ENGI 1313 Mechanics I
Faculty of Engineering and Applied Science
Shawn Kenny, Ph.D., P.Eng.
September 2007
Revision 0
TUTORIAL PROBLEM SET #3 (WEEK 39: SEPTEMBER 23, 2007)
The problem set provides a representative sample of questions on relevant courses material and
concepts covered in the lectures. The tutorial problems sets are intended to develop good study habits
and become engaged in the learning process.
1: Problem 2-110 (page 73)
Determine the magnitude of the projected component of r1 along r2, and the projection of r2 along r1.
Given:
r1 = 9 m
r2 = 6 m
α = 60 deg
β = 45 deg
γ = 120 deg
φ = 30 deg
ε = 40 deg
Solution:
Write the vectors and unit vectors
⎛⎜ sin ( ε ) cos ( φ ) ⎞⎟
r1v = r1 ⎜ −sin ( ε ) sin ( φ ) ⎟
⎜ cos ( ε )
⎟
⎝
⎠
⎛ 5.01 ⎞
r1v = ⎜ −2.89 ⎟ m
⎜
⎟
⎝ 6.89 ⎠
⎛⎜ cos ( α ) ⎟⎞
r2v = r2 ⎜ cos ( β ) ⎟
⎜ cos ( γ ) ⎟
⎝
⎠
⎛ 3 ⎞
r2v = ⎜ 4.24 ⎟ m
⎜
⎟
⎝ −3 ⎠
u1 =
r1v
r1v
u2 =
r2v
r2v
⎛ 0.557 ⎞
u1 = ⎜ −0.321 ⎟
⎜
⎟
⎝ 0.766 ⎠
⎛ 0.5 ⎞
u2 = ⎜ 0.707 ⎟
⎜
⎟
⎝ −0.5 ⎠
The magnitude of the projection of r1 along r2.
r1v⋅ u2 = 2.99m
The magnitude of the projection of r2 along r1.
r2v⋅ u1 = 1.99m
Tutorial Problem Set #3
Page 1 of 6
ENGI 1313 Mechanics I
Faculty of Engineering and Applied Science
Shawn Kenny, Ph.D., P.Eng.
September 2007
Revision 0
2: Problem 2-116 (page 74)
Determine the components of F that act along rod AC and perpendicular to it. Point B is located a
distance 3 m along the rod from end C.
Given:
F = 600 N
c = 4m
a = 4m
d = 3m
b = 6m
e = 4m
f = 3m
Solution :
f
r=
2
2
d +e +a
2
Find the force vector and the unit vector uAC.
rBD
⎡ c + d( 1 − r) ⎤
⎢
⎥
= b − e( 1 − r)
⎢
⎥
⎣ −a r
⎦
Fv = F
rAC
rBD
rBD
⎛ −d ⎞
⎜ e ⎟
=
⎜ ⎟
⎝ −a ⎠
⎛ 5.5944 ⎞
⎜
⎟
rBD = 3.8741 m
⎜
⎟
⎝ −1.8741 ⎠
⎛ 475.6 ⎞
⎜
⎟
Fv = 329.3 N
⎜
⎟
⎝ −159.3 ⎠
⎛ −3 ⎞
⎜ ⎟
rAC = 4 m
⎜ ⎟
⎝ −4 ⎠
uAC =
rAC
rAC
⎛ −0.5 ⎞
⎜
⎟
uAC = 0.6
⎜
⎟
⎝ −0.6 ⎠
Now find the component parallel to AC.
Fparallel = Fv⋅ uAC
Fparallel = 82.4 N
The perpendicular component is now found
Fperpendicular =
Tutorial Problem Set #3
Fv⋅ Fv − F parallel
2
Fperpendicular = 594.3 N
Page 2 of 6
ENGI 1313 Mechanics I
Faculty of Engineering and Applied Science
Shawn Kenny, Ph.D., P.Eng.
