91Ó°ÊÓ

Concept Questions The drawing shows two identical systems of objects; each consists of three small balls (masses \(m_{1}, m_{2}\), and \(m_{3}\) ) connected by massless rods. In both systems the axis is perpendicular to the page, but it is located at a different place, as shown. (a) Do the systems necessarily have the same moments of inertia? If not, why not? (b) The same force of magnitude \(F\) is applied to the same ball in each system (see the drawing). Is the magnitude of the torque created by the applied force greater for system A or for system B? Or is the magnitude the same in the two cases? Explain. (c) The two systems start from rest. Will system A or system B have the greater angular speed at the same later time? Or will they have the same angular speeds? Justify your answer. Problem The masses of the balls are \(m_{1}=9.00 \mathrm{~kg}, m_{2}=6.00 \mathrm{~kg}\), and \(m_{3}=7.00 \mathrm{~kg}\). The magnitude of the force is \(F=424 \mathrm{~N}\). (a) For each of the two systems, determine the moment of inertia about the given axis of rotation. (b) Calculate the torque (magnitude and direction) acting on each system. (c) Both systems start from rest, and the direction of the force moves with the system and always points along the \(4.00-\mathrm{m}\) rod. What is the angular velocity of each system

Step by step solution

01

Calculate the Moment of Inertia for System A

The moment of inertia for a system of masses is calculated using the formula:\[I = \sum_{i} m_{i} r_{i}^{2}\]where \(m_{i}\) is the mass and \(r_{i}\) is the distance of the mass from the axis of rotation. For System A, let's consider the distances for each mass to the axis of rotation. If given specific distances, plug them into the equation to find \(I_A\).

02

Calculate the Moment of Inertia for System B

Repeat the calculation for System B, using the same formula:\[I = \sum_{i} m_{i} r_{i}^{2}\]Ensure that you use the correct distances \(r_i\) from each mass to System B's axis of rotation. Compute \(I_B\).

03

Compare Moments of Inertia

Based on calculations from Step 1 and Step 2, compare \(I_A\) and \(I_B\). Determine whether \(I_A = I_B\) or not, and explain the reason related to the axis location.

04

Calculate Torque for System A

Torque \(\tau\) is defined by:\[\tau = rF\sin(\theta)\]Assuming the force \(F\) is applied perpendicular to the distance \(r\) from the axis, \(\theta = 90^\circ\) and \(\sin(\theta) = 1\). Use the given length \(4.00\,\text{m}\) to find the torque for System A: \(\tau_A = rF = 4.00 \times 424\). Calculate \(\tau_A\).

05

Calculate Torque for System B

Repeat the torque calculation for System B using the same force \(F\). Identify the distance \(r\) in System B from the force application point to the axis. Use \(\tau_B = rF\). Compute \(\tau_B\).

06

Compare Torques for Systems A and B

Compare \(\tau_A\) and \(\tau_B\) from Steps 4 and 5. Determine which is greater and why, based on the location of the axis and the applied force's distance.

07

Determine Angular Velocity for Both Systems

Using the relation \[\tau = I\alpha\]where \(\alpha\) is the angular acceleration, solving for \(\alpha\) gives: \[\alpha = \frac{\tau}{I}\]Calculate \(\alpha_A = \frac{\tau_A}{I_A}\) and \(\alpha_B = \frac{\tau_B}{I_B}\). With both systems starting from rest, use constant acceleration for the angular velocity \(\omega\):\[\omega = \alpha t\]Assuming same time \(t\) elapsed for both systems, compare \(\omega_A\) and \(\omega_B\) using their respective \(\alpha\).

08

Conclusion on Angular Velocity

The acceleration \(\alpha\) of each system indicates the rate of change of angular velocity. Compare \(\omega_A\) and \(\omega_B\) to see which system has a greater or same angular velocity based on the preceding calculations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Torque Calculation

When we talk about torque, we're looking at how a force causes an object to rotate around an axis. Torque, often described as the rotational equivalent of linear force, depends on several factors. Let's break it down.

Firstly, torque (\( \tau \)) is calculated using the formula: \[ \tau = rF\sin(\theta) \]where:

• \( r \) is the distance from the rotation axis to the point where the force is applied.
• \( F \) is the magnitude of the force.
• \( \theta \) is the angle between the force vector and the lever arm, which is perpendicular when maximized.

When \( \theta = 90^\circ \), \( \sin(\theta) = 1 \), meaning maximum torque occurs when force is applied perpendicularly to the lever arm. The system in question assumes forces applied at this angle, simplifying calculations.

Understanding the setup is crucial: System A and System B each have forces applied, but the axes of rotation differ. This difference means \( r \) is different for each system, affecting the torque. By calculating \( \tau_A \) and \( \tau_B \), and comparing them, we can see which arrangement results in greater rotational effect.

Angular Velocity

Angular velocity is all about how fast something is spinning. In essence, it's the rotational version of linear velocity. For this exercise, angular velocity is determined by the angular acceleration and the time a system has been rotating.

To dive in deeper, angular velocity (\( \omega \)) is derived from the relation:\[ \tau = I\alpha \]where:

• \( \tau \) is the torque applied on the system.
• \( I \) is the moment of inertia, representing rotational inertia depending on the mass distribution related to the axis.
• \( \alpha \) is the angular acceleration.

From this, \( \alpha = \frac{\tau}{I} \) helps us see that the greater the torque or the smaller the moment of inertia, the greater the angular acceleration. Since both systems begin from a resting state and are subject to a constant \( F \), we calculate angular velocities \( \omega_A \) and \( \omega_B \) with \( \omega = \alpha t \). Here, time \( t \) is consistent for both systems, but \( \alpha \)'s differences, due to torque and moments of inertia, result in differing angular speeds. We compare \( \omega_A \) and \( \omega_B \) to find which system achieves higher rotational speed.

Systems of Masses

The exercise involves examining two systems, each comprising three masses (\( m_1, m_2, m_3 \)) linked by rods. Understanding how these masses and their arrangement affect movement is key.

Moments of inertia are crucial indicators of how much torque is required for a desired rotational effect. It's computed by:\[ I = \sum m_i r_i^2 \]where:

• \( m_i \) are the individual masses.
• \( r_i \) are the distances from each mass to the axis of rotation.

Since moments of inertia depend on both the arrangement of masses and the axis's location, System A and System B, though consisting of identical masses, may exhibit different inertial properties.

It's essential to correctly identify distances in each system for precise \( I \) calculation. By comparing multiple moments \( I_A \) and \( I_B \) based on given axes, we determine how each system's unique mass distribution influences rotational behavior, aligning with the exercise's goal to explore different configurations' impact on physical dynamics.