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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = x, y, z is F(r) = Kr/| r |3 where K is a constant. (See Example 16.1.5.) Find the work done as the particle moves along a straight line from (2, 0, 0) to (2, 1, 5).

To determine

To find: The work done by the particle when it moves along a straight line from the point (2,0,0) to (2,1,5).

Explanation

Given:

The force field is F(r)=Kr|r|3.

The position vector is r=x,y,z.

The particle moves along a straight line from the point (2,0,0) to (2,1,5).

Formula used:

The work done by the force field F(x,y,z) along the line segment.

CFdr=abF(r)rdt (1)

Where, r is the position vector of the particle, r is the derivative of position vector, a is the lower limit of scalar a parameter, and b is the upper limit of the scalar parameter.

The parametric equations for a line segment through the point (x0,y0,z0) and parallel to the direction vector v=a,b,c.

x=x0+at,y=y0+bt,z=z0+ct (2)

The director vector v=a,b,c for a line segment from the point (x0,y0,z0) to (x1,y1,z1).

a,b,c=x1x0,y1y0,z1z0 (3)

To calculate the vector v=a,b,c substitute 2 for x0, 0 for y0, 0 for z0, 2 for x1, 1 for y1, and 5 for z1 in equation (3),

a,b,c=22,10,50=0,1,5

Substitute 2 for x0, 0 for y0, 0 for z0, 0 for a, 1 for b, and 5 for c in equation (2),

x=2+(0)t,y=0+(1)t,z=0+(5)tx=2,y=t,z=5t

Consider the limits of scalar parameter t are from 0 to 1 that is 0t1.

Write the position vector from the parametric equations as follows.

r=2,t,5t

Write the equation of force field as follows.

F(r)=Kr|r|3 (4)

Substitute the position vector 2,t,5t for r in equation (4),

F(r)=K2,t,5t|2,t,5t|3=K2,t,5t|22+t2+(5t)2|3=K(4+26t2)322,t,5t

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=2,t,5t as follows

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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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