Domanda del libro di testo
Spring work
b. It takes 50 N of force to stretch a spring 0.2 m from its equilibrium position. How much work is needed to stretch it an additional 0.5 m?
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 6.R.77b
Pumping water A water tank has the shape of a box that is 2 m wide, 4 m long, and 6 m high.
b. If the water in the tank is 2 m deep, how much work is required to pump the water to a level of 1 m above the top of the tank?
Guida verificata passo dopo passo1
First, identify the dimensions of the tank and the depth of the water. The tank is 2 m wide, 4 m long, and 6 m high. The water depth is 2 m.
Set up a coordinate system to represent the height of the water. Let \( y = 0 \) be at the bottom of the tank and \( y = 6 \) be at the top. The water occupies from \( y = 0 \) to \( y = 2 \).
Consider a thin horizontal slice of water at height \( y \) with thickness \( dy \). The volume of this slice is the area of the base times the thickness: \( dV = (2 \times 4) dy = 8 dy \) cubic meters.
Calculate the weight of this slice. Since the density of water is \( \rho = 1000 \, \text{kg/m}^3 \) and gravitational acceleration is \( g = 9.8 \, \text{m/s}^2 \), the weight is \( dW = \rho g dV = 1000 \times 9.8 \times 8 dy = 78400 dy \) newtons.
Determine the distance the slice must be lifted. The water must be pumped to 1 m above the top of the tank, so the lifting height for the slice at height \( y \) is \( (6 + 1) - y = 7 - y \) meters.
Set up the integral for the work done to pump all the water: \[ W = \int_0^2 \text{(weight of slice)} \times \text{(distance lifted)} \, dy = \int_0^2 78400 (7 - y) dy. \] This integral represents the total work required.
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Work Done by a Variable Force
Work is calculated as the integral of force over distance when the force varies. In pumping water, the force depends on the weight of each water layer, and the distance varies with depth, requiring integration to sum the work done on all layers.
Video consigliato:
Percorso guidato
05:40
Work Done On A Spring (Hooke's Law)
Volume and Weight of Water
The volume of water is found by multiplying the tank's cross-sectional area by the water depth. The weight is then calculated by multiplying the volume by the density of water and gravitational acceleration, which gives the force needed to move the water.
Video consigliato:
04:48Finding Volume Using Disks
Setting up the Integral for Work
To find the total work, divide the water into thin horizontal slices, calculate the work to lift each slice to the target height, and integrate over the depth of the water. This involves expressing the distance each slice is lifted as a function of its depth.
Video consigliato:
Percorso guidato
06:22Introduction To Work
Pratica correlata
Domanda del libro di testo
Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:
b. Apply the shell method and integrate with respect to x.
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Domanda del libro di testo
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?
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Domanda del libro di testo
An oscillator The acceleration of an object moving along a line is given by a(t) = 2 sin πt/4. The initial velocity and position are v(0)= −8/π and s(0)=0.
a. Find the velocity and position for t≥0.
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Domanda del libro di testo
58–61. Arc length Find the length of the following curves.
y = x³/6 + 1/2x on [1,2]
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Domanda del libro di testo
Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?
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