Download Lecture 13 - University of Oklahoma

Physics 2514
Lecture 13
P. Gutierrez
Department of Physics & Astronomy
University of Oklahoma
Physics 2514 – p. 1/18
Goals
We will discuss some examples that involve equilibrium.
We then move on to a discussion of friction.
Physics 2514 – p. 2/18
Newton’s Laws of Motion
1)
An object that is at rest will remain at rest, or an object that
is moving will continue to move (in a straight line) with
constant velocity, if and only if the net force acting on the
object is zero.
2)
~ 1, F
~ 2, F
~ 3 , . . . will
An object of mass m subjected to forces F
undergo an acceleration ~a given by
~ net = m~a
F
where
~ net =
F
n
X
~i
F
i=1
3)
To every action there is always opposed an equal reaction;
or, the mutual actions of two bodies upon each other are
always equal and directed to contrary parts.
Physics 2514 – p. 3/18
Equilibrium
Let’s first consider problems of static equilibrium
Equilibrium corresponds to the case that the object has zero
~ net = 0
net force acting on it F
Two types of equilibrium conditions
Static equilibrium corresponds to the case when the
object is at rest
Dynamic equilibrium corresponds to the case when the
object moves at a constant velocity
~ net = 0.
Both correspond to F
Physics 2514 – p. 4/18
Solving Problems
To solve problems using Newton’s laws of motion, we will follow
the same problem solving procedure as for the case of
kinematics.
Rewrite the problem in a couple of sentences;
1. State the situation;
2. What variables are we solving for;
Draw a force diagram Free Body Diagram
State the known and unknown variables;
Solve the problem algebraically;
Substitute numbers into the problem.
Physics 2514 – p. 5/18
Example 1
Consider a lead block of mass 5 Kg held against a frictionless
wall with a force applied at an angle of 30◦ as shown in the
figure. What is the magnitude of the force required to keep the
block from sliding?
~
F
θ
PSfrag replacements
Physics 2514 – p. 6/18
Solution
Brief description A 5 Kg block is held against a frictionless wall
with a force acting at 30◦ relative to the horizontal. What is the
~ required to keep the block in static equilibrium?
magnitude of P
Sfrag replacements
Free body diagram
y
PSfrag replacements
~
P
~
P
~
n
θ
x
θ
m~
g
Physics 2514 – p. 7/18
Solution
Known
θ = 30◦
replacements
~ = −mg ĵ = −49PSfrag
w
ĵ N
y
~
P
~
n
θ
Unknown
~ , ~n
P
x
m~
g
Equations:
X
X
Fx = n − Px = n − P cos θ = 0


Fy = Py − mg = P sin θ − mg = 0
⇒
⇒
(
n = P cos θ
P = mg/ sin θ
)
P = mg/ sin(30) = 98 N
Physics 2514 – p. 8/18
Example 2
A 1000 Kg beam is supported by two ropes. Calculate the
tension in each rope.
y
PSfrag replacements
~1
T
~2
T
α
β
x
m~
g
Physics 2514 – p. 9/18
Example 3
A 1000 Kg beam is supported by two ropes. Calculate the
tension in each rope.
y
Known
~1
T
α = 20◦
PSfrag replacements
◦
β = 30
~ = −mg ĵ = −9800 ĵ N
w
Unknown
~ 1, T
~2
T
~2
T
α
β
x
m~
g
Physics 2514 – p. 10/18
Example
Known
α = 20◦
PSfrag replacements T
~1
β = 30◦
~ = −mg ĵ = −9800 ĵ N
w
y
~2
T
α β
Unknown
~ 1, T
~2
T
T2x − T1x = T1 sin α − T2 sin β = 0
T1y + T2y − mg = 0 = T1 cos α + T2 cos β − mg = 0
x
m~
g
)
⇒
(
T1 = 6396 N
T 2 = 4375 N
Physics 2514 – p. 11/18
Clicker
Joe and Bill are playing tug-o-war. Joe is pulling with a force of
200 N. Bill is simply hanging on, but skidding towards Joe at a
constant velocity. What is the magnitude of the force of friction
between Bill’s feet and the ground.
1.
2.
3.
4.
5.
200 N
400 N
0N
300 N
None of the above
Physics 2514 – p. 12/18
Friction
Next to gravity, friction is the most common force that we interact
with. It allows us to walk, slows us down, and prevents us from
moving.
There are 3 types of friction (all act to oppose motion)
Static fraction fs ≤ µs N when no motion occurs;
Kinetic friction fk = µk N , when object is moving;
Rolling friction fr = µr N , when objects rolls.
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Clicker
For which situation is the frictional force the largest.
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Solution
Physics 2514 – p. 15/18
Example 3
A 50 kg steel box is in the back of a dump truck. The truck’s bed,
also made of steel, is slowly tilted. At what angle will the file
cabinet begin to slide?
Brief description: A 50 kg steel box is on a steel incline plane.
What is the maximum angle the incline can have for the box to
remain in static equilibrium?
Physics 2514 – p. 16/18
Solution
Known
µs = 0.80, m = 50 kg
Unknown
angle θ of incline
Normal force n
Equations of motion:
mg sin θ − µs n = 0
− mg cos θ + n = 0
)
⇒
µs = tan θ
⇒
θ = tan−1 µs = 38.7◦
Physics 2514 – p. 17/18
Assignment
Read the sections on Newton’s second law and its applications:
Sections 5.2, 5.3, 5.6
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