is the angular acceleration, and \( t \) is the time.
Step 3: Determine the tangential acceleration \( a_t \) of the point on the rim using the formula \( a_t = r \cdot \alpha \), where \( r \) is the radius of the turntable (half the diameter) and \( \alpha \) is the angular acceleration in radians per second squared.
Step 4: Calculate the centripetal acceleration \( a_c \) using the formula \( a_c = r \cdot \omega^2 \), where \( \omega \) is the angular velocity at time t = 0.200 s and \( r \) is the radius of the turntable.
Step 5: Find the magnitude of the resultant acceleration \( a_{res} \) by combining the tangential and centripetal accelerations using the Pythagorean theorem: \( a_{res} = \sqrt{a_t^2 + a_c^2} \).

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Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second or revolutions per second. In this question, the initial angular velocity of the turntable is given as 0.250 rev/s, indicating the rate at which the turntable is spinning at the start of the observation.
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Intro to Angular Momentum

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time, typically measured in revolutions per second squared or radians per second squared. The constant angular acceleration of 0.900 rev/s² in this scenario indicates that the turntable's rotation speed is increasing uniformly, which is crucial for calculating the angular velocity at a specific time.
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Intro to Acceleration

Resultant Acceleration

Resultant acceleration is the vector sum of tangential and centripetal accelerations acting on a point in circular motion. For a point on the rim of the turntable, tangential acceleration arises from angular acceleration, while centripetal acceleration is due to the circular path. Understanding both components is essential to determine the total acceleration at a given time.
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An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s2. What is the tangential speed of a point on the rim of the turntable at t = 0.200 s?

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