Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

Problem 58

Chapter 10, Problem 58

(III) Integrate to derive the formula for the moment of inertia of a uniform thin rod of length ℓ about an axis through its center, perpendicular to the rod (see Fig. 10–21f).

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Start by recalling the formula for the moment of inertia: \( I = \int r^2 \, dm \), where \( r \) is the perpendicular distance from the axis of rotation to the mass element \( dm \).

Define the mass element \( dm \) in terms of the linear mass density \( \lambda \), where \( \lambda = \frac{M}{\ell} \) (\( M \) is the total mass of the rod and \( \ell \) is its length). Thus, \( dm = \lambda \, dx \), where \( dx \) is a small segment of the rod.

Set up the coordinate system: Place the center of the rod at the origin of the \( x \)-axis, so the rod extends from \( -\ell/2 \) to \( \ell/2 \). The distance \( r \) from the axis of rotation to a small segment of the rod is simply \( |x| \).

Substitute \( r \) and \( dm \) into the moment of inertia formula: \( I = \int_{-\ell/2}^{\ell/2} x^2 \lambda \, dx \). Replace \( \lambda \) with \( \frac{M}{\ell} \), giving \( I = \frac{M}{\ell} \int_{-\ell/2}^{\ell/2} x^2 \, dx \).

Evaluate the integral: \( \int_{-\ell/2}^{\ell/2} x^2 \, dx \). Use the symmetry of the function \( x^2 \) about the origin to simplify the integral to \( 2 \int_{0}^{\ell/2} x^2 \, dx \). Solve this integral and multiply by \( \frac{M}{\ell} \) to find the final expression for \( I \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It depends on the mass distribution relative to that axis, with greater distances from the axis resulting in a higher moment of inertia. For a uniform thin rod, this concept is crucial as it quantifies how the rod will respond to applied torques.

Integration in Physics

Integration is a mathematical process used to calculate quantities that accumulate over a continuous range, such as mass distribution along a rod. In deriving the moment of inertia, integration allows us to sum the contributions of infinitesimally small mass elements across the length of the rod, providing a comprehensive measure of its rotational inertia.

Axis of Rotation

The axis of rotation is the line about which an object rotates. In this case, the axis is through the center of the rod and perpendicular to its length. The choice of axis significantly affects the moment of inertia, as it determines how the mass is distributed relative to that axis, influencing the rod's rotational dynamics.

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Textbook Question

To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 10–61. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 28 rpm in 5.0 min, starting from rest?

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Textbook Question

Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia I. The blocks move (towards the right) with an acceleration of 1.00 m/s² along their frictionless inclines (see Fig. 10–62). Find the net torque acting on the pulley, and determine its moment of inertia, I.

Textbook Question

A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.7 m/s. Calculate its total kinetic energy.

Textbook Question

Suppose the force Fₜ in the cord hanging from the pulley of Example 10–10, Fig. 10–22, is given by the relation Fₜ = 3.00 t ― 0.20 t² (newtons) where t is in seconds. If the pulley starts from rest, what is the linear speed of a point on its rim 9.0 s later? Ignore friction and use the moment of inertia, calculated in Example 10–10.

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Textbook Question

A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]

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Textbook Question

II) A uniform solid sphere of radius r₀ = 24.5 cm and mass m = 1.60 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long. Calculate its translational and rotational speeds when it reaches the bottom. Avoid putting in numbers until the end so you can answer.

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