Transcript Chapter-5-Forces

Chapter 5 – Force and Motion I
I.
Newton’s first law.
II. Newton’s second law.
III. Particular forces:
- Gravitational
- Weight
- Normal
- Friction
- Tension
IV. Newton’s third law.
Force
• Forces are what cause any change in the
velocity of an object
– A force is that which causes an acceleration
• The net force is the vector sum of all the forces
acting on an object
– Also called total force, resultant force, or
unbalanced force
Zero Net Force
• When the net force is equal to zero:
– The acceleration is equal to zero
– The velocity is constant
• Equilibrium occurs when the net force is equal
to zero
– The object, if at rest, will remain at rest
– If the object is moving, it will continue to move
at a constant velocity
Classes of Forces
• Contact forces
involve physical
contact between
two objects
• Field forces act
through empty
space
– No physical
contact is required
Fundamental Forces
• Gravitational force
– Between two objects
• Electromagnetic forces
– Between two charges
• Nuclear (strong) force
– Between subatomic particles
• Weak forces
– Arise in certain radioactive decay processes
• A spring can be used
to calibrate the
magnitude of a force
• Forces are vectors,
so you must use the
rules for vector
addition to find the net
force acting on an
object
Newton mechanics laws cannot be applied when:
1) The speed of the interacting bodies are a fraction
of the speed of light Einstein’s special theory of
relativity.
2) The interacting bodies are on the scale of the
atomic structure  Quantum mechanics
I. Newton’s first law: If no net force acts on a body, then
the body’s velocity cannot change;
the body cannot accelerate 
v = constant in magnitude and
direction.
Principle of superposition: when two or more forces act
on a body, the net force can be obtained by adding the
individual forces as vectors.
Newton’s First Law
• If an object does not interact with other
objects, it is possible to identify a
reference frame in which the object has
zero acceleration
– This is also called the law of inertia
– It defines a special set of reference frames
called inertial frames,
• We call this an inertial frame of reference
Inertial reference frame: where Newton’s laws hold.
Inertial Frames
• Any reference frame that moves with constant
velocity relative to an inertial frame is itself an
inertial frame
• A reference frame that moves with constant
velocity relative to the distant stars is the best
approximation of an inertial frame
– We can consider the Earth to be such an inertial
frame although it has a small centripetal acceleration
associated with its motion
Newton’s First Law – Alternative Statement
• In the absence of external forces, when viewed
from an inertial reference frame, an object at rest
remains at rest and an object in motion
continues in motion with a constant velocity
– Newton’s First Law describes what happens in the
absence of a force
– Also tells us that when no force acts on an object, the
acceleration of the object is zero
Inertia and Mass
• The tendency of an object to resist any
attempt to change its velocity is called
inertia
• Mass is that property of an object that
specifies how much resistance an object
exhibits to changes in its velocity
Mass and Weight
The weight of an object is the force of gravity on the
object and may be defined as the mass times the
acceleration of gravity, W = mg. Since the weight is a force,
its SI unit is the newton. Density is mass/volume.
Mass and Weight
The mass of an object is a fundamental property of
the object; a numerical measure of its inertia; a fundamental
measure of the amount of matter in the object. Definitions of
mass often seem circular because it is such a fundamental
quantity that it is hard to define in terms of something else.
All mechanical quantities can be defined in terms of
mass, length, and time. The usual symbol for mass is m and
its SI unit is the kilogram. While the mass is normally
considered to be an unchanging property of an object, at
speeds approaching the speed of light one must consider the
increase in the relativistic mass.
• Mass is an inherent property of an object
• Mass is independent of the object’s
surroundings
• Mass is independent of the method used
to measure it
• Mass is a scalar quantity
• The SI unit of mass is kg
Mass vs. Weight
• Mass and weight are two different
quantities
• Weight is equal to the magnitude of the
gravitational force exerted on the object
– Weight will vary with location
Newton’s Second Law
• When viewed from an inertial frame, the
acceleration of an object is directly
proportional to the net force acting on it
and inversely proportional to its mass
– Force is the cause of change in motion, as
measured by the acceleration
• Algebraically, SF = m a
II. Newton’s second law: The net force on a body is
equal to the product of the body’s mass and its
acceleration.


