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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Question
Chapter 10.1, Problem 10.5P

(a)

To determine

Find the force in the spring and the vertical motion of point G when a vertical load of 120N force is applied at point C.

(a)

Expert Solution
Check Mark

Answer to Problem 10.5P

The force in the spring is 60N(C)_.

The vertical motion of point G is 8mm()_.

Explanation of Solution

Given information:

The spring constant is k=15kN/m.

Calculation:

Show the free-body diagram of the spring assembly as in Figure 1.

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 10.1, Problem 10.5P

Write the relation of the deflections at point G, H, F, E, D with C as follows;

yG=4yC;δyG=4δyC

yH=4yC;δyH=4δyCyF=3yC;δyF=3δyCyD=2yC;δyD=2δyCyE=2yC;δyE=2δyC

The deflection Δ of the spring is;

Δ=yFyC=3yCyC=2yC

Assume the spring force Q is in tension.

Find the force in the spring Q using the relation.

Q=+kΔ

Here, the spring constant is k.

Substitute 15kN/m for k and 2yC for Δ.

Q=15(2yC)=30yC (1)

Use the virtual work principle:

δU=0CδyC+QδyCQδyFFδyFHδyHEδyE=0

Here, E=0;F=0;H=0;C=120N

Substitute 120 N for C, 3δyC for δyF, 0 for F, 0 for H, 4δyC for δyH, 0 for E, and 2δyC for δyE.

120δyC+QδyCQ(3δyC)(0)(3δyC)(0)(4δyC)(0)(2δyC)=0120+Q3Q000=02Q=120Q=60N

Q=60N(C)

The spring force Q is in compression. The assumption is incorrect.

Therefore, the force in the spring is 60N(C)_.

Substitute –60 N for Q in Equation (1).

60=30yCyC=2mm

Find the vertical motion (yG) of point G using the relation.

yG=4yC

Substitute –2 mm for yC.

yG=4(2)=8mm=8mm()

Therefore, the vertical motion of point G is 8mm()_.

(b)

To determine

Find the force in the spring and the vertical motion of point G when a vertical load of 120N force is applied at point C and H.

(b)

Expert Solution
Check Mark

Answer to Problem 10.5P

The force in the spring is 300N(C)_.

The vertical motion of point G is 40mm()_.

Explanation of Solution

Given information:

The spring constant is k=15kN/m.

Calculation:

Use the virtual work principle:

δU=0CδyC+QδyCQδyFFδyFHδyHEδyE=0

Here, E=0;F=0;H=120N;C=120N

Substitute 120 N for C, 3δyC for δyF, 0 for F, 120 N for H, 4δyC for δyH, 0 for E, and 2δyC for δyE.

120δyC+QδyCQ(3δyC)(0)(3δyC)(120)(4δyC)(0)(2δyC)=0120+Q3Q04800=02Q=600Q=300N

Q=300N(C)

The spring force Q is in compression. The assumption is incorrect.

Therefore, the force in the spring is 300N(C)_.

Substitute –300 N for Q in Equation (1).

300=30yCyC=10mm

Find the vertical motion (yG) of point G using the relation.

yG=4yC

Substitute –10 mm for yC.

yG=4(10)=40mm=40mm()

Therefore, the vertical motion of point G is 40mm()_.

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Chapter 10 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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