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Chapter 10: Problem 37

Calculate the rotational inertia of a meter stick, with mass \(0.56 \mathrm{~kg}\), about an axis perpendicular to the stick and located at the \(20 \mathrm{~cm}\) mark. (Treat the stick as a thin rod.)

Short Answer

Expert verified
The rotational inertia is \( 0.0691 \, \text{kg} \, \text{m}^2 \).

Step by step solution

01

Identify the Rod's Mass and Length

The meter stick is treated as a thin rod with a mass \( m = 0.56 \, \text{kg} \) and a length \( L = 1 \, \text{m} \). The rotational inertia needs to be found about an axis located at \( d = 0.20 \, \text{m} \) from one end.
02

Use the Parallel Axis Theorem

The rotational inertia \( I \) for a thin rod about an axis through its center and perpendicular to its length is \( I_c = \frac{1}{12} mL^2 \). Using the parallel axis theorem, the inertia about a parallel axis \( d \) away can be written as:\[I = I_c + md^2\]
03

Calculate the Rotational Inertia About the Center Axis

Calculate \( I_c \) for the rod using the formula:\[I_c = \frac{1}{12} mL^2 = \frac{1}{12} \times 0.56 \, \text{kg} \times (1 \, \text{m})^2 = 0.0467 \, \text{kg} \, \text{m}^2\]
04

Calculate the Rotational Inertia About the 20 cm Axis

Using the result from Step 3 and the parallel axis theorem:\[I = I_c + md^2 = 0.0467 \, \text{kg} \, \text{m}^2 + 0.56 \, \text{kg} \times (0.20 \, \text{m})^2 = 0.0467 \, \text{kg} \, \text{m}^2 + 0.0224 \, \text{kg} \, \text{m}^2 = 0.0691 \, \text{kg} \, \text{m}^2\]
05

Final Result

The rotational inertia of the meter stick about the axis at the \( 20 \, \text{cm} \) mark is \( 0.0691 \, \text{kg} \, \text{m}^2 \).

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Key Concepts

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

Parallel Axis Theorem
Understanding the Parallel Axis Theorem is crucial when dealing with rotational inertia, especially when the axis of rotation is not through the object's center. The theorem states that you can find the moment of inertia about any axis parallel to one through the center of mass by adding the product of the object's mass and the square of the distance between the two axes.
This is particularly useful in the physics problem involving our meter stick.
  • Find the inertia around the center axis first.
  • Use the theorem to adjust this inertia to the new, off-center axis.
This involves applying the formula:\[I = I_c + md^2\]where:
  • \(I\) is the inertia about the given axis,
  • \(I_c\) is the inertia about the center of mass,
  • \(m\) is the mass,
  • \(d\) is the distance between the center and the new axis.
For the thin rod with the off-center axis, this adjustment allows accurate calculations of rotational inertia.
Thin Rod
In our physics problem, we treat the meter stick as a thin rod. This simplifies calculations, allowing us to use common formulas to determine rotational inertia efficiently.
A thin rod has uniform mass distribution and simplified symmetry, which aids in consistent calculations.
  • Use the formula for rotational inertia about the center axis for a thin rod:\[I_c = \frac{1}{12} mL^2\]
  • This abbreviation assumes the rod's cross-sectional dimensions are negligible.
Why does this matter? In essence, by treating objects as thin, we apply simple yet accurate models that make complex problems more manageable. This assumption, in our meter stick problem, ensures straightforward application of the parallel axis theorem.
Physics Problems
Solving physics problems like rotational inertia requires attention to basic principles and formulas, which then build up to more complex calculations.
The key is breaking down the problem into understandable pieces.
  • First, identify the fundamental parameters such as mass and length.
  • Next, understand the question's spatial arrangement, such as where the axis of rotation lies.
  • Finally, apply relevant formulas, adjusting as necessary with tools like the parallel axis theorem.
Practicing these steps strengthens problem-solving skills and builds confidence when facing various challenges in physics. The methodology used in this meter stick problem is not just about getting to a solution but about understanding the process that leads there.

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Most popular questions from this chapter

Attached to each end of a thin steel rod of length \(1.20 \mathrm{~m}\) and mass \(6.40 \mathrm{~kg}\) is a small ball of mass \(1.06 \mathrm{~kg} .\) The rod is constrained to rotate in a horizontal plane about a vertical axis through its midpoint. At a certain instant, it is rotating at \(39.0 \mathrm{rev} / \mathrm{s}\). Because of friction, it slows to a stop in \(32.0\) s Assuming a constant retarding torque due to friction, compute (a) the angular acceleration, (b) the retarding torque, (c) the total energy transferred from mechanical energy to thermal energy by friction, and (d) the number of revolutions rotated during the \(32.0 \mathrm{~s}\) (e) Now suppose that the retarding torque is known not to be constant. If any of the quantities (a), (b), (c), and (d) can still be computed without additional information, give its value.

An automobile crankshaft transfers energy from the engine to the axle at the rate of \(100 \mathrm{hp}(=74.6 \mathrm{~kW})\) when rotating at \(\mathrm{a}\) speed of 1800 rev/min. What torque (in newton-meters) does the crankshaft deliver?

The length of a bicycle pedal arm is \(0.152 \mathrm{~m}\), and a downward force of \(111 \mathrm{~N}\) is applied to the pedal by the rider. What is the magnitude of the torque about the pedal arm's pivot when the arm is at angle (a) \(30^{\circ}\), (b) \(90^{\circ}\), and (c) \(180^{\circ}\) with the vertical?

A record turntable rotating at \(33 \frac{1}{3}\) rev/min slows down and stops in \(30 \mathrm{~s}\) after the motor is turned off. (a) Find its (constant) angular acceleration in revolutions per minute-squared. (b) How many revolutions does it make in this time?

A thin spherical shell has a radius of \(1.90 \mathrm{~m}\). An applied torque of \(960 \mathrm{~N} \cdot \mathrm{m}\) gives the shell an angular acceleration of \(6.20 \mathrm{rad} / \mathrm{s}^{2}\) about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?
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