   Chapter 16.2, Problem 44E

Chapter
Section
Textbook Problem

An object with mass m moves with position function r(t) = a sin t i + b cos t j + ct k, 0 ⩽ t ⩽ π/2. Find the work done on the object during this time period.

To determine

To find: The work done on the object.

Explanation

Given data:

An object with mass m moves with the position vector r(t)=asinti+bcostj+ctk,0tπ2 .

Formula used:

Write the expression to find the work done on the object. by the force field F(x,y,z) along the line segment.

W=abF(t)r(t)dt (1)

Here,

F(t) is the force vector on the object,

r(t) is the velocity vector of the object,

a is the lower limit of the scalar parameter, which is 0, and

b is the upper limit of the scalar parameter, which is π2 .

Write the expression to find the force vector.

F(t)=ma(t) (2)

Here,

m is the mass of the object and

a(t) is the acceleration vector of the object.

Write the expression to find acceleration vector of the object.

a(t)=ddt[v(t)] (3)

Here,

v(t) is the velocity vector of the object.

Write the expression to find velocity vector of the object.

v(t)=ddt[r(t)] (4)

Calculation of v(t) :

Substitute asinti+bcostj+ctk for r(t) in equation (4),

v(t)=ddt(asinti+bcostj+ctk)=ddt(asint)i+ddt(bcost)j+ddt(ct)k=(acost)i+(bsint)j+ck=acostibsintj+ck

Notice that, the vector r(t)=v(t)

Calculation of a(t) :

Substitute acostibsintj+ck for v(t) in equation (3),

a(t)=ddt(acostibsintj+ck)=ddt(acost)iddt(bsint)j+ddt(c)k=asintibcostj+0k=asintibcostj

Calculation of force vector F(t) :

Substitute (asintibcostj) for a(t) in equation (2),

F(t)=m(asintibcostj)=masintimbcostj

Calculation of work done on object:

Substitute

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