Ch 09: Rotation of Rigid Bodies

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 09: Rotation of Rigid Bodies

Problem 46

Chapter 9, Problem 46

A uniform 2.00-m ladder of mass 9.00 kg is leaning against a vertical wall while making an angle of 53.0° with the floor. A worker pushes the ladder up against the wall until it is vertical. What is the increase in the gravitational potential energy of the ladder?

Verified step by step guidance

Step 1: Understand the problem. The gravitational potential energy (U) of an object is given by the formula:

U = m g h

, where

m

is the mass of the object,

g

is the acceleration due to gravity, and

h

is the height of the center of mass above a reference point. The goal is to calculate the change in gravitational potential energy as the ladder moves from its initial inclined position to a vertical position.

Step 2: Determine the initial height of the ladder's center of mass. Since the ladder is uniform, its center of mass is located at its midpoint. Initially, the ladder makes an angle of 53.0° with the floor. The height of the center of mass can be calculated as:

h

_initial = (L/2)sin(θ), where

L

is the length of the ladder and

θ

is the angle with the floor.

Step 3: Determine the final height of the ladder's center of mass. When the ladder is vertical, its center of mass is directly above the floor at a height of:

h

_final = L/2, since the center of mass is still at the midpoint of the ladder.

Step 4: Calculate the change in gravitational potential energy. The change in gravitational potential energy is given by:

ΔU = m g (h

_final - h_initial). Substitute the expressions for

h

_final and

h

_initial into this equation.

Step 5: Plug in the known values. Use

m = 9.00

kg,

g = 9.8

m/s²,

L = 2.00

m, and

θ = 53.0°

. Perform the calculations to find the change in gravitational potential energy.

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Key Concepts

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

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula GPE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height above a reference point. In this scenario, as the ladder is raised, its height increases, leading to an increase in GPE.

Height Calculation

To determine the change in gravitational potential energy, it is essential to calculate the height of the ladder's center of mass at both the initial and final positions. When the ladder is leaning at an angle, the height can be found using trigonometric functions, specifically h = L * sin(θ), where L is the length of the ladder and θ is the angle with the horizontal. This height change is crucial for calculating the increase in GPE.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its energy. In this context, as the worker pushes the ladder up, work is done against gravity, resulting in an increase in the ladder's gravitational potential energy. Understanding this principle helps in analyzing the energy transformations occurring during the ladder's movement.

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