Ch 10: Dynamics of Rotational Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 10: Dynamics of Rotational Motion

문제 46

10장, 문제 46

Suppose that an asteroid traveling straight toward the center of the earth were to collide with our planet at the equator and bury itself just below the surface. What would have to be the mass of this asteroid, in terms of the earth's mass M, for the day to become 25.0% longer than it presently is as a result of the collision? Assume that the asteroid is very small compared to the earth and that the earth is uniform throughout.

검증된 단계별 안내

Understand the problem: We need to find the mass of the asteroid that would cause the Earth's rotation period to increase by 25%. This involves the conservation of angular momentum, as the collision is inelastic and the asteroid embeds itself into the Earth.

Identify the initial and final states: Initially, the Earth is rotating with an angular velocity $ \omega_i $ and has a moment of inertia $ I_i = \frac{2}{5}MR^2 $, where $ M $ is the Earth's mass and $ R $ is its radius. After the collision, the new angular velocity is $ \omega_f $ and the moment of inertia becomes $ I_f = \frac{2}{5}MR^2 + mR^2 $, where $ m $ is the mass of the asteroid.

Apply the conservation of angular momentum: The initial angular momentum $ L_i = I_i \omega_i $ must equal the final angular momentum $ L_f = I_f \omega_f $. Therefore, $ \frac{2}{5}MR^2 \omega_i = \left( \frac{2}{5}MR^2 + mR^2 \right) \omega_f $.

Relate the change in the Earth's rotation period: The period of rotation $ T $ is inversely proportional to the angular velocity, so $ T_f = 1.25T_i $ implies $ \omega_f = \frac{\omega_i}{1.25} $. Substitute this into the angular momentum equation.

Solve for the mass of the asteroid $ m $: Rearrange the equation from step 3 to solve for $ m $ in terms of $ M $. This will give you the mass of the asteroid needed to increase the Earth's day by 25%.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

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주요 개념

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

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torques act on a system, the total angular momentum remains constant. In this scenario, the earth-asteroid system's angular momentum before and after the collision must be equal, allowing us to relate the asteroid's mass to the change in the earth's rotational period.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. For a uniform sphere like the earth, it is calculated as I = (2/5)MR², where M is the mass and R is the radius. Understanding how the asteroid's mass affects the earth's moment of inertia is crucial for determining the change in rotational speed.

Rotational Kinematics

Rotational kinematics involves the study of rotational motion without considering the forces that cause it. The earth's rotational period is related to its angular velocity, and a change in this period due to the collision can be analyzed using the relationship between angular velocity and moment of inertia, helping to find the asteroid's required mass.

관련 실천

교과서 질문

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교과서 질문

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교과서 질문

A small 10.0-g bug stands at one end of a thin uniform bar that is initially at rest on a smooth horizontal table. The other end of the bar pivots about a nail driven into the table and can rotate freely, without friction. The bar has mass 50.0 g and is 100 cm in length. The bug jumps off in the horizontal direction, perpendicular to the bar, with a speed of 20.0 cm/s relative to the table. What is the angular speed of the bar just after the frisky insect leaps?

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교과서 질문

A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.)

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