A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3.00 cm. It is spinning about a frictionless axle through its center, but at one point on its equator, it is scraping against metal, resulting in a friction force of 0.0200 N at that point. Find its angular acceleration.

검증된 단계별 안내

First, convert the mass of the sphere from grams to kilograms. Since 1 g = 0.001 kg, the mass of the sphere is 225 g * 0.001 kg/g = 0.225 kg.

Next, calculate the radius of the sphere. The diameter is given as 3.00 cm, so the radius is half of that: 3.00 cm / 2 = 1.50 cm. Convert this to meters: 1.50 cm * 0.01 m/cm = 0.015 m.

Determine the moment of inertia (I) for a solid sphere rotating about an axis through its center. The formula for the moment of inertia of a solid sphere is:

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

, where m is the mass and r is the radius.

Calculate the torque (τ) caused by the friction force. Torque is given by the formula:

τ = F r

, where F is the friction force and r is the radius of the sphere.

Finally, use Newton's second law for rotation to find the angular acceleration (α). The formula is:

α = \frac{τ}{I}

, where τ is the torque and I is the moment of inertia.

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주요 개념

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

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. For a solid sphere, it is calculated using the formula I = 2/5 * m * r^2, where m is the mass and r is the radius. This concept is crucial for determining how the sphere's mass distribution affects its angular acceleration.

Torque

Torque is the rotational equivalent of force, causing an object to rotate around an axis. It is calculated as the product of the force applied and the distance from the axis of rotation, T = F * r. In this scenario, the friction force at the equator generates torque, influencing the sphere's angular acceleration.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is determined by the net torque acting on an object divided by its moment of inertia, α = T/I. Understanding this concept is essential for calculating how the friction force affects the sphere's rotational speed.