The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. 10–52. The wheel has angular velocity ω₁ = 48.0 rad/s about its axle, and the turntable has angular velocity ω₂ = 35.0 rad/s about a vertical axis. (Note arrows showing these motions in the figure.) What is the magnitude and direction of the angular acceleration of the wheel at the instant shown? Take the 𝒵 axis vertically upward and the direction of the axle at the moment shown to be the 𝓍 axis pointing to the right.

검증된 단계별 안내

Understand the problem: The wheel has two angular velocities, one about its own axle (ω₁) and another due to the rotation of the turntable (ω₂). We are tasked with finding the angular acceleration of the wheel, which involves determining the rate of change of the angular velocity vector.

Break down the angular velocity vectors: Represent ω₁ and ω₂ in terms of their components. ω₁ is along the 𝓍-axis (to the right), so ω₁ = (48.0 rad/s)𝓍̂. ω₂ is about the vertical 𝒵-axis, so ω₂ = (35.0 rad/s)𝒵̂.

Determine the angular velocity vector of the wheel: The total angular velocity of the wheel is the vector sum of ω₁ and ω₂. Write this as ω = ω₁𝓍̂ + ω₂𝒵̂ = (48.0 rad/s)𝓍̂ + (35.0 rad/s)𝒵̂.

Find the angular acceleration: The angular acceleration is the time derivative of the angular velocity vector, α = dω/dt. Since ω₁ is constant, its derivative is zero. However, ω₂ causes the direction of ω₁ to change as the turntable rotates. This results in a change in the angular velocity vector, which contributes to α.

Calculate the magnitude and direction of α: Use the cross product of ω₂ and ω₁ to find the angular acceleration vector, α = ω₂ × ω₁. The magnitude of α is |α| = |ω₂||ω₁|sin(θ), where θ = 90° (since ω₁ and ω₂ are perpendicular). The direction of α is given by the right-hand rule, perpendicular to both ω₁ and ω₂.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis. It is measured in radians per second (rad/s) and indicates both the speed of rotation and the direction of the axis of rotation. In this problem, the wheel's angular velocity (ω₁) and the turntable's angular velocity (ω₂) are crucial for determining the overall motion of the system.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, also expressed as a vector. It indicates how quickly an object is speeding up or slowing down its rotation. In this scenario, calculating the angular acceleration of the wheel involves understanding how the wheel's rotation is affected by both its own motion and the motion of the turntable.

Reference Frames

Reference frames are coordinate systems used to measure and describe the motion of objects. In this question, the Z-axis is defined as vertically upward, and the X-axis is aligned with the axle of the wheel. Understanding the reference frame is essential for accurately determining the direction of angular acceleration and how it relates to the angular velocities of both the wheel and the turntable.