An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm (rev/min) about an axis through its center. You can model the propeller as a slender rod.
(a) What is its rotational kinetic energy?
(b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to 75.0% of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

검증된 단계별 안내

Step 1: Understand the problem and identify the given values. The propeller is modeled as a slender rod, which means its moment of inertia can be calculated using the formula for a slender rod rotating about its center: I = (1/12) * m * L^2, where m is the mass and L is the length. The given values are: length L = 2.08 m, mass m = 117 kg, and angular velocity ω = 2400 rpm. Convert ω to radians per second using the relation ω (rad/s) = ω (rpm) * (2π / 60).

Step 2: Calculate the moment of inertia (I) of the propeller using the formula I = (1/12) * m * L^2. Substitute the values of m and L into the formula to find I.

Step 3: Use the formula for rotational kinetic energy, K = (1/2) * I * ω^2, to calculate the rotational kinetic energy of the propeller. Substitute the values of I and ω (converted to rad/s) into the formula to find K.

Step 4: For part (b), reduce the mass to 75.0% of its original value. The new mass m' = 0.75 * m. To maintain the same rotational kinetic energy, use the formula K = (1/2) * I' * ω'^2, where I' = (1/12) * m' * L^2. Solve for the new angular velocity ω' in rad/s using the equation K = (1/2) * I' * ω'^2.

Step 5: Convert the new angular velocity ω' from rad/s back to rpm using the relation ω' (rpm) = ω' (rad/s) * (60 / 2π). This gives the angular speed in rpm required to maintain the same kinetic energy with the reduced mass.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

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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 = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity in radians per second. For a slender rod rotating about its center, the moment of inertia is I = (1/12) m L², where m is the mass and L is the length of the rod. Understanding this concept is crucial for calculating the kinetic energy of the airplane propeller.

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 slender rod rotating about its center, the moment of inertia is given by I = (1/12) m L². This concept is essential for determining how the mass and shape of the propeller affect its rotational kinetic energy and angular speed.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second or revolutions per minute (rpm). It is related to the rotational kinetic energy and moment of inertia, as changes in mass or shape can affect the required angular velocity to maintain a specific kinetic energy. In this problem, understanding how to convert between mass and angular velocity while keeping kinetic energy constant is key to solving part (b).