Fnet  ma
Fnet , x  max , Fnet , y  ma y , Fnet , z  maz
The acceleration component along a given axis is caused
only by the sum of the force components along the same
axis, and not by force components along any other axis.
Units of Force
System: collection of bodies.
External force: any force on the bodies inside the system.
III. Particular forces:
Gravitational: pull directed towards a second body, normally
the Earth 


Fg  mg
Weight: magnitude of the upward force needed to balance
the gravitational force on the body due to an astronomical
body 
W  mg
Normal force: perpendicular force on a body from a surface
against which the body presses.
N  mg
Frictional force: force on a
body when the body
attempts to slide along a
surface. It is parallel to the
surface and opposite to the
motion.
Tension: pull on a body directed away from the body
along a massless cord.
Newton’s Third Law
• If two objects interact, the force F12
exerted by object 1 on object 2 is equal in
magnitude and opposite in direction to the
force F21 exerted by object 2 on object 1
• F12 = - F21
– Note on notation: FAB is the force exerted by
A on B
Newton’s Third Law, Alternative
Statements
• Forces always occur in pairs
• A single isolated force cannot exist
• The action force is equal in magnitude to the
reaction force and opposite in direction
– One of the forces is the action force, the other is the
reaction force
– It doesn’t matter which is considered the action and
which the reaction
– The action and reaction forces must act on different
objects and be of the same type
Action-Reaction Examples
• The force F12 exerted
by object 1 on object
2 is equal in
magnitude and
opposite in direction
to F21 exerted by
object 2 on object 1
• F12 = - F21
Newton’s third law: When two bodies interact,
the forces on the bodies from each other are always
equal in magnitude and opposite in direction.


