The three 200 g masses in FIGURE EX12.11 are connected by massless, rigid rods. What is the triangle's kinetic energy if it rotates about the axis at 5.0 rev/s?

검증된 단계별 안내

Step 1: Identify the given values from the problem. The masses of the three objects are 200 g each, which can be converted to kilograms (0.2 kg). The radius of rotation is given as 45 cm, which can be converted to meters (0.45 m). The rotational speed is 5.0 revolutions per second.

Step 2: Calculate the moment of inertia (I) for the system. Since the masses are connected by massless rods and are equidistant from the axis of rotation, the moment of inertia for each mass is given by \( I = m r^2 \). For three masses, the total moment of inertia is \( I_{total} = 3 m r^2 \). Substitute the values of mass (m = 0.2 kg) and radius (r = 0.45 m) into the formula.

Step 3: Convert the rotational speed from revolutions per second to angular velocity in radians per second. Use the formula \( \omega = 2 \pi f \), where \( f \) is the frequency in revolutions per second. Substitute \( f = 5.0 \) into the formula to find \( \omega \).

Step 4: Use the formula for rotational kinetic energy \( KE = \frac{1}{2} I \omega^2 \). Substitute the values of \( I \) (calculated in Step 2) and \( \omega \) (calculated in Step 3) into the formula.

Step 5: Simplify the expression to find the kinetic energy. Ensure all units are consistent (mass in kg, radius in meters, angular velocity in radians per second) before performing the calculation.

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주요 개념

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

Kinetic Energy of Rotation

The kinetic energy of a rotating object is given by the formula KE = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity in radians per second. This concept is crucial for understanding how the mass distribution and rotation speed affect the energy of the system.

Moment of Inertia

The moment of inertia (I) quantifies how mass is distributed relative to the axis of rotation. For point masses, it is calculated as I = Σ mᵢ rᵢ², where mᵢ is the mass and rᵢ is the distance from the axis. In this case, the three 200 g masses contribute to the total moment of inertia based on their distances from the rotation axis.

Angular Velocity

Angular velocity (ω) measures how fast an object rotates and is expressed in radians per second. To convert revolutions per second to radians per second, multiply by 2π (ω = rev/s × 2π). This is essential for calculating the kinetic energy of the rotating triangle in the given problem.