Vector Mechanics for Engineers: Statics and Dynamics
Vector Mechanics for Engineers: Statics and Dynamics
12th Edition
ISBN: 9781259638091
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 11.5, Problem 11.179P

The three-dimensional motion of a particle is defined by the cylindrical coordinates R = A/(t + 1), θ = Bt, and z = Ct/(t + 1). Determine the magnitudes of the velocity and acceleration when (a) t = 0, (b) t = ∞.

(a)

Expert Solution
Check Mark
To determine

The magnitudes of the velocity (v0) and acceleration (a0) when time (t) is 0.

Answer to Problem 11.179P

The magnitudes of the velocity (v0) and acceleration (a0) when time (t) is 0 are A2+A2B2+C2_ and 4A2+A2B4+4C2_ respectively.

Explanation of Solution

Given Information:

The three dimensional motion of a particle is defined by the cylindrical coordinates (R) is A(t+1), (θ) is Bt and (z) is Ct(t+1).

Calculation:

The three dimensional motion of a particle is defined by the cylindrical coordinates (R):

R=A(t+1) (1)

The three dimensional motion of a particle is defined by the cylindrical coordinates (θ):

θ=Bt (2)

The three dimensional motion of a particle is defined by the cylindrical coordinates (z):

z=Ct(t+1) (3)

Differentiate the equation (1) with respective to time (t),

R˙=A(t+1)2 (4)

Differentiate the equation (4) with respective to time (t),

R¨=2A(t+1)3 (5)

Differentiate the equation (2) with respective to time (t),

θ˙=B (6)

Differentiate the equation (5) with respective to time (t),

θ¨=0

Differentiate the equation (3) with respective to time (t),

z˙=C(t+1)Ct(t+1)2=C(t+1)2 (7)

Differentiate the equation (6) with respective to time (t),

z¨=2C(t+1)3 (8)

Calculate the value (R):

Substitute 0 for t in equation (1).

R=A(0+1)=A

Calculate the value (R˙):

Substitute 0 for t in equation (4).

R˙=A(0+1)2=A

Calculate the value (R¨):

Substitute 0 for t in equation (4).

R¨=2A((0)+1)3=2A

Calculate the value (θ).

Substitute 0 for t in equation (2).

θ=B(0)=0

Calculate the value (z):

Substitute 0 for t in equation (3).

z=C(0)((0)+1)=0

Calculate the value (z˙):

Substitute 0 for t in equation (7).

z˙=C((0)+1)2=C

Calculate the value (z¨):

Substitute 0 for t in equation (8).

z¨=2C((0)+1)3=2C

Write the expression for radial component of velocity (vr) as below:

vr=R˙

Substitute A for R˙.

vr=A

Write the expression for transverse component of velocity (vθ):

vθ=Rθ˙

Substitute A for R and B for θ˙.

vθ=AB

Write the expression for axial component of velocity (vz):

vz=z˙

Substitute C for z˙.

vz=C

Calculate the magnitude of the velocity (v) using the relation:

v02=vr2+vθ2+vz2

Substitute A for vr, AB for vθ and C for vz.

v02=A2+(AB)2+C2v0=A2+A2B2+C2

Write the expression for radial component of acceleration (ar):

ar=R¨Rθ˙2

Substitute 2A for R¨, A for R and B for θ˙.

ar=2AA(B)2=2AAB2

Write the expression for transverse component of acceleration (aθ):

aθ=Rθ¨+2R˙θ˙

Substitute A for R, B for θ˙, A for R˙, and 0 for θ¨.

aθ=A(0)+2(A)(B)=2AB

Write the expression for axial component of acceleration (az):

az=z¨

Substitute 2C for z¨.

az=2C

Calculate the magnitude of the acceleration (a) using the relation:

a02=ar2+aθ2+az2

Substitute 2AAB2 for ar, 2AB for aθ and 2C for az.

a02=(2AAB2)2+(2AB)2+(2C)2a02=4A2+A2B44A2B2+4A2B2+4C2a02=4A2+A2B4+4C2a0=4A2+A2B4+4C2

Therefore, the magnitudes of the velocity (v) and acceleration (a) when time (t) is 0 are A2+A2B2+C2_ and 4A2+A2B4+4C2_ respectively.

(b)

Expert Solution
Check Mark
To determine

The magnitudes of the velocity (v) and acceleration (a) when time (t) is .

Answer to Problem 11.179P

The magnitudes of the velocity (v) and acceleration (a) when time (t) is are 0_ and 0_ respectively.

Explanation of Solution

Given Information:

The three dimensional motion of a particle is defined by the cylindrical coordinates (R) is A(t+1), (θ) is Bt and (z) is Ct(t+1).

Calculation:

Calculate the value (R):

Substitute for t in equation (1).

R=A(+1)=0

Calculate the value (R˙):

Substitute for t in equation (4).

R˙=A(+1)2=0

Calculate the value (R¨):

Substitute for t in equation (4).

R¨=2A(()+1)3=0

Calculate the value (θ).

Substitute for t in equation (2).

θ=B()=

Calculate the value (z):

Substitute for t in equation (3).

z=C()(()+1)=C

Calculate the value (z˙):

Substitute for t in equation (7).

z˙=C(()+1)2=0

Calculate the value (z¨) :

Substitute for t in equation (8).

z¨=2C(()+1)3=0

Write the expression for radial component of velocity (vr):

vr=R˙

Substitute 0 for R˙.

vr=0

Write the expression for transverse component of velocity (vθ):

vθ=Rθ˙

Substitute 0 for R and B for θ˙.

vθ=(0)B=0

Write the expression for axial component of velocity (vz):

vz=z˙

Substitute 0 for z˙.

vz=C

Calculate the magnitude of the velocity (v) using the relation:

v2=vr2+vθ2+vz2

Substitute 0 for vr, 0 for vθ and 0 for vz.

v2=(0)2+(0)2+(0)2v=0

Write the expression for radial component of acceleration (ar):

ar=R¨Rθ˙2

Substitute 0 for R¨, 0 for R and B for θ˙.

ar=00(B)2=0

Write the expression for transverse component of acceleration (aθ):

aθ=Rθ¨+2R˙θ˙

Substitute 0 for R, B for θ˙, 0 for R˙ and 0 for θ¨.

aθ=0+2(0)(B)=0

Write the expression for axial component of acceleration (az):

az=z¨

Substitute 0 for z¨.

az=0

Calculate the magnitude of the acceleration (a) using the relation:

a2=ar2+aθ2+az2

Substitute 0 for ar, 0 for aθ and 0 for az.

a2=(0)2+(0)2+(0)2a=0

Therefore, the magnitudes of the velocity (v) and acceleration (a) when time (t) is are 0_ and 0_ respectively.

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Chapter 11 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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