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 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]

Hooke's Law for Springs

Definition: Hooke's Law

When a spring is stretched or compressed, so that its length changes by an amount $x$$x$ from its equilibrium length, $x=c$$x=c$, then it exerts a force $f(x)=-kx$$f(x) = -kx$, where $k$$k$ is the force constant for the particular spring, in a direction towards its equilibrium position.

The work performed on a spring to stretch it from $x=a$$x=a$ to $x=b$$x=b$ is $w={\int }_{a-c}^{b-c}-kxdx$$w=\int_{a-c}^{b-c}{-kx\,dx}$.

Investigation 04

You may need to convert to the correct units.

1. A $10\text{cm}$$10\,\text{cm}$ long spring is attached to a wall. When pulled horizontally with a force of $80\text{N}$$80 \,\text{N}$, the spring stretches to a length of $13\text{cm}$$13\,\text{cm}$. What is the value of the spring constant?
2. A force of $20\text{lbs}$$20\,\text{lbs}$ stretches a spring from a natural length of $8$$8$ inches to a length of $11$$11$ inches. How much work was performed in stretching the spring to this length?
3. A force of $10\text{N}$$10\,\text{N}$ stretches a spring from a natural length of $5\text{cm}$$5\,\text{cm}$ to a length of $13\text{cm}$$13\,\text{cm}$. How much work was performed in stretching the spring to this length?
4. A force of $25\text{lbs}$$25\,\text{lbs}$ is required to hold a spring stretched $8$$8$ inches beyond its natural length. How much work is done in stretching it from its natural length to $15$$15$ inches beyond its natural length?
5. A force of $200\text{N}$$200 \,\text{N}$ compresses a spring $30\text{cm}$$30\,\text{cm}$ from its natural length. How much work is performed in compressing the spring $45\text{cm}$$45\,\text{cm}$ from its natural length?
6. A force of $35\text{lb}$$35\,\text{lb}$ compresses a spring from a natural length of $17\text{in}$$17\,\text{in}$ to $13\text{in}$$13\,\text{in}$. How much work is performed in compressing the spring $7\text{in}$$7\,\text{in}$ from its natural length?
7. A force of $20\text{lb}$$20\,\text{lb}$ stretches a spring from a natural length of $6\text{in}$$6\,\text{in}$ to $8\text{in}$$8\,\text{in}$. How much work is performed in stretching the spring from length of $9\text{in}$$9\,\text{in}$ to $15\text{in}$$15\,\text{in}$?
8. A force of $7\text{N}$$7\,\text{N}$ stretches a spring from a natural length of $11\text{cm}$$11\,\text{cm}$ to $21\text{cm}$$21\,\text{cm}$. How much work is performed in stretching the spring from a length of $16\text{cm}$$16\,\text{cm}$ to $21\text{cm}$$21\,\text{cm}$?
9. A $40\text{lb}$$40\,\text{lb}$ weight is attached to a spring. The weight stretches the spring from a natural length of $18\text{in}$$18\,\text{in}$ to $32\text{in}$$32\,\text{in}$. How much work is done in lifting the box $12\text{in}$$12\,\text{in}$?
10. $6\text{J}$$6\,\text{J}$ of work is needed to stretch a spring from its natural length of $10\text{cm}$$10\,\text{cm}$ to $25\text{cm}$$25\,\text{cm}$. Find the spring constant.

Homework

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

Section 6.4: # 7, 9.

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: Friday, 15 July 2022 - 11:29 (EDT)