/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 A can of sardines is made to mov... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 40

A can of sardines is made to move along an \(x\) axis from \(x=0.25 \mathrm{~m}\) to \(x=1.25 \mathrm{~m}\) by a force with a magnitude given by \(F=\exp \left(-4 x^{2}\right)\), with \(x\) in meters and \(F\) in newtons (Here exp is the exponential function.) How much work is done on the can by the force?

Short Answer

Expert verified
The work done is approximately 0.3435 joules.

Step by step solution

01

Understanding the Problem

We are tasked with finding the work done on a can of sardines by a force as it moves along a path from \(x = 0.25\) meters to \(x = 1.25\) meters. The force is described by the function \(F(x) = \exp(-4x^2)\).
02

Work Formula

The work \(W\) done by a force \(F(x)\) as an object moves from position \(x_1\) to \(x_2\) is given by the integral: \[ W = \int_{x_1}^{x_2} F(x) \, dx \] In this problem, \(x_1 = 0.25\) and \(x_2 = 1.25\), so we need to evaluate the integral: \[ W = \int_{0.25}^{1.25} \exp(-4x^2) \, dx \]
03

Evaluate the Integral Numerically

The function \(\exp(-4x^2)\) does not have a simple antiderivative that can be expressed in terms of elementary functions. Therefore, we use numerical techniques or a calculator to evaluate the integral:\[ W \approx 0.3435 \] This is the approximate numeric result of the integral from 0.25 to 1.25.
04

Conclusion

The work done on the can by the force as it moves from \(x = 0.25\)m to \(x = 1.25\)m is approximately \(0.3435\) joules.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus is a branch of mathematics focused on the concept of integration, which involves summing infinitesimal parts to find whole quantities, such as areas under curves or accumulated quantities like work done. In this exercise, we use integral calculus to find the work done by a force described by a function. To calculate the work done, we integrate the force function over a specific path or interval. The force function given here is exponential, noted as \( F(x) = \exp(-4x^2) \). The integration of this function over the interval from \( x = 0.25 \) meters to \( x = 1.25 \) meters helps determine the total work involved in moving the object along this path.
  • Definite Integrals: These are used to find quantities like work done by evaluating the integral over specified limits - here from 0.25 to 1.25 meters.
  • Integration in Physics: It allows us to analyze forces and motions where exact algebraic solutions aren't available, using numerical results instead.
Understanding and calculating these integrals are pivotal in physics problems and help in understanding how various physical forces interact over space.
Physics Problem-Solving
Physics problem-solving requires employing mathematical principles to resolve questions about physical situations. In this exercise, we focus on the concept of work, which is defined as the product of force and displacement in the direction of the force. The work done by a force can be calculated using integral calculus when the force varies along the path of movement.
  • Physical Interpretation of Work: Work quantifies how much energy is transferred by a force across a displacement.
  • Using Functions for Forces: The exercise illustrates that forces can be expressed as mathematical functions like \( \exp(-4x^2) \), modeling how they vary across different positions.
When tackling such problems, determining the work done involves setting up the integral properly and interpreting the resulting values within the context of the problem.
Numerical Integration
Numerical integration is a practical technique used to evaluate integrals that do not have closed-form solutions鈥攖hose that cannot be expressed with an elementary antiderivative. The given function \( \exp(-4x^2) \) is one such function. In this scenario, numerical methods are used for calculating the integral, which provides us with an approximate value for the work done over the specified interval.
  • Techniques Used: Methods such as the trapezoidal rule, Simpson's rule, or using computational tools help in estimating the integral.
  • Precision and Approximation: Numerical results offer a balance between precision and feasibility, enabling us to work with complex problems when analytic solutions aren't possible.
Applying numerical integration techniques is essential in physics to understand systems influenced by complex force patterns, ensuring that we can still reach usable results without exact functions.

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Most popular questions from this chapter

An iceboat is at rest on a frictionless frozen lake when a sudden wind exerts a constant force of \(200 \mathrm{~N}\), toward the east, on the boat. Due to the angle of the sail, the wind causes the boat to slide in a straight line for a distance of \(8.0 \mathrm{~m}\) in a direction \(20^{\circ}\) north of east. What is the kinetic energy of the iceboat at the end of that \(8.0 \mathrm{~m} ?\)

During spring semester at MIT, residents of the parallel buildings of the East Campus dorms battle one another with large catapults that are made with surgical hose mounted on a window frame. A balloon filled with dyed water is placed in a pouch attached to the hose, which is then stretched through the width of the room. Assume that the stretching of the hose obeys Hooke's law with a spring constant of \(100 \mathrm{~N} / \mathrm{m}\). If the hose is stretched by \(5.00 \mathrm{~m}\) and then released, how much work does the force from the hose do on the balloon in the pouch by the time the hose reaches its relaxed length?

A cord is used to vertically lower an initially stationary block of mass \(M\) at a constant downward acceleration of \(g / 4\). When the block has fallen a distance \(d\), find (a) the work done by the cord's force on the block, (b) the work done by the gravitational force on the block, (c) the kinetic energy of the block, and (d) the speed of the block.

If a Saturn \(V\) rocket with an Apollo spacecraft attached had a combined mass of \(2.9 \times 10^{5} \mathrm{~kg}\) and reached a speed of \(11.2 \mathrm{~km} / \mathrm{s}\), how much kinetic energy would it then have?

A particle moves along a straight path through displacement \(\vec{d}=(8 \mathrm{~m}) \hat{\mathrm{i}}+c \hat{\mathrm{j}}\) while force \(\vec{F}=(2 \mathrm{~N}) \hat{\mathrm{i}}-(4 \mathrm{~N}) \hat{\mathrm{j}}\) acts on it. (Other forces also act on the particle.) What is the value of \(c\) if the work done by \(\vec{F}\) on the particle is (a) zero, (b) positive, and (c) negative?
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