VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
11th Edition
ISBN: 9781259633133
Author: BEER
Publisher: MCG
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Chapter 13.2, Problem 13.83P

(a)

To determine

The work done from D to O by the force P (UOD).

(a)

Expert Solution
Check Mark

Answer to Problem 13.83P

The work done from D to O by the force P (UOD) is 3a_.

Explanation of Solution

Given information:

The potential function associated with force P is V(x,y,z)=(x2+y2+z2)1/2.

Calculation:

Write the potential function associated with force P:

V(x,y,z)=(x2+y2+z2)1/2 (1)

Calculate the force P in ‘x’ component using the relation:

Partial differentiate the equation (1) with respect to ‘x’.

Px=Vx

Substitute (x2+y2+z2)1/2 for V.

Px=((x2+y2+z2)1/2)x=12x2+y2+z2×(2x)=x(x2+y2+z2)1/2

Calculate the force P in ‘y’ component using the relation:

Partial differentiate the equation (1) with respect to ‘y’.

Py=Vy

Substitute (x2+y2+z2)1/2 for V.

Py=((x2+y2+z2)1/2)y=12x2+y2+z2×(2y)=y(x2+y2+z2)1/2

Calculate the force P in ‘z’ component using the relation:

Partial differentiate the equation (1) with respect to ‘z’.

Pz=Vz

Substitute (x2+y2+z2)1/2 for V.

Pz=((x2+y2+z2)1/2)x=12x2+y2+z2×(2z)=z(x2+y2+z2)1/2

Calculate the force (P) using the relation:

P=Px+Py+Pz

Substitute x(x2+y2+z2)1/2 for Px, y(x2+y2+z2)1/2 for Py and z(x2+y2+z2)1/2 for Pz.

P=x(x2+y2+z2)1/2+y(x2+y2+z2)1/2+z(x2+y2+z2)1/2=xi+yj+zk(x2+y2+z2)1/2

Write the expression for the position vector of the cube (r) as follows:

r=xi+yj+zk

Differentiate the above equation.

dr=dxi+dyj+dzk

Calculate the work done from D to O by the force P (UOD) using the relation:

UOD=0aPdr

Substitute xi+yj+zk(x2+y2+z2)1/2 for P and dxi+dyj+dzk for dr.

UOD=0axi+yj+zk(x2+y2+z2)1/2(dxi+dyj+dzk)

Over the diagonal, x=y=z.

Substitute x for y and x for z.

UOD=0axi+xj+xk(x2+x2+x2)1/2(dxi+dxj+dxk)=0a3x(3x2)1/2dx=0a3dx

=3[x]0a=3[a0]=3a

Therefore, the work done from D to O by the force P (UOD) is 3a_.

(b)

To determine

Verify the work done by a conservative force around the closed path OABDO (UOABDO) is zero.

(b)

Expert Solution
Check Mark

Explanation of Solution

Given information:

The potential function associated with force P is V(x,y,z)=(x2+y2+z2)1/2.

Calculation:

Write the coordinates of O-A using the relation:

The line O-A is lies on the z-axis. The term x and y are zero.

Calculate the work done on O-A (UOA) using the relation:

UOA=0zPzdz

Since the force Px and Py are perpendicular to the O-A and do not work.

Substitute z(x2+y2+z2)1/2 for Pz.

UOA=0az(x2+y2+z2)dz

Substitute 0 for x and 0 for y.

UOA=0az((0)2+(0)2+z2)dz=0az(z2)dz=0azzdz

=0adz=[z]0a=a0=a

The co-ordinates A-B is lies on the x-axis. Then, the term y is zero and z tends to ‘a’.

Calculate the work done on A-B (UAB) using the relation:

UAB=0aPxdz

Since the force Pz and Py are perpendicular to the A-B and do not work.

Substitute x(x2+y2+z2)1/2 for Px.

UAB=0ax(x2+y2+z2)dx

Substitute a for z and 0 for y.

UAB=0ax((x)2+(0)2+(a)2)dx=0ax(x2+a2)dx=0a12x2+a2dx

Simplify the Equation:

UAB=12[(x2+a2)12+112+1]0a=[12(x2+a2)12×21]0a=[(x2+a2)]0a

=(a2+a2)(02+a2)=2a2a2=2a2a2=2aa=a(21)

The co-ordinates B-D is lies on the y-axis. Then, the term x is zero and z tends to ‘a’.

Calculate the work done on B-D (UBD) using the relation:

UBD=0aPydz

Since the force Pz and Px are perpendicular to the B-D and do not work.

Substitute y(x2+y2+z2)1/2 for Py.

UBD=0ay(x2+y2+z2)dy

Substitute 2a for z and 0 for x.

UBD=0ay((0)2+(y)2+(2a)2)dx=0ay(y2+2a2)dx=0a12y2+2a2dx

Simplify the Equation,

UBD=12[(y2+2a2)12+112+1]0a=[12(y2+2a2)12×21]0a=[(y2+2a2)]0a

=(a2+2a2)(02+2a2)=3a22a2=3a22a2=3a2a=a(32)

Calculate the work done by P (UOABD) using the relation:

UOABD=UOA+UAB+UBD

Substitute a for UOA, a(21) for UAB, and a(32) for UBD.

UOABD=a+[a(21)]+[a(32)]=a+2aa+3a2a=a3

The work done from D to O along the diagonal is negative of the work done from O to D.

UOD=3aUOD=3a

Calculate the work done by a conservative force around the closed path OABDO (UOABDO) using the relation:

UOABDO=UOABD+UOD

Substitute a3 for UOABD and 3a for UOD.

UOABDO=3a3a=0

Therefore, the work done by a conservative force around the closed path OABDO (UOABDO) is zero.

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Chapter 13 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

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