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 66b

12장, 문제 66b

Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm. How much energy is stored in the flywheel?

검증된 단계별 안내

Determine the moment of inertia of the flywheel. Since the flywheel is a solid disk, its moment of inertia is given by the formula:

I = \frac{1}{2} m r^{2}

, where

m

is the mass of the flywheel and

r

is its radius. The radius is half the diameter, so

r = \frac{1.5}{2} = 0.75 m

Convert the maximum angular velocity from rpm (revolutions per minute) to radians per second. Use the conversion factor:

1 \(\text{ revolution }\) = 2 π \(\text{ radians }\)

and

1 \(\text{ minute }\) = 60 \(\text{ seconds }\)

. The angular velocity in radians per second is:

ω = 1200 imes \frac{2π}{60}

Calculate the rotational kinetic energy stored in the flywheel using the formula:

E = \frac{1}{2} I ω^{2}

, where

I

is the moment of inertia and

ω

is the angular velocity in radians per second.

Substitute the values of

I

and

ω

into the formula for rotational kinetic energy. Ensure that all units are consistent (mass in kilograms, radius in meters, angular velocity in radians per second).

Simplify the expression to find the total energy stored in the flywheel. This will give the energy in joules (J), which is the standard unit of energy in the SI system.

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 possessed by an object due to its rotation. It is calculated using the formula KE_rot = 0.5 * I * ω², where I is the moment of inertia and ω is the angular velocity in radians per second. This concept is crucial for determining how much energy is stored in a flywheel as it spins.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a solid disk, it is calculated using the formula I = 0.5 * m * r², where m is the mass and r is the radius. Understanding the moment of inertia is essential for calculating the rotational kinetic energy of the flywheel.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second. To convert from revolutions per minute (rpm) to radians per second, the formula ω = (rpm * 2π) / 60 is used. Knowing the angular velocity is necessary for calculating the energy stored in the flywheel.

관련 실천

교과서 질문

Blocks of mass m₁ and m₂ are connected by a massless string that passes over the pulley in FIGURE P12.64. The pulley turns on frictionless bearings. Mass m₁ slides on a horizontal, frictionless surface. Mass m₂ is released while the blocks are at rest. Assume the pulley is massless. Find the acceleration of m₁ and the tension in the string. This is a Chapter 7 review problem.

1654

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교과서 질문

Blocks of mass m₁ and m₂ are connected by a massless string that passes over the pulley in FIGURE P12.64. The pulley turns on frictionless bearings. Mass m₁ slides on a horizontal, frictionless surface. Mass m₂ is released while the blocks are at rest. Suppose the pulley has mass m_p and radius R. Find the acceleration of m₁ and the tensions in the upper and lower portions of the string. Verify that your answers agree with part a if you set m_p = 0.

185

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교과서 질문

A 30-cm-diameter, 1.2 kg solid turntable rotates on a 1.2-cm-diameter, 450 g shaft at a constant 33 rpm. When you hit the stop switch, a brake pad presses against the shaft and brings the turntable to a halt in 15 seconds. How much friction force does the brake pad apply to the shaft?

1789

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교과서 질문

The 2.0 kg, 30-cm-diameter disk in FIGURE P12.65 is spinning at 300 rpm. How much friction force must the brake apply to the rim to bring the disk to a halt in 3.0 s?

A 750 g disk and a 760 g ring, both 15 cm in diameter, are rolling along a horizontal surface at 1.5 m/s when they encounter a 15° slope. How far up the slope does each travel before rolling back down?

2376

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교과서 질문

Your engineering team has been assigned the task of measuring the properties of a new jet-engine turbine. You've previously determined that the turbine's moment of inertia is 2.6 kg m². The next job is to measure the frictional torque of the bearings. Your plan is to run the turbine up to a predetermined rotation speed, cut the power, and time how long it takes the turbine to reduce its rotation speed by 50%. Your data are given in the table. Draw an appropriate graph of the data and, from the slope of the best-fit line, determine the frictional torque.

1333

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