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Chapter 10: Problem 42

The masses and coordinates of four particles are as follows: \(50 \mathrm{~g}, x=2.0 \mathrm{~cm}, y=2.0 \mathrm{~cm} ; 25 \mathrm{~g}, x=0, y=4.0 \mathrm{~cm} ; 25 \mathrm{~g}\) \(x=-3.0 \mathrm{~cm}, \quad y=-3.0 \mathrm{~cm} ; 30 \mathrm{~g}, x=-2.0 \mathrm{~cm}, \quad y=4.0 \mathrm{~cm} .\) are the rotational inertias of this collection about the (a) \(x,\) (b) \(y\), and (c) \(z\) axes? (d) Suppose that we symbolize the answers to (a) and (b) as \(A\) and \(B\), respectively. Then what is the answer to (c) in terms of \(A\) and \(B ?\)

Short Answer

Expert verified
(a) 1305 g cm虏, (b) 545 g cm虏, (c) 1850 g cm虏, (d) I_z = A + B.

Step by step solution

01

Calculate Rotational Inertia about the X-axis

The rotational inertia about the x-axis, \( I_x \), is computed by summing the products of each mass and the square of its distance from the x-axis. The formula is \( I_x = \sum m_i y_i^2 \):- For the first particle: \( 50 \text{ g} \times (2.0 \text{ cm})^2 = 200 \text{ g cm}^2 \).- For the second particle: \( 25 \text{ g} \times (4.0 \text{ cm})^2 = 400 \text{ g cm}^2 \).- For the third particle: \( 25 \text{ g} \times (-3.0 \text{ cm})^2 = 225 \text{ g cm}^2 \).- For the fourth particle: \( 30 \text{ g} \times (4.0 \text{ cm})^2 = 480 \text{ g cm}^2 \).Total \( I_x = 200 + 400 + 225 + 480 = 1305 \text{ g cm}^2 \).
02

Calculate Rotational Inertia about the Y-axis

The rotational inertia about the y-axis, \( I_y \), is found by summing the products of each mass and the square of its distance from the y-axis. Use the formula \( I_y = \sum m_i x_i^2 \):- For the first particle: \( 50 \text{ g} \times (2.0 \text{ cm})^2 = 200 \text{ g cm}^2 \).- For the second particle: \( 25 \text{ g} \times (0 \text{ cm})^2 = 0 \text{ g cm}^2 \).- For the third particle: \( 25 \text{ g} \times (-3.0 \text{ cm})^2 = 225 \text{ g cm}^2 \).- For the fourth particle: \( 30 \text{ g} \times (-2.0 \text{ cm})^2 = 120 \text{ g cm}^2 \).Total \( I_y = 200 + 0 + 225 + 120 = 545 \text{ g cm}^2 \).
03

Calculate Rotational Inertia about the Z-axis

The rotational inertia about the z-axis, \( I_z \), is the sum of the products of each mass and the square of its distance from the origin (both x and y components). The formula is \( I_z = \sum m_i (x_i^2 + y_i^2) \):Using \( I_z = I_x + I_y \), since both components are orthogonal:\( I_z = 1305 + 545 = 1850 \text{ g cm}^2 \).
04

Express Z-axis Rotational Inertia in terms of A and B

Given that \( A = I_x \) and \( B = I_y \), express \( I_z \) as:\( I_z = A + B \).Thus, \( I_z = 1305 + 545 = 1850 \text{ g cm}^2 \), where \( A = 1305 \text{ g cm}^2 \) and \( B = 545 \text{ g cm}^2 \).

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Key Concepts

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

Moment of Inertia Calculations
Moment of inertia, also known as rotational inertia, is a measure of how difficult it is to change the rotational state of an object. Think of it as the rotational equivalent of mass. In physics, calculating the moment of inertia is crucial to understand how different forces can affect the motion of particles or rigid body systems.

The basic formula for calculating the moment of inertia (\( I \)) in particle systems is expressed as:
  • \( I = \sum m_i r_i^2 \)
where \( m_i \) is the mass of the particle and \( r_i \) is the distance from the axis of rotation.

