Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

문제 73b

10장, 문제 73b

A solid rubber ball rests on the floor of a railroad car when the car begins moving with acceleration a. Assuming the ball rolls without slipping, what is its acceleration relative to the ground?

검증된 단계별 안내

Identify the forces acting on the ball: The ball experiences a static friction force from the floor of the railroad car, which prevents it from slipping. This friction force is responsible for both the translational and rotational motion of the ball.

Apply Newton's second law for translational motion: The net force acting on the ball in the horizontal direction is the static friction force \( F_f \). Using \( F = ma \), we can write \( F_f = m a_{ball} \), where \( a_{ball} \) is the acceleration of the ball relative to the ground.

Apply the rotational dynamics equation: The torque caused by the static friction force \( F_f \) about the center of the ball is \( \tau = F_f R \), where \( R \) is the radius of the ball. Using the rotational analog of Newton's second law, \( \tau = I \alpha \), where \( I \) is the moment of inertia of the ball and \( \alpha \) is its angular acceleration, we can write \( F_f R = I \alpha \).

Relate angular acceleration to linear acceleration: For rolling without slipping, the angular acceleration \( \alpha \) and the linear acceleration \( a_{ball} \) are related by \( \alpha = \frac{a_{ball}}{R} \). Substitute this into the rotational dynamics equation to express \( F_f \) in terms of \( a_{ball} \).

Combine the equations: Use the expressions for \( F_f \) from both the translational and rotational dynamics equations to solve for \( a_{ball} \) in terms of the acceleration of the railroad car \( a \). The result will show how the ball's acceleration relative to the ground depends on the car's acceleration and the ball's physical properties (e.g., its moment of inertia).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is crucial for understanding how forces affect the motion of the rubber ball as the railroad car accelerates. The net force acting on the ball will determine its acceleration relative to the ground.

Rolling Motion

Rolling motion occurs when an object rotates about an axis while translating along a surface. For the rubber ball, rolling without slipping means that the point of contact with the ground does not slide. This concept is essential for analyzing the relationship between the ball's linear acceleration and its angular acceleration as it rolls in the accelerating railroad car.

Relative Motion

Relative motion refers to the calculation of the motion of an object as observed from a particular reference frame. In this scenario, understanding the ball's acceleration relative to the ground involves considering both the acceleration of the railroad car and the ball's own motion. This concept helps in determining how the ball's acceleration appears to an observer on the ground.

관련 실천

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교과서 질문

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교과서 질문

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교과서 질문

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