Domanda del libro di testo
A fan blade rotates with angular velocity given by ωz(t) = g - bt2, where g = 5.00 rad/s and b = 0.800 rad/s3. Calculate the angular acceleration as a function of time.
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Ch 09: Rotation of Rigid Bodies
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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TOC.tableOfContents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Capitolo 9, Problema 3b
The angular velocity of a flywheel obeys the equation ωz(t) = A + Bt2, where t is in seconds and A and B are constants having numerical values 2.75 (for A) and 1.50 (for B). What is the angular acceleration of the wheel at (i) t = 0 and (ii) t = 5.00 s?
Guida verificata passo dopo passo1
Understand the problem: The angular velocity of the flywheel is given as a function of time, \( \omega_z(t) = A + B t^2 \), where \( A = 2.75 \) and \( B = 1.50 \). Angular acceleration is the time derivative of angular velocity, \( \alpha(t) = \frac{d\omega_z(t)}{dt} \). We need to calculate \( \alpha(t) \) at \( t = 0 \) and \( t = 5.00 \ \text{s} \).
Differentiate the angular velocity equation with respect to time to find the angular acceleration: \( \alpha(t) = \frac{d}{dt}(A + B t^2) \). Since \( A \) is a constant, its derivative is zero, and the derivative of \( B t^2 \) is \( 2 B t \). Thus, \( \alpha(t) = 2 B t \).
Substitute the value of \( B = 1.50 \) into the angular acceleration equation: \( \alpha(t) = 2 (1.50) t = 3.00 t \). This is the expression for angular acceleration as a function of time.
To find the angular acceleration at \( t = 0 \), substitute \( t = 0 \) into \( \alpha(t) = 3.00 t \): \( \alpha(0) = 3.00 (0) \). Simplify to find the angular acceleration at \( t = 0 \).
To find the angular acceleration at \( t = 5.00 \ \text{s} \), substitute \( t = 5.00 \) into \( \alpha(t) = 3.00 t \): \( \alpha(5.00) = 3.00 (5.00) \). Simplify to find the angular acceleration at \( t = 5.00 \ \text{s} \).
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Angular Velocity
Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second. In the given equation, ω_z(t) = A + Bt², the angular velocity depends on time and is influenced by constants A and B. Understanding this concept is crucial for analyzing rotational motion and determining how the speed of rotation changes over time.
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Percorso guidato
06:18
Intro to Angular Momentum
Angular Acceleration
Angular acceleration refers to the rate of change of angular velocity over time, usually expressed in radians per second squared. It can be calculated as the derivative of angular velocity with respect to time. In this case, to find the angular acceleration, we differentiate the given equation for angular velocity, which will provide insights into how the rotation speed of the flywheel changes at specific moments.
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12:12Conservation of Angular Momentum
Differentiation in Physics
Differentiation is a fundamental mathematical process used in physics to determine rates of change. In the context of the problem, differentiating the angular velocity function with respect to time allows us to find the angular acceleration. This concept is essential for analyzing dynamic systems, as it helps relate position, velocity, and acceleration in both linear and rotational motion.
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Percorso guidato
13:04Gravitational Force from a Solid Disk
Pratica correlata
Domanda del libro di testo
A fan blade rotates with angular velocity given by ωz(t) = g - bt2, where g = 5.00 rad/s and b = 0.800 rad/s3. Calculate the instantaneous angular acceleration αz at t = 3.00 s and the average angular acceleration αav-z for the time interval t = 0 to t = 3.00 s. How do these two quantities compare? If they are different, why?
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Domanda del libro di testo
The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct3, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = π/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s2. Find a, b, and c, including their units.
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