Domanda del libro di testo
Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)
b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy
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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 6.7.28b
Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.
b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position
Guida verificata passo dopo passo1
Recall that the work done to stretch or compress a spring from its equilibrium position is given by the formula \(W = \frac{1}{2} k x^2\), where \(k\) is the spring constant and \(x\) is the displacement from equilibrium.
Use the information given for the spring: it requires 2 J of work to stretch it 0.1 m. Substitute these values into the work formula to find the spring constant \(k\): \(2 = \frac{1}{2} k (0.1)^2\).
Solve the equation for \(k\) by isolating it: multiply both sides by 2 and divide by \((0.1)^2\) to get \(k = \frac{2 \times 2}{(0.1)^2}\).
Now that you have the spring constant \(k\), use it to calculate the work required to stretch the spring 0.4 m by substituting into the work formula: \(W = \frac{1}{2} k (0.4)^2\).
Simplify the expression to find the work done for the 0.4 m stretch, which will give you the answer to the problem.
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Hooke's Law
Hooke's Law states that the force required to stretch or compress a spring is proportional to the displacement from its equilibrium position, expressed as F = kx, where k is the spring constant and x is the displacement. This linear relationship is fundamental for calculating forces and work in spring problems.
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Percorso guidato
05:40
Work Done On A Spring (Hooke's Law)
Work Done by a Variable Force
When stretching a spring, the force varies with displacement, so work is calculated as the integral of force over distance. For springs, work done W = (1/2)kx², representing the area under the force-displacement curve, which is essential for determining energy stored or work required.
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Percorso guidato
05:40Work Done On A Spring (Hooke's Law)
Determining the Spring Constant from Work
Given the work done to stretch a spring a certain distance, the spring constant k can be found by rearranging the work formula: k = 2W / x². This allows solving for k when work and displacement are known, enabling calculation of work for other displacements.
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Percorso guidato
05:40Work Done On A Spring (Hooke's Law)
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Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).
b. How is this formula changed if x0>b?
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Domanda del libro di testo
Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.
b. How far does the car travel in the time it takes to come to rest?
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Domanda del libro di testo
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
b. Find the displacement over the given interval.
v(t) = 3t²−6t on [0, 3]
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Domanda del libro di testo
40–43. Population growth
When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.
b. Find the population P(t) at any time t≥0.
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Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.
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