A long, thin rod of mass M and length L is standing straight up on a table. Its lower end rotates on a frictionless pivot. A very slight push causes the rod to fall over. As it hits the table, what are the angular velocity

검증된 단계별 안내

Step 1: Recognize that the problem involves rotational motion and energy conservation. The rod starts in a vertical position and falls due to gravity, converting potential energy into rotational kinetic energy.

Step 2: Write the expression for the initial gravitational potential energy of the rod. Since the rod is uniform, its center of mass is located at a height of L/2. The potential energy is given by \( U = M g \frac{L}{2} \), where \( g \) is the acceleration due to gravity.

Step 3: Write the expression for the rotational kinetic energy of the rod as it hits the table. The rotational kinetic energy is given by \( K = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia of the rod about the pivot point, and \( \omega \) is the angular velocity.

Step 4: Determine the moment of inertia \( I \) of the rod about the pivot point. For a thin rod rotating about one end, \( I = \frac{1}{3} M L^2 \). Substitute this into the rotational kinetic energy expression.

Step 5: Apply the principle of conservation of energy. The initial potential energy \( U \) is completely converted into rotational kinetic energy \( K \) as the rod hits the table. Set \( M g \frac{L}{2} = \frac{1}{2} \left( \frac{1}{3} M L^2 \right) \omega^2 \) and solve for \( \omega \), the angular velocity.

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주요 개념

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

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a pivot point, expressed in radians per second. In the context of the falling rod, it describes the rate at which the angle of the rod changes as it falls. Understanding angular velocity is crucial for analyzing the motion of rotating bodies and can be calculated using the formula ω = θ/t, where θ is the angle in radians and t is the time taken.

Moment of Inertia

Moment of inertia is a property of a body that quantifies its resistance to angular acceleration about a pivot point. For a rod rotating about one end, the moment of inertia is given by I = (1/3)ML², where M is the mass and L is the length of the rod. This concept is essential for determining how the mass distribution affects the rod's rotational motion as it falls.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In the case of the falling rod, potential energy is converted into kinetic energy as it falls. This concept helps in calculating the angular velocity at the moment of impact by equating the initial potential energy at the upright position to the kinetic energy just before it hits the table.