September 2007
Revision 0
3: Problem 2-124 (page 75)
Determine the angle θ between the two cables attached to the pipe.
Given:
F1 = 30 lb
β = 30 deg
F2 = 25 lb
γ = 60 deg
α = 30 deg
ε = 60 deg
Solution:
We first need to find the third
angle ( > 90 deg) that locates
force F2.
Initial Guesses:
φ = 120 deg
Given
cos ( ε ) + cos ( γ ) + cos ( φ ) = 1
2
2
2
φ = Find( φ )
φ = 135 deg
Find the unit vectors u1 and u2.
⎛⎜ cos ( α ) sin ( β ) ⎟⎞
u1 = ⎜ cos ( α ) cos ( β ) ⎟
⎜ −sin ( α ) ⎟
⎝
⎠
⎛ 0.433 ⎞
⎜
⎟
u1 = 0.75
⎜
⎟
⎝ −0.5 ⎠
⎛⎜ cos ( φ ) ⎟⎞
u2 = ⎜ cos ( ε ) ⎟
⎜ cos ( γ ) ⎟
⎝
⎠
⎛ −0.707 ⎞
⎜ 0.5 ⎟
u2 =
⎜
⎟
⎝ 0.5 ⎠
Find the angle using the dot product
Tutorial Problem Set #3
θ = acos ( u1 ⋅ u2 )
θ = 100.4 deg
Page 3 of 6
ENGI 1313 Mechanics I
Faculty of Engineering and Applied Science
Shawn Kenny, Ph.D., P.Eng.
4: Problem 3-2 (page 92)
Determine the magnitude and direction
September 2007
Revision 0
of F so that the particle is in equilibrium.
Units Used:
3
kN = 10 N
Given:
F1 = 7kN
F2 = 3kN
c = 4
d = 3
Solution:
F = 1kN
The initial guesses:
θ = 30deg
Given
Equations of equilibrium :
+
Σ Fx = 0;
→
+
↑Σ F y = 0 ;
⎛ −d ⎞ F + F cos ( θ ) =
⎜ 2 2⎟ 1
⎝ c +d ⎠
0
c
⎛
⎞
⎜ 2 2 ⎟ F1 − F2 − F sin ( θ ) =
⎝ c +d ⎠
0
⎛F⎞
⎜ ⎟ = Find( F , θ )
⎝θ ⎠
F = 4.94kN
θ = 31.8deg
Tutorial Problem Set #3
Page 4 of 6
ENGI 1313 Mechanics I
Faculty of Engineering and Applied Science
Shawn Kenny, Ph.D., P.Eng.
September 2007
Revision 0
5: Problem 3-9 (page 94)
Cords AB and AC can each sustain a maximum tension T. If the drum has weight W, determine the
smallest angle at which they can be attached to the drum.
Given:
T = 800 lb
W = 900 lb
Solution:
+
↑ Σ Fy = 0;
W − 2T sin ( θ ) = 0
θ = asin⎛⎜
⎝
W⎞
⎟
2T ⎠
θ = 34.2 deg
Tutorial Problem Set #3
Page 5 of 6
ENGI 1313 Mechanics I
Faculty of Engineering and Applied Science
Shawn Kenny, Ph.D., P.Eng.
September 2007
Revision 0
6: Problem 2-129
Determine the angle θ between pipe segments BA and BC.
Given:
F = 100 lb
a = 3 ft
b = 8 ft
c = 6 ft
d = 4 ft
e = 2 ft
Solution:
rBC
⎛ c ⎞
⎜ ⎟
= d
⎜ ⎟
⎝ −e ⎠
rBA
Tutorial Problem Set #3
⎛ −a ⎞
⎜ 0 ⎟
=
⎜ ⎟
⎝ 0 ⎠
⎛ rBC ⋅rBA ⎞
⎟
⎝ rBC rBA ⎠
θ = acos ⎜
θ = 143.30deg
Page 6 of 6