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 car?

검증된 단계별 안내

Identify the forces acting on the ball: The ball experiences a static friction force due to the floor of the railroad car, which prevents slipping. This friction force is responsible for both the linear acceleration of the ball's center of mass and the torque causing it to roll.

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

Apply the rotational dynamics equation: The torque due to the static friction force causes the ball to roll. Using \( \tau = I \alpha \), where \( I \) is the moment of inertia of the ball and \( \alpha \) is its angular acceleration, we can write \( f_s R = I \alpha \), where \( R \) is the radius of the ball.

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

Combine the equations: Substitute the expression for \( f_s \) from the rotational dynamics equation into the linear motion equation. Solve for \( a_{\text{ball}} \) in terms of the car's acceleration \( a \), the ball's mass \( m \), and its moment of inertia \( I \). Use the moment of inertia for a solid sphere, \( I = \frac{2}{5} m R^2 \), to simplify the result.

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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 law can be expressed with the formula F = ma, where F is the net force, m is the mass, and a is the acceleration.

Rolling Motion

Rolling motion occurs when an object rotates about an axis while simultaneously translating along a surface. For the rubber ball, this means that as it rolls without slipping, its point of contact with the ground does not slide. This concept is essential for analyzing the relationship between the ball's linear and angular accelerations, which are linked through the radius of the ball.

Relative Acceleration

Relative acceleration refers to the acceleration of one object as observed from another object. In this scenario, understanding the ball's acceleration relative to the accelerating railroad car is key. The ball experiences a different acceleration than the car due to the forces acting on it, and this difference must be calculated to determine how the ball behaves in the moving frame of reference.