For each axis鈥攛, y, and z鈥攚e calculate the distance of each particle from that specific axis. For the x-axis, only the y-coordinates matter, for the y-axis, only the x-coordinates, and for the z-axis, both coordinates contribute. In the exercise, we broke down these steps clearly to ensure all students could follow:
  • Calculate \( I_x \) by summing \( m_i y_i^2 \)
  • Calculate \( I_y \) by summing \( m_i x_i^2 \)
  • Calculate \( I_z \) by adding \( I_x \) and \( I_y \), which accounts for both x and y components.
Coordinate System
Understanding the coordinate system is key to calculating moments of inertia in exercises like these. In our problem, each particle has specific coordinates that help determine their positions relative to the axes. The Cartesian coordinate system is used here and consists of the x, y, and z dimensions.
  • X-axis: Horizontal line where the x-coordinate increases to the right.
  • Y-axis: Vertical line where the y-coordinate increases upwards.
  • Z-axis: Perpendicular to both x and y, often viewed as the depth.
The position of each particle is given by coordinates \((x, y)\). In our case, the z-coordinates are not explicitly given as this exercise is in two dimensions. When calculating moments of inertia:
  • For the x-axis, the distance of particles is determined by their y-position.
  • For the y-axis, the distance of particles is determined by their x-position.
  • For the z-axis, the 3D perspective is calculated as the combination of the distances from both x and y.
Knowing how a coordinate system functions helps simplify complex calculations involving rotational inertia.
Particles in Physics
In physics, particles are treated as small objects with mass. Understanding how they behave and interact with forces is crucial to understanding motion. In this exercise, each particle is assigned a specific mass and coordinate location that contributes to the calculation of the system鈥檚 rotational inertia.

There are essential points to remember when dealing with particles in physics:
  • Mass is critical: Each particle has its own mass \( m_i \), which directly affects the overall moment of inertia.
  • Coordinate influence: The position (coordinates) of each particle determines its impact on inertia relating to each axis.
  • Sum equals whole: The system鈥檚 moment of inertia is the sum of contributions from all particles, showing how even small particles can collectively influence larger rigid body rotations.
For students, this reminder highlights the interconnectedness of mass and position in calculating and understanding rotational dynamics in any physical system.

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Most popular questions from this chapter

In Fig. 10-61, four pulleys are connected by two belts. Pulley \(A\) (radius \(15 \mathrm{~cm}\) ) is the drive pulley, and it rotates at \(10 \mathrm{rad} / \mathrm{s}\). Pulley \(B\) (radius \(10 \mathrm{~cm})\) is connected by belt 1 to pulley \(A .\) Pulley \(B^{\prime}\) (radius \(5 \mathrm{~cm})\) is concentric with pulley \(B\) and is rigidly attached to it. Pulley \(C\) (radius \(25 \mathrm{~cm}\) ) is connected by belt 2 to pulley \(B^{\prime}\). Calculate (a) the linear speed of a point on belt \(1,\) (b) the angular speed of pulley \(B\), (c) the angular speed of pulley \(B^{\prime},(\mathrm{d})\) the linear speed of a point on belt \(2,\) and \((\mathrm{e})\) the angular speed of pulley \(C\). (Hint: If the belt between two pulleys does not slip, the linear speeds at the rims of the two pulleys must be equal.)

If a \(32.0 \mathrm{~N} \cdot \mathrm{m}\) torque on a wheel causes angular acceleration \(25.0 \mathrm{rad} / \mathrm{s}^{2}\), what is the wheel's rotational inertia?

Four identical particles of mass \(0.50 \mathrm{~kg}\) each are placed at the vertices of a \(2.0 \mathrm{~m} \times 2.0 \mathrm{~m}\) square and held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is perpendicular to the plane of the square, and (c) lies in the plane of the square and passes through two diagonally opposite particles?

A small ball of mass \(0.75 \mathrm{~kg}\) is attached to one end of a 1.25 -m-long massless rod, and the other end of the rod is hung from a pivot. When the resulting pendulum is \(30^{\circ}\) from the vertical, what is the magnitude of the gravitational torque calculated about the pivot?

The angular position of a point on the rim of a rotating wheel is given by \(\theta=4.0 t-3.0 t^{2}+t^{3},\) where \(\theta\) is in radians and \(t\) is in seconds. What are the angular velocities at (a) \(t=2.0 \mathrm{~s}\) and \((\mathrm{b}) t=4.0 \mathrm{~s} ?\) (c) What is the average angular acceleration for the time interval that begins at \(t=2.0 \mathrm{~s}\) and ends at \(t=4.0 \mathrm{~s} ?\) What are the instantaneous angular accelerations at (d) the beginning and (e) the end of this time interval?
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