To determine

To find:

The work done in moving the particle from x=1 to x=2 and interpret the answer by considering the work done from x=1 to x=1.5 and from x=1.5 to x=2

Answer

_0 J

Interpretation: Work done from x=1 to x=1.5 is positive; it means kinetic energy of particle is increasing. Work done from x=1.5 to x=2 is negative; it means kinetic energy is decreasing.

Explanation

1) Concept:

Use the formula to calculate the work done.

2) Formula:

The work done in moving the object from a to b is given by

W=∫abf(x)dx

3) Given:

fx=cos(πx3)

4) Calculation:

The work done in moving the particle from x=1 to x=2 is given by using the formula

W=∫12cos(πx3)dx

Simplify,

=sin⁡πx3π312

=3πsin⁡πx312

=3πsin⁡π(2)3-sin⁡π(1)3

=3π32-32

=0 N.m

=0 J

Interpretation:

The work done in moving the particle from x=1 to x=1.5 is given by using the formula

W=∫11.5cos(πx3)dx

Simplify,

=sin⁡πx3π311.5

=3πsin⁡πx311.5

=3πsin⁡π(1.5)3-sin⁡π(1)3

=3πsin⁡π2-sin⁡π(1)3

=3π1-32

=32π2-3 N·m

=32π2-3  J

Work done is positive which means that kinetic energy is increasing.

Now,

The work done in moving the particle from x=1.5 to x=2 is given by using the formula

W=∫1.52cosπx3dx

Simplify,

=sin⁡πx3π31.52

=3πsin⁡πx31.52

=3πsin⁡π(2)3-sin⁡π(1.5)3

=3πsin⁡2π3-sin⁡π2

=3π32-1

=32π(3-2) N.m

=-32π2-3  J

The force opposes the motion of the particle, the kinetic energy is decreasing, and hence, the work done is negative.

W=∫12cosπx3dx=∫11.5cosπx3dx+∫1.52cosπx3dx

=32π2-3-32π2-3

_=0 J

Therefore, the work done is W=0 J

Conclusion:

The work done is W=0 J

Interpretation: Work done from x=1 to x=1.5 is positive; it means kinetic energy is increasing. Work done from x=1.5 to x=2 is negative; it means kinetic energy is decreasing.