Question

A small lead sphere of mass \(25 \mathrm{~g}\) is attached to the origin by a thin rod of length \(74 \mathrm{~cm}\) and negligible mass. The rod pivots about the \(z\) axis in the \(x y\) plane. A constant force of \(22 \mathrm{~N}\) in the \(y\) direction acts on the sphere. (a) Considering the sphere to be a particle, what is the rotational inertia about the origin? (b) If the rod makes an angle of \(40^{\circ}\) with the positive \(x\) axis, find the angular acceleration of the rod.

Short answer

Rotational inertia and torque first need to be calculated. The angular acceleration is then calculated using these values.

Step-by-step solution

Calculation of Rotational Inertia

Rotational inertia of a particle is given by \(I=mr^2\), where \(m\) is the mass of the particle and \(r\) is the distance from the axis of rotation to the particle. Given \(m = 25g = 0.025kg\) and \(r = 74cm = 0.74m\), the rotational inertia \(I\) then can be calculated as \(I = 0.025kg * (0.74m)^2\).

Calculation of Torque

Torque is given by the equation \(\tau = r \cdot F \cdot sin\theta \), where \(\theta \) is the angle between the force vector and the position vector. Given that the force \(F = 22N\), distance \(r = 0.74m\) from the axis, and angle \(\theta = 40^{\circ} = 40 \cdot \pi/180 \, rad\), the torque \(\tau \) can be calculated as \(\tau = 0.74m * 22N * sin(40 \cdot \pi/180 \, rad)\).

Calculation of Angular Acceleration

The angular acceleration \( \alpha \) can be calculated using the formula \( \tau = I \cdot \alpha \). Solving for \( \alpha \) gives \( \alpha = \tau/I \). Using the previously calculated values for torque and rotational inertia, the angular acceleration can be calculated.

Key concepts

Rotational Inertia

Understanding the concept of rotational inertia, often referred to as the moment of inertia, is fundamental in physics, especially when dealing with rotational motion. It is the rotational equivalent of mass for linear motion. In other words, it's a measure of an object's resistance to changes in its rotation rate. When a force is applied, an object with greater rotational inertia will accelerate less than an object with less rotational inertia.

For a point mass, the rotational inertia is calculated using the formula: \(I = mr^2\), where \(m\) is the mass and \(r\) is the radial distance from the axis of rotation. The units of rotational inertia are kilogram meter squared (\(kg \times m^2\)). In our exercise, the sphere attached to the rod represents such a point mass, and its rotational inertia is a crucial element for understanding how the system behaves under applied forces.

Torque

The concept of torque is pivotal in rotational dynamics. It's defined as the tendency of a force to rotate an object about an axis. The magnitude of torque depends not only on the magnitude of the force but also on the distance from the pivot point (lever arm) and the angle at which the force is applied.

To calculate torque (\(\tau\)), we use the formula \(\tau = r \times F \times \text{sin}\theta\), where \(r\) is the distance to the pivot point, \(F\) is the force's magnitude, and \(\theta\) is the angle between the force vector and the lever arm. It's important to note that torque is a vector quantity, meaning it has both magnitude and direction. In our problem, the torque exerted on the sphere is what causes it to accelerate rotationally around the pivot point.

Angular Acceleration

Angular acceleration (\(\alpha\)) is a measure of how quickly an object increases its rotation speed. It's similar to linear acceleration but for rotating systems. When a torque is applied to an object, it results in an angular acceleration proportional to the torque and inversely proportional to the rotational inertia of the object, as given by the formula \(\alpha = \frac{\tau}{I}\).

This relationship is the rotational version of Newton's second law of motion, which states that force equals mass times acceleration (\(F = ma\)). In our exercise, once we calculate the torque exerted on the rod, we can determine the angular acceleration using the sphere's rotational inertia. This explains how the forces in play affect the rotational speed of the rod-sphere system.

Physics Problem Solving

Effective problem-solving in physics is a skill that develops over time. It involves understanding concepts, identifying relevant principles, applying mathematical equations correctly, and interpreting results in the context of physical phenomena. In our scenario, we first identified the rotational inertia and torque, and then related them to find the angular acceleration.

To solve physics problems, it's often useful to follow a methodical approach: carefully read the problem, visualize the situation, identify known and unknown quantities, choose the right physics laws and formulas, perform the calculations, and finally, check if the solution makes sense physically. When improving your problem solving, focus on the interplay between theory and application, as understanding one reinforces the comprehension of the other.

Rotational Dynamics

Rotational dynamics is the branch of physics that describes the behavior of rotating systems, where forces cause objects to spin around a fixed axis. The principles governing rotational motion are analogous to those of linear motion but applied to rotation. Key concepts in rotational dynamics include rotational inertia, torque, and angular acceleration, as discussed in the previous sections.

When analyzing the motion of rotating objects, it's essential to know how mass distribution affects rotational inertia and how forces applied at various points and angles produce different torques. By understanding these principles and their mathematical descriptions, we can predict the behavior of any rotating system, from simple toys to complex celestial mechanics.