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In Fig. \(10-42 a\), an irregularly shaped plastic plate with uniform thickness and density (mass per unit volume) is to be rotated around an axle that is perpendicular to the plate face and through point \(O\). The rotational inertia of the plate about that axle is measured with the following method. A circular disk of mass \(0.500 \mathrm{~kg}\) and radius \(2.00\) \(\mathrm{cm}\) is glued to the plate, with its center aligned with point \(O\) (Fig. \(10-42 b)\). A string is wrapped around the edge of the disk the way a string is wrapped around a top. Then the string is pulled for \(5.00 \mathrm{~s}\). As a result, the disk and plate are rotated by a constant force of \(0.400 \mathrm{~N}\) that is applied by the string tangentially to the edge of the disk. The resulting angular speed is \(114 \mathrm{rad} / \mathrm{s}\). What is the rotational inertia of the plate about the axle?

Step by step solution

01

Understand the Problem

We need to find the rotational inertia (\( I \)) of the plastic plate. A disk is glued at point \( O \), and a force is applied by a string to rotate the disk and the plate. The force, radius of the disk, mass of the disk, duration of the force, and resulting angular speed are known.

02

Calculate the Torque

The torque (\( \tau \)) applied on the disk is given by the formula \( \tau = r \times F \), where \( r \) is the radius of the disk and \( F \) is the force. Here, \( r = 0.02 \, \mathrm{m} \) and \( F = 0.400 \, \mathrm{N} \). Thus, \( \tau = 0.02 \times 0.400 = 0.008 \, \mathrm{N \cdot m} \).

03

Apply Torque-Angular Acceleration Relationship

Since torque is related to angular acceleration (\( \alpha \)) by \( \tau = I_{\text{total}} \cdot \alpha \), first calculate \( \alpha \) using the angular speed (\( \omega = 114 \, \mathrm{rad/s} \)) and time \( t = 5.00 \, \mathrm{s} \): \( \alpha = \frac{\omega}{t} = \frac{114}{5} = 22.8 \, \mathrm{rad/s^2} \).

04

Calculate Total Rotational Inertia

Rearrange the torque equation to find \( I_{\text{total}} \): \( I_{\text{total}} = \frac{\tau}{\alpha} = \frac{0.008}{22.8} \approx 0.00035 \, \mathrm{kg \cdot m^2} \). This includes the rotational inertia of the disk and the plate.

05

Calculate Rotational Inertia of the Disk

The rotational inertia of the disk \( I_{\text{disk}} \) about its center is \( \frac{1}{2} m r^2 \), where \( m = 0.500 \, \mathrm{kg} \) and \( r = 0.02 \, \mathrm{m} \). So, \( I_{\text{disk}} = \frac{1}{2} \times 0.500 \times (0.02)^2 = 0.0001 \mathrm{kg \cdot m^2} \).

06

Calculate Rotational Inertia of the Plate

Subtract the rotational inertia of the disk from the total rotational inertia to find the rotational inertia of the plate \( I_{\text{plate}} \): \( I_{\text{plate}} = I_{\text{total}} - I_{\text{disk}} = 0.00035 - 0.0001 = 0.00025 \, \mathrm{kg \cdot m^2} \).

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Key Concepts

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

Torque

Torque is a fundamental concept in rotational motion, very much like force is to linear motion. It denotes the rotational effect of a force applied at a specific distance from the axis of rotation. To calculate torque, we use the equation \( \tau = r \times F \), where \( \tau \) represents torque, \( r \) is the radius or distance from the axis, and \( F \) is the force applied. In the given problem, a force of \(0.400 \, \mathrm{N}\) was applied tangentially at a radius of \(0.02 \, \mathrm{m}\) to the edge of the disk. This illustrates how torque leads to rotational motion. Key Takeaways on Torque:

• Torque depends on both the force applied and the distance from the axis of rotation.
• The direction of the torque vector is perpendicular to the plane formed by \(r\) and \(F\).
• Torque causes angular acceleration, similar to how force causes linear acceleration.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It highlights how quickly a rotating object is speeding up or slowing down. In our exercise, the constant torque applied on the disk caused a change in the disk's angular velocity. Angular acceleration \( \alpha \) can be determined using \( \alpha = \frac{\omega}{t} \), where \( \omega \) is the angular speed and \( t \) is time. Here, after applying torque for \(5 \, \mathrm{s}\), the disk reached an angular speed of \(114 \, \mathrm{rad/s}\), which gave it an angular acceleration of \(22.8 \, \mathrm{rad/s^2}\).Understanding Angular Acceleration:

• Angular acceleration indicates how fast angular speed is changing over time.
• It is calculated by dividing the change in angular speed by the time taken.
• This concept links rotational motion with torque through the relation \( \tau = I \cdot \alpha \).

Physics Problem Solving

Physics problems, particularly involving rotation, require a strategic approach in identifying given quantities and determining unknowns. In the exercise, the problem was broken down into steps to ensure a clear path to the solution. Steps Taken in Solving the Problem:

• Identifying Known Values: Force, radius, mass, time, and resulting angular speed were recognized as given data.
• Finding Torque: Calculated using the radius and force applied to the edge of the disk.
• Determining Angular Acceleration: Utilized to find the total rotational inertia using the relationship with torque.
• Solving for Unknowns: Each calculation was checked for physical relevance and unit consistency.

By identifying critical variables and applying the right equations, the rotational inertia of the plastic plate was accurately determined, demonstrating the effectiveness of structured physics problem-solving.

Inertia Calculation

Calculating rotational inertia involves understanding how mass is distributed relative to the axis of rotation. It is vital in determining an object's resistance to changes in its rotational motion.Steps in Calculating Inertia for Combined Systems:

• Total Rotational Inertia: First, the total inertia \( I_{\text{total}} \) of the disk-plate system was calculated with \( \tau = I \cdot \alpha \).
• Rotational Inertia of the Disk: The disk's own inertia \( I_{\text{disk}} \) was calculated using \( \frac{1}{2} m r^2 \), a formula that applies to circular objects.
• Rotational Inertia of the Plate: Subtracted disk's inertia from the total inertia to get the plate's inertia \( I_{\text{plate}} = I_{\text{total}} - I_{\text{disk}} \).

Each part of this calculation reflects the significant understanding of inertia in distributed mass systems, emphasizing how various factors like mass and radius affect rotational properties.