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Physics 170

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tePROBLEM 2-44 (page 41, 13th edition) • The magnitude of the resultant force acting on the bracket ! is 400 N. Determine the magnitude of F1 . Take ! = 30 o . Disregard the u-axis.

2-44 (page 41) 13th edition COMPLETION OF PROBLEM • Determine F1 so that FR = 400 N :

! ! FR = ! F

• Cartesian Vector Method (suppressing units)

! ! ! FR = FRx i + FRy j

# 3& FRx = ! Fx = " 650 % ( + F1 cos30! + 500cos45! $ 5' " 4% FRy = ! Fy = 650 $ ' + F1 sin30! ( 500sin45! # 5&
2 2 FR = FRx + FRy

400 = (!390+F1 cos 30 o +500 cos 45 o )2 +(520+F1 sin 30 o !500 sin 45 o )2
• Solution Solve using Solver on a TI-83 Plus calculator:

F1 = 314 N
The second solution F1 = ! 417 N is rejected since F1 must be positive.

PROBLEM 2-51 (page 41, 13th edition) PROBLEM 2-40 (page 40, 12th edition) • Determine the magnitude and direction measured counterclockwise from the positive x-axis of the resultant force of the three forces acting on the ring A. Take F1 = 500 N and ! = 20o.

2-51 (page 41, 13th edition)

2-40 (page 40, 12th edition)

COMPLETION OF PROBLEM

! ! • Determine the magnitude and direction of FR = ! F .
Cartesian Vector Method (suppressing units):

! ! ! ! ! FR = FRx i + FRy j = FR (cos! i + sin! j )
4 FRx = ! Fx = 400cos30! + 500cos70! " 600( ) = 37.42 = A 5 3 FRy = ! Fy = 400sin30! + 500sin70! + 600( ) = 1030 = B 5
• Answers
2 2 FR = FRx + FRy = A 2 + B 2 = 1.03 kN

! = tan"1(FRy / FRx ) = tan"1(B / A) = 87.9!

PROBLEM 2-53 (page 42, 13th edition) PROBLEM 2-57 (page 42, 12th edition)

! • Determine the magnitude of force F so that the resultant of the three forces is a small as possible. • Determine the magnitude of the resultant force.

2-53 (page 42) 13th edition

2-57 (page 42) 12th edition

COMPLETION OF PROBLEM

! ! • Determine F so that FR is a small as possible: FR = ! F .
Cartesian Vector Method (suppressing units): ! ! ! FR = FRx i + FRy j

FRx = ! Fx = 8 " F cos45! "14cos30!

FRy = ! Fy = " Fsin45! +14sin30!
2 2 FR = FRx + FRy

FR (F) = (8!F cos 45! !14 cos 30! )2 +(!F sin 45! +14 sin 30! )2
We can determine the minimum of FR (F) using calculus. Alternatively, since we want numerical answers, we determine the minimum of FR (F) using the minimum function on a graphing calculator. This yields the minimum point:

(F, FR ) = (2.03, 7.87) kN

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