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 100

Chapter 10, Problem 100

A 5.0-m-long ladder is leaning against the side of a building making a 35° angle with the building. When a person is about 1/3 of the way up, the ladder slips and falls to the ground in 3.0 s. What is the average angular acceleration of the ladder as it falls?

Verified step by step guidance

Identify the given values: The length of the ladder is 5.0 m, the angle with the building is 35°, and the time it takes for the ladder to fall is 3.0 s. The goal is to find the average angular acceleration (α).

Understand the relationship between angular displacement (θ), angular acceleration (α), and time (t). The ladder starts from rest, so its initial angular velocity (ω₀) is 0. Use the kinematic equation for rotational motion:

θ = ω_{0} t + \frac{1}{2} α t 2

. Since

ω_{0} = 0

, the equation simplifies to

θ = \frac{1}{2} α t 2

Determine the angular displacement (θ) of the ladder as it falls. The ladder starts upright (90° or

\frac{π}{2}

radians) and ends flat on the ground (0 radians). Therefore, the total angular displacement is

\frac{π}{2}

radians.

Rearrange the simplified kinematic equation to solve for angular acceleration (α):

α = \frac{2}{θ} \frac{1}{t^{2}}

. Substitute the known values:

θ = \frac{π}{2}

radians and

t = 3.0

Simplify the expression to calculate the average angular acceleration (α). Ensure the units are consistent, and the result will be in radians per second squared (

rad / s^{2}

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Key Concepts

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

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is a vector quantity that indicates how quickly an object is rotating and how that rotation is changing. In this scenario, it is crucial to determine how fast the ladder's rotation increases as it falls, which can be calculated using the change in angular velocity divided by the time taken for the fall.

Kinematics of Rotational Motion

Kinematics of rotational motion describes the motion of objects that rotate around an axis. Key equations relate angular displacement, angular velocity, and angular acceleration, similar to linear motion. Understanding these relationships is essential for analyzing the ladder's fall, as it allows us to apply the appropriate kinematic equations to find the average angular acceleration during the fall.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on the mass distribution relative to the axis of rotation. For the ladder, its moment of inertia will influence how it accelerates as it falls. Knowing the moment of inertia is important for calculating the dynamics of the ladder's fall and understanding how mass and shape affect its angular acceleration.

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