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. (a) Find its angular acceleration.

Verified step by step guidance

1. Identify the given values: mass (m) = 225 g = 0.225 kg, diameter = 3.00 cm = 0.03 m, radius (r) = diameter / 2 = 0.015 m, friction force (F) = 0.0200 N.
2. Calculate the torque (\(\tau\)) produced by the friction force. Torque is given by \(\tau = r \times F\), where r is the radius of the sphere and F is the friction force.
3. Determine the moment of inertia (I) for a solid sphere about an axis through its center. The formula for the moment of inertia of a solid sphere is \(I = \frac{2}{5}mr^2\), where m is the mass and r is the radius of the sphere.
4. Use Newton's second law for rotation, \(\tau = I \alpha\), to find the angular acceleration (\(\alpha\)). Rearrange the equation to solve for \(\alpha\): \(\alpha = \frac{\tau}{I}\).
5. Substitute the values of \(\tau\) and I from the previous steps to calculate the angular acceleration (\(\alpha\)).

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a solid uniform sphere, it is calculated using the formula I = (2/5) m r², where m is the mass and r is the radius. This concept is crucial for determining how the mass distribution affects the sphere's angular acceleration when subjected to a torque.

Torque

Torque is the rotational equivalent of linear force and is calculated as the product of the force applied and the distance from the pivot point (lever arm). In this scenario, the friction force creates a torque that opposes the sphere's rotation. Understanding torque is essential for calculating the angular acceleration resulting from the applied friction force.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically denoted by the symbol α. It can be calculated using the formula α = τ/I, where τ is the torque and I is the moment of inertia. This concept is key to solving the problem, as it allows us to determine how quickly the sphere's rotation is changing due to the friction force acting on it.