FBC   FCB
Q. A body suspended by a rope has a weigh W of 75N. Is T
equal to, greater than, or less than 75N when the body is
moving downward at (a) increasing speed and (b)
decreasing speed?
Fg
5. There are two forces on the 2 kg box in the overhead view of
the figure below but only one is shown. The figure also shows
the acceleration of the box. Find the second force (a) in unitvector notation and as (b) magnitude and (c) direction.
F2
Rules to solve Dynamic problems
- Select a reference system.
- Make a drawing of the particle system.
- Isolate the particles within the system.
- Draw the forces that act on each of the isolated bodies.
- Find the components of the forces present.
- Apply Newton’s second law (F=ma) to each isolated particle.
23. An electron with a speed of 1.2x107m/s moves horizontally into a region
where a constant vertical force of 4.5x10-16N acts on it. The mass of the
electron is m=9.11x10-31kg. Determine the vertical distance the electron is
deflected during the time it has moved 30 mm horizontally.
F
Fg
dy
v0
dx=0.03m
13. In the figure below, mblock=8.5kg and θ=30º, and the surface
of incline is frictionless. Find (a) Tension in the cord. (b) Normal
force acting on the block. (c) If the cord is cut, find the
magnitude of the block’s acceleration.
N
T
Fg
Friction
• Related to microscopic interactions of surfaces
• Friction is related to force holding surfaces
together
• Frictional forces are different depending on
whether surfaces are static or moving
Kinetic Friction
• When a body slides over a surface there is a
force exerted by the surface on the body in the
opposite direction to the motion of the body
• This force is called kinetic friction
• The magnitude of the force depends on the
nature of the two touching surfaces
• For a given pair of surfaces, the magnitude of
the kinetic frictional force is proportional to the
normal force exerted by the surface on the body.
Kinetic Friction
The kinetic frictional
force can be written as
F fr  m k FN
Where mk is a
constant called the
coefficient of kinetic
friction
The value of mk
depends on the
surfaces involved
+
+
FN = mg
Fp
m=
Ffr = mFN
20.0 kg
Fg = -mg
Static Friction
A frictional force can arise even if the body
remains stationary
– A block on the floor - no frictional force
FN = -mg
m
Fg = mg
Static Friction
– If someone pushes the desk (but it does not
move) then a static frictional force is exerted
by the floor on the desk to balance the force
of the person on the desk
FN = -mg
Fp
m
Ffr = mFN
Fg = mg
Static Friction
– If the person pushes with a greater force and
still does not manage to move the block, the
static frictional force still balances it
FN = -mg
Fp
m
Ffr = mFN
Fg = mg
Static Friction
– If the person pushes hard enough, the block
will move.
• The maximum force of static friction has been
exceeded
FN = -mg
Fp
m
Ffr = mFN
Fg = mg
Static Friction
• The maximum force of static friction is
given by
F fr (max)  m s FN
• Static friction can take any value from zero
to msFN
– In other words
F fr  m s FN
Forces on an incline
• Often when solving
problems involving
Newton’s laws we
will need to deal
with resolving
acceleration due to
gravity on an
inclined surface
Forces on an incline
y
What normal force
does the surface exert?
mgsinq
mgcosq
W = mg
q
x
Forces on an incline
Forces on an incline
F
F
x
 mg sin q
y
 N  mg cos q
Forces on an incline
F
F
x
 mg sin q  ma
y
 N  mg cos q  0
Equilibrium
Forces on an incline
• If the car is just
stationary on the
incline what is the
(max) coefficient of
static friction?
 Fx  mg sin q  ms N  ma  0
F
y
 N  mg cos q  0
mg sin q  m s N  m s mg cos q
sin q
ms 
 tan q
cos q
‘Equilibrium’ problems (SF = 0)
‘Equilibrium’ problems (SF = 0)
0
0
F

T
cos
53

T
cos
37
0
 x 2
1
0
0
F

T
sin
37

T
sin
53
 T3  0
 y 1
2
Connected Object problems
• One problem often posed is how to work
out acceleration of a system of masses
connected via strings and/or pulleys
– for example - Two blocks are fastened to the
ceiling of an elevator.
Connected masses
• Two 10 kg blocks are
strung from an elevator
roof, which is accelerating
up at 2 m/s2.
• Find T1 and T2
T1
m1=10kg
T2
m2=10kg
a
2 m/s2
Connected masses
• What is the acceleration of the system
below, if T is 1000 N?
• What is T*?
T*
m2=10kg
m1=1000kg
a
T
Connected masses
• θ is 250, what are T* and a now?
a
Connected masses
• Add in friction: μ = 0.4
• What are T* and a now?
T*
a
Motion over pulleys
a
• We know that the
magnitude and
direction of the
acceleration should
be the same for the
whole system
T
a
10kg
θ = 300
μ = 0.4
55. The figure below gives as a function of time t, the force component
Fx that acts on a 3kg ice block, which can move only along the x axis.
At t=0, the block is moving in the positive direction of the axis, with a
speed of 3m/s. What are (a) its speed and (b) direction of travel at
t=11s?
A block of mass 3.00 kg is pushed up against a wall by a
force P that makes a 50.0 angle with the horizontal as shown
in Figure. The coefficient of static friction between the block
and the wall is 0.250. Determine the possible values for the
magnitude of P that allow the block to remain stationary.
Two bodies, m1= 1kg and m2=2kg are connected over a
massless pulley. The coefficient of kinetic friction between
m2 and the incline is 0.1. The angle θ of the incline is 20º.
Calculate: a) Acceleration of the blocks. (b) Tension of the
cord.
N
f
T
m2
T
m1
20º
m2g
m1g
The three blocks in the figure below are connected by massless cords
and pulleys. Data: m1=5kg, m2=3kg, m3=2kg. Assume that the incline plane
is frictionless.
(i) Show all the forces that act on each block.
N
T2
(ii) Calculate the acceleration of m1, m2, m3. T
m2
Fg2x
2
m3
(iii) Calculate the tensions on the cords.
(iv) Calculate the normal force acting on
m2
Fg2y
30º
T1
T1
m3g
m2g
m1
m1g
1B. (a) What should be the magnitude of F in the figure below if the body of
mass m=10kg is to slide up along a frictionless incline plane with constant
acceleration a=1.98 m/s2? (b) What is the
y
magnitude of the Normal force?
N
x
20º F
30º
Fg




