Applications to Physics and Engineering

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 6.4 – Work
• Section 7.7 – Approximate Integration
• Section 8.3 – Applications to Physics and Engineering

Expected Educational Results

• Objective 14–1: Given the velocity of an object along a straight line, I can set up the integral to compute the displacement and total distance traveled of the object.
• Objective 14–2: Given the acceleration of an object along a straight line, I can set up the integral to compute the velocity, speed, displacement, and total distance traveled of the object.
• Objective 14–3: I can estimate a definite integral using a Riemann Sum.
• Objective 14–4: I can use the shape of a function to determine if a Riemann Sum is an over- or under-estimate of a definite integral.
• Objective 14–5: I can set up the integral to compute the moments and center of mass of a lamina.
• Objective 14–6: I can set up and evaluate the integral to compute the work on an object.
• Objective 14–7: I can set up and evaluate the integral using Hooke's Law to compute the work on a spring.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Work

Definition: Work with Constant Force

$W=Fd$$W=Fd$

where $W$$W$ is work, $F$$F$ is force on an object, $d$$d$ is distance.

Definition: Work with Variable Force

$W={\int }_a^{b}f(x)dx$$W=\int_a^b{f(x)\,dx}$

where $W$$W$ is work, $f(x)$$f(x)$ is force on an object.

Units

English

• Distance: foot [abbreviated ft]
• Force: pound [abbreviated lb or #]
• Work: foot-pound [abbreviated ft-lb]

SI

• Distance: meter [abbreviated m]
• Force: newton [abbreviated N]
• Work: newton-meter or joule [abbreviated Nm or J]

Investigation 03

1. How much work is done in lifting a $5\text{lb}$$5\,\text{lb}$ weight vertically a distance of $3$$3$ feet?

2. A $20\text{m}$$20\,\text{m}$ long steel cable has density $2\text{N/m}$$2\,\text{N/m}$, and is hanging straight down. How much work is required to lift the entire cable vertically $20\text{m}$$20\,\text{m}$?

3. A $50\text{ft}$$50\,\text{ft}$ rope, weighing $0.2\text{lb/ft}$$0.2\,\text{lb/ft}$, hangs over the edge of a tall building.

   1. How much work is done pulling the entire rope to the top of the building?
   2. How much work is done pulling in the first $20\text{m}$$20\,\text{m}$?
   3. How much work is done pulling in the last $20\text{m}$$20\,\text{m}$?

4. We have a cable that weighs $2\text{lbs/ft}$$2\,\text{lbs/ft}$ attached to a bucket filled with coal that weighs $800\text{lbs}$$800\,\text{lbs}$. The bucket is initially at the bottom of a $500\text{ft}$$500\,\text{ft}$ mine shaft. Determine the amount of work required to lift the bucket all the way up the shaft.

5. A $50\text{lb}$$50\,\text{lb}$ bag of flour is lifted uniformly $3\text{ft}$$3\,\text{ft}$ in $15$$15$ seconds. The flour leaks from the bag at a rate of $1/2\text{lb/s}$$1/2\,\,\text{lb/s}$. What is the total work done in lifting the bag?

Homework

At this time, you should be able to complete the following assignments:

Section 6.4: # 3, 13, 15, 17, 19.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Monday, 19 October 2020 8:23 EDT Last Modified: Friday, 15 July 2022 - 11:26 (EDT)