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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 21b
Capitolo 9, Problema 21b

A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of 3.00 rad/s2. Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship arad = v2/r.

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Step 1: Start by determining the angular displacement (θ) for two revolutions. Since one revolution corresponds to an angular displacement of 2π radians, two revolutions correspond to θ = 4π radians.
Step 2: Use the kinematic equation for rotational motion to find the angular velocity (ω) at the instant the wheel completes its second revolution. The equation is: ω^2 = ω_0^2 + 2αθ, where ω_0 is the initial angular velocity (0 rad/s), α is the angular acceleration (3.00 rad/s²), and θ is the angular displacement (4π radians).
Step 3: Calculate the linear velocity (v) of a point on the rim of the wheel using the relationship between linear and angular velocity: v = rω, where r is the radius of the wheel. The radius can be found from the diameter: r = 40.0 cm / 2 = 20.0 cm = 0.200 m.
Step 4: Substitute the value of v into the formula for radial acceleration: a_{rad} = \(\frac{v^2}{r}\). Use the radius (r = 0.200 m) and the linear velocity (v) calculated in the previous step.
Step 5: Simplify the expression to find the radial acceleration (a_rad). Ensure all units are consistent (meters, seconds, etc.) throughout the calculation.

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Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). In this scenario, the wheel has a constant angular acceleration of 3.00 rad/s², which means its angular velocity increases uniformly as it rotates. This concept is crucial for determining the wheel's angular velocity at any given time.
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Conservation of Angular Momentum

Tangential Velocity

Tangential velocity is the linear speed of a point on the circumference of a rotating object, calculated as the product of the radius and the angular velocity (v = rω). As the wheel rotates, its tangential velocity increases due to the constant angular acceleration. This velocity is essential for calculating the radial acceleration at a specific point on the wheel.
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Radial (Centripetal) Acceleration

Radial acceleration, also known as centripetal acceleration, is the acceleration directed towards the center of a circular path, necessary for an object to maintain its circular motion. It can be calculated using the formula a_rad = v²/r, where v is the tangential velocity and r is the radius of the circular path. Understanding this concept is vital for solving the problem regarding the acceleration of a point on the wheel's rim.
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Intro to Centripetal Forces
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CA compact disc (CD) stores music in a coded pattern of tiny pits 10-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line?

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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.

(a) Compute the angular velocity of the turntable after 0.200 s.

(b) Through how many revolutions has the turntable spun in this time interval?

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A compact disc (CD) stores music in a coded pattern of tiny pits 10-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. What is the average angular acceleration of a maximum duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

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A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of 3.00 rad/s2. Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship arad = ω2r.

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Domanda del libro di testo

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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Domanda del libro di testo

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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