Three forces, given by , F1  2.00ˆi  2.00 ˆj N , F2  5.00 ˆi  3.00 ˆj N


and , F3  45.0ˆi N act on an object to give it an acceleration of
magnitude 3.75 m/s2. (a) What is the direction of the acceleration?
(b) What is the mass of the object? (c) If the object is initially at rest,
what is its speed after 10.0 s? (d) What are the velocity components of
the object after 10.0 s?
A bag of cement of weight 325 N hangs from three wires as
suggested in Figure. Two of the wires make angles q1 = 60.0
and q2 = 25.0 with the horizontal. If the system is in
equilibrium, find the tensions T1, T2, and T3 in the wires.
A block of mass 3.00 kg is pushed up against a wall by a
force P that makes a 50.0 angle with the horizontal as
shown in Figure. The coefficient of static friction between
the block and the wall is 0.250. Determine the possible
values for the magnitude of P that allow the block to
remain stationary.
Review problem. A block of mass m = 2.00 kg is released from rest at h
= 0.500 m above the surface of a table, at the top of a 30.0 incline as
shown in Figure. The frictionless incline is fixed on a table of height H =
2.00 m. (a) Determine the acceleration of the block as it slides down the
incline. (b) What is the velocity of the block as it leaves the incline? (c)
How far from the table will the block hit the floor? (d) How much time has
elapsed between when the block is released and when it hits the floor? (e)
Does the mass of the block affect any of the above calculations?
A block is given an initial velocity of 5.00 m/s up a
frictionless 20.0° incline. How far up the incline does the
block slide before coming to rest?
A woman at an airport is towing her 20.0-kg suitcase at
constant speed by pulling on a strap at an angle q above
the horizontal. She pulls on the strap with a 35.0-N force,
and the friction force on the suitcase is 20.0 N. Draw a
free-body diagram of the suitcase. (a) What angle does
the strap make with the horizontal? (b) What normal force
does the ground exert on the suitcase?
Three objects are connected on the table as shown in
Figure. The table is rough and has a coefficient of kinetic
friction of 0.350. The objects have masses 4.00 kg, 1.00
kg and 2.00 kg, as shown, and the pulleys are frictionless.
Draw free-body diagrams of each of the objects. (a)
Determine the acceleration of each object and their
directions. (b) Determine the tensions in the two cords.
Gravitational Mass vs. Inertial
Mass
• In Newton’s Laws, the mass is the inertial mass
and measures the resistance to a change in the
object’s motion
• In the gravitational force, the mass is
determining the gravitational attraction between
the object and the Earth
• Experiments show that gravitational mass and
inertial mass have the same value
Applications of Newton’s Law
• Assumptions
– Objects can be modeled as particles
– Masses of strings or ropes are negligible
• When a rope attached to an object is pulling it,
the magnitude of that force, T, is the tension in
the rope
– Interested only in the external forces acting
on the object
• can neglect reaction forces
– Initially dealing with frictionless surfaces
Objects in Equilibrium
• If the acceleration of an object that can be
modeled as a particle is zero, the object is
said to be in equilibrium
• Mathematically, the net force acting on the
object is zero
F 0
 F  0 and  F
x
y
0
Equilibrium, Example 1a
• A lamp is suspended from
a chain of negligible mass
• The forces acting on the
lamp are
– the force of gravity (Fg)
– the tension in the chain (T)
• Equilibrium gives
F
y
 0  T  Fg  0
T  Fg
Equilibrium, Example 1b
• The forces acting on the
chain are T’ and T”
• T” is the force exerted by
the ceiling
• T’ is the force exerted by
