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Ch 04: Kinematics in Two Dimensions
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 04: Kinematics in Two DimensionsProblema 74b
Capitolo 4, Problema 74b

A 6.0-cm-diameter gear rotates with angular velocity ω = ( 20 ─ ½ t² ) rad/s where t is in seconds. At t = 4.0 s, what are: The tangential acceleration of a tooth on the gear?

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Step 1: Understand the relationship between tangential acceleration and angular acceleration. Tangential acceleration (aₜ) is related to angular acceleration (α) by the formula: at=rα, where r is the radius of the gear.
Step 2: Calculate the angular acceleration (α). Angular acceleration is the time derivative of angular velocity (ω). Start by differentiating the given angular velocity equation ω=20−12t2 with respect to time t. This gives: α=dωdt=−t.
Step 3: Evaluate the angular acceleration at t = 4.0 s. Substitute t = 4.0 s into the expression for α: α=−t. This will give the angular acceleration at that specific time.
Step 4: Determine the radius of the gear. The diameter of the gear is given as 6.0 cm, so the radius (r) is half of the diameter: r=6.02=3.0 cm or 0.03 m (convert to meters for SI units).
Step 5: Calculate the tangential acceleration. Use the formula at=rα. Substitute the values of r and α (calculated in the previous steps) to find the tangential acceleration of a tooth on the gear at t = 4.0 s.

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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. In this question, the angular velocity ω is given as a function of time, indicating that it changes as time progresses. Understanding angular velocity is crucial for determining how the rotational motion of the gear affects the linear motion of its teeth.
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Intro to Angular Momentum

Tangential Acceleration

Tangential acceleration refers to the rate of change of linear velocity of a point on a rotating object, directed along the tangent to the circular path. It can be calculated using the formula a_t = r * α, where r is the radius and α is the angular acceleration. In this scenario, finding the tangential acceleration of a tooth on the gear requires knowledge of both the radius and the angular acceleration derived from the angular velocity function.
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Acceleration in 2D

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically expressed in radians per second squared. It can be determined by differentiating the angular velocity function with respect to time. In this problem, calculating the angular acceleration at t = 4.0 s is essential for finding the tangential acceleration, as it directly influences how quickly the linear speed of the gear's teeth is changing.
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Conservation of Angular Momentum
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