Ch 12: Rotation of a Rigid Body

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 12: Rotation of a Rigid Body

문제 9

12장, 문제 9

What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth can be found inside the back cover of the book.

검증된 단계별 안내

Start by recalling the formula for rotational kinetic energy:

K = \frac{1}{2} I ω^{2}

, where

I

is the moment of inertia and

ω

is the angular velocity.

For a uniform sphere, the moment of inertia is given by

I = \frac{2}{5} m r^{2}

, where

m

is the mass of the sphere (Earth) and

r

is its radius.

Substitute the mass and radius of the Earth into the moment of inertia formula. The mass of the Earth is approximately

5.97 \times 10^{24}

kg, and the radius is approximately

6.37 \times 10^{6}

Determine the angular velocity of the Earth. The Earth completes one rotation in approximately 24 hours. Convert this period into seconds and use the formula

ω = \frac{2}{π} T

, where

T

is the period of rotation.

Finally, substitute the values of

I

and

ω

into the rotational kinetic energy formula to calculate the result.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation. It is calculated using the formula KE_rot = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity. For a uniform sphere, the moment of inertia can be determined using I = 2/5 m r², where m is the mass and r is the radius of the sphere.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a uniform sphere, the moment of inertia is given by I = 2/5 m r², which reflects how the mass is distributed throughout the sphere, affecting its rotational dynamics.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis. It is typically measured in radians per second. For the Earth, angular velocity can be calculated based on its rotation period, which is approximately 24 hours for a complete rotation, allowing us to determine how quickly it spins on its axis.