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 46

Chapter 10, Problem 46

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.

Verified step by step guidance

Identify the given information: The force in the cord is given as Fₜ = 3.00t - 0.20t² (in newtons), where t is in seconds. The pulley starts from rest, and we are tasked with finding the linear speed of a point on its rim after 9.0 seconds. The moment of inertia of the pulley is provided in Example 10–10.

Relate the force in the cord to the torque on the pulley. Torque (τ) is given by τ = r × Fₜ, where r is the radius of the pulley. Use this to express the torque as a function of time: τ(t) = r × (3.00t - 0.20t²).

Use the relationship between torque and angular acceleration: τ = Iα, where I is the moment of inertia of the pulley and α is the angular acceleration. Solve for α(t): α(t) = τ(t) / I = [r × (3.00t - 0.20t²)] / I.

Integrate the angular acceleration α(t) with respect to time to find the angular velocity ω(t). Since the pulley starts from rest, the initial angular velocity ω₀ = 0. The angular velocity is given by ω(t) = ∫α(t) dt.

Relate the angular velocity ω(t) to the linear speed v(t) of a point on the rim of the pulley using the formula v(t) = r × ω(t). Evaluate v(t) at t = 9.0 seconds to find the linear speed of the point on the rim.

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Key Concepts

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is crucial for understanding how forces affect the motion of objects, including the relationship between force, mass, and acceleration in the context of the pulley system.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. In this problem, the moment of inertia of the pulley will influence how the applied force affects its angular acceleration and, consequently, the linear speed of a point on its rim.

Kinematics of Rotational Motion

Kinematics of rotational motion describes the relationship between angular displacement, angular velocity, and angular acceleration. For the pulley, the linear speed of a point on its rim can be determined from its angular velocity, which is related to the angular acceleration produced by the net torque from the force in the cord.

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Related Practice

Textbook Question

Let us treat a helicopter rotor blade as a long thin rod, as shown in Fig. 10–60. If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.

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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

Calculate the moment of inertia of the array of point objects shown in Fig. 10–58 about the y axis, and the x axis. Assume m = 22kg, M = 3.2kg, and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the x axis. About which axis would it be harder to accelerate this array?

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Textbook Question

(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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Textbook Question

A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 330 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?

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