engineering mechanice lecture 05
TRANSCRIPT
Lecture 03
BY
Engr Muhammad Usman
Mechanical Engineering
Department
CECOS University
COURSE OUTLINE
• Fundamental concept and principle of
mechanics, important vector quantity
• Force system i.e concurrent
• Non concurrent and parallel force system
• Resultant of forces
• Moment and couple
• Equilibrium of forces (law & type) concept of
free body diagram
Resultants
Vectors in opposite directions:
6 m s-1 10 m s-1 = 4 m s-1
6 N 10 N = 4 N
Resultant of Two Vectors
Vectors in the same direction:
6 N 4 N = 10 N
6 m
= 10 m
4 m
The resultant is the sum or the combined effect of two vector quantities
Graphical Vector Addition
2. Parallelogram method
8 N
4 N
3 N
3 forces act on an object at the same
time. Fnet is not 15 N because these
forces aren’t working together. But
they’re not completely opposing each
either. So how do find Fnet ? The
answer is to add the vectors ... not their
magnitudes, but the vectors themselves.
There are two basic ways to add
vectors: 1. Tip to tail method
Head to Tail Methodin-line examples
Place the tail of one vector at the
tip of the other. The vector sum
(also called the resultant) is shown
in red. It starts where the black
vector began and goes to the tip of
the blue one. In these cases, the
vector sum represents the net force.
You can only add or subtract
magnitudes when the vectors are
in-line!
16 N
20 N
4 N
20 N
16 N
12 N9 N
9 N
12 N
21 N
Head to Tail – 2 Vectors
5 m
2 m
To add the red and blue displacement vectors first note:
• Vectors can only be added if they are of the same
quantity—in this case, displacement.
• The magnitude of the resultant must be less than
7 m (5 + 2 = 7) and greater than 3 m (5 - 2 = 3).
Interpretation: Walking 5 m in the direction
of the blue vector and then 2 m in the
direction of the black one is equivalent to
walking in the direction of the red vector.
The distance walked this way is the red
vector’s magnitude.
The Parallelogram Law When two vectors are joined
tail to tail
Complete the parallelogram
The resultant is found by drawing the diagonal
When two vectors are joined head to tail
Draw the resultant vector by completing the triangle
Parallelogram Method1. Create parallelogram using
“copies” of the two vectors
2. Draw Resultant vector from
tail of first vector to tip of last
vector.
3. You cannot add more than 2
vectors at a time with this
method, so…how do you add 3
or 4 vectors with this
method???
Note: Opposite sides of a
parallelogram are congruent.
Comparison of Methods
Tip to tail method
Parallelogram method
The resultant has
the same
magnitude and
direction
regardless of the
method used.
Law of Cosines
a b
c
R
A B
Law of Cosines: R2 = A2 + B2 - 2 AB cos Θ
This side is always opposite this angle.
These two sides are repeated.
It matters not which side is called A, B, and R, so long as the two rules above are
followed. This law is like the Pythagorean theorem with a built in correction term of -2
AB cos Θ . This term allows us to work with non-right triangles. Note if Θ = 90, this
term drops out (cos 90 = 0), and we have the normal Pythagorean theorem.
Θ
• Here we see a table being
pulled by a force of 50 N at
a 30º angle to the horizontal
• When resolved we see that
this is the same as pulling
the table up with a force of
25 N and pulling it
horizontally with a force of
43.3 N
Practical Applications
y=25 N
x=43.3 N30º
We can see that it would be more efficient to pull the table with a horizontal force of 50 N
Determine the resultant of the three forces below.
25o
45o
350 N
800 N600 N
60o
y
x
Solution F x = 350 cos 25o + 800 cos 70o - 600 cos 60o
= 317.2 + 273.6 - 300 = 290.8 N
F y = 350 sin 25o + 800 sin 70o + 600 sin 60o
= 147.9 + 751 + 519.6 = 1419.3 N
i.e. F = 290.8 N i + 1419.3 N j
Resultant, F
F N
290 8 1419 3 1449
1419 3
290 878 4
2 2
1 0
. .
tan.
..
F = 1449 N 78.4o 25o
45o
350 N
800 N600 N
60o
y
x
Determine the resultant of the four forces and one
couple which act on the plate shown.