Problem 50 - Fundamentals of Physics

Question

The rotor of an electric motor has rotational inertia \(I_{m}=\) \(2.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\) about its central axis. The motor is used to change the orientation of the space probe in which it is mounted. The motor axis is mounted along the central axis of the probe; the probe has rotational inertia \(I_{p}=12 \mathrm{~kg} \cdot \mathrm{m}^{2}\) about this axis. Calculate the number of revolutions of the rotor required to turn the probe through \(30^{\circ}\) about its central axis.

Step-by-Step Solution

Answer

The rotor must complete approximately 572 revolutions.

Step 1: Understand the Concept of Conservation of Angular Momentum

In this problem, since there are no external torques acting on the system (motor and probe), the total angular momentum is conserved. This means that the angular momentum of the rotor and the probe together remains constant.

Step 2: Write the Angular Momentum Conservation Equation

The total angular momentum of the system can be defined as the sum of the individual angular momentums of the rotor and the probe. Mathematically, this can be expressed as:\[ I_m \omega_m = I_p \omega_p \]where \( \omega_m \) is the angular velocity of the rotor and \( \omega_p \) is the angular velocity of the probe.

Step 3: Relate Angular Displacement to Angular Velocity

Given that the probe needs to be turned through \( 30^{\circ} \). Convert this to radians as \( \theta_p = \frac{\pi}{6} \text{ rad} \). If the rotor completes \( n \) revolutions, then \( \theta_m = 2\pi n \), knowing that one revolution equals \( 2\pi \) radians.

Step 4: Solve for Number of Revolutions

Using the relationship from Step 3, substitute into the angular momentum equation:\[ \frac{2\pi n}{\pi/6} = \frac{I_p}{I_m} \]Rearranging the equation to find \( n \):\[ n = \frac{1}{2\pi} \times \frac{6 I_p}{I_m}\]Substitute the given values \(I_m = 2.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\) and \(I_p = 12 \mathrm{~kg} \cdot \mathrm{m}^{2}\):\[ n = \frac{6 \times 12}{2 \times 10^{-3}} \cdot \frac{1}{2\pi} \approx 572\]

Step 5: Finalize the Calculation

The number of revolutions of the rotor required to turn the probe through \( 30^{\circ} \) is approximately 572 revolutions.

Key Concepts

Understanding Rotational Inertia

Rotational inertia, also known as the moment of inertia, is a measure of an object's resistance to changes in its rotation. It's similar to mass in linear motion, but specifically for rotational motion.
It depends not only on the mass of the object but also on how that mass is distributed relative to the axis of rotation.
\( I = \sum m_i r_i^2 \)
where \( I \) is the rotational inertia, \( m_i \) is the mass of each point in the object, and \( r_i \) is the distance of each point from the axis of rotation.
Key points about rotational inertia:

• Objects with a larger mass or mass further from the axis have a higher rotational inertia.
• Rotational inertia affects how much torque is needed for a desired angular acceleration.
• For complex shapes, the rotational inertia can be calculated by adding up contributions from all constituent parts.

This property plays a crucial role in the conservation of angular momentum, especially in systems like motors and space probes where changes in momentum affect motion directly.

Exploring Angular Velocity

Angular velocity is an essential concept in rotational dynamics, indicating how quickly an object rotates around a specific point or axis.
It is the rate of change of angular displacement with respect to time, measured in radians per second (rad/s).
The formula for angular velocity \( \omega \) is:
\( \omega = \frac{\Delta \theta}{\Delta t} \)
where:\( \Delta \theta \) is the change in angular displacement,\( \Delta t \) is the change in time.
Important aspects of angular velocity include:

• Angular velocity impacts how fast a rotating object will spin based on its shape and mass distribution.
• In systems adhering to the conservation of angular momentum, the angular velocity can be influenced by external forces like torques.
• For rotating systems without external torques, angular velocity remains constant.

This concept is critical when calculating how systems like electric motors modulate their spin to adjust the orientation of linked structures like space probes.

Decoding Angular Displacement

Angular displacement refers to the angle through which a point, line, or body is rotated in a specified direction and about a specified axis.
It is defined in terms of radians, degrees, or revolutions, commonly used in problems involving rotational movements.
The formula for angular displacement \( \theta \) is:
\( \theta = \omega \times t \)
where \( \omega \) is the angular velocity and \( t \) is the time.
Key points about angular displacement include:

• It provides a measure of the angle by which an object has rotated from its original position.
• In physics problems, it helps relate angular velocity and time to quantify rotation.
• Angular displacement is particularly essential in calculations involving rotational kinetics, such as determining the number of revolutions based on changes in orientation or spin rates.

In exercises concerning space probes and motors, angular displacement helps determine how much rotation is needed or achieved, as illustrated by the necessary revolutions to reorient a probe.