the lamp
• T’ is the reaction force to
T
• Only T is in the free body
diagram of the lamp,
since T’ and T” do not act
on the lamp
Equilibrium, Example 2a
• Example 5.4
• Conceptualize the
traffic light
• Categorize as an
equilibrium problem
– No movement, so
acceleration is zero
Equilibrium, Example 2b
• Analyze
– Need two free-body
diagrams
– Apply equilibrium
equation to the light
and find T3
– Apply equilibrium
equations to the knot
and find T1 and T2
Objects Experiencing a Net
Force
• If an object that can be modeled as a
particle experiences an acceleration, there
must be a nonzero net force acting on it.
• Draw a free-body diagram
• Apply Newton’s Second Law in component
form
Newton’s Second Law, Example
1a
• Forces acting on the
crate:
– A tension, the
magnitude of force T
– The gravitational force,
Fg
– The normal force, n,
exerted by the floor
Newton’s Second Law, Example
1b
• Apply Newton’s Second Law in component form:
 F  T  ma
 Fy  n  Fg  0  n  Fg
x
x
• Solve for the unknown(s)
• If T is constant, then a is constant and the
kinematic equations can be used to more fully
describe the motion of the crate
Note About the Normal Force
• The normal force is not
always equal to the
gravitational force of the
object
• For example, in this case
F
y
 n  Fg  F  0
and n  Fg  F
• n may also be less than
Fg
Multiple Objects, Example 3
• Draw the free-body diagram for each object
– One cord, so tension is the same for both objects
– Connected, so acceleration is the same for both objects
• Apply Newton’s Laws
• Solve for the unknown(s)
Forces of Friction
• When an object is in motion on a surface
or through a viscous medium, there will be
a resistance to the motion
– This is due to the interactions between the
object and its environment
• This resistance is called the force of
friction
Forces of Friction, cont.
• Friction is proportional to the normal force
– ƒs  µs n and ƒk= µk n
– These equations relate the magnitudes of the forces,
they are not vector equations
• The force of static friction is generally greater
than the force of kinetic friction
• The coefficient of friction (µ) depends on the
surfaces in contact
Forces of Friction, final
• The direction of the frictional force is
opposite the direction of motion and
parallel to the surfaces in contact
• The coefficients of friction are nearly
independent of the area of contact
Static Friction
• Static friction acts to keep
the object from moving
• If F increases, so does ƒs
• If F decreases, so does
ƒs
• ƒs  µs n where the
equality holds when the
surfaces are on the verge
of slipping
– Called impending motion
Kinetic Friction
• The force of kinetic
friction acts when the
object is in motion
• Although µk can vary
with speed, we shall
neglect any such
variations
• ƒk = µk n
Some Coefficients of Friction
Friction in Newton’s Laws
Problems
• Friction is a force, so it simply is included
in the SF in Newton’s Laws
• The rules of friction allow you to determine
the direction and magnitude of the force of
friction
Friction Example, 1
• The block is sliding down
the plane, so friction acts up
the plane
• This setup can be used to
experimentally determine
the coefficient of friction
• µ = tan q
– For µs, use the angle where
the block just slips
– For µk, use the angle where
the block slides down at a
constant speed
Friction, Example 2
• Draw the free-body
diagram, including
the force of kinetic
friction
– Opposes the motion
– Is parallel to the
surfaces in contact
• Continue with the
solution as with any
Newton’s Law
problem
Friction, Example 3
• Friction acts only on the object in contact with another
surface
• Draw the free-body diagrams
• Apply Newton’s Laws as in any other multiple object
system problem