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Chapter 4
The Laws of Motion
Classical Mechanics
Describes the relationship between the motion of objects in our everyday world and the forces acting on them
Conditions when Classical Mechanics does not apply
very tiny objects (< atomic sizes)
objects moving near the speed of light
Forces
Usually think of a force as a push or pull
Vector quantity
May be a contact force or a field force Contact forces result from physical contact
between two objects
Field forces act between disconnected objects Also called “action at a distance”
Contact and Field Forces
Fundamental Forces
Types Strong nuclear force
Electromagnetic force
Weak nuclear force
Gravity
Characteristics All field forces
Listed in order of decreasing strength
Only gravity and electromagnetic in mechanics
External and Internal Forces
External force Any force that results from the
interaction between the object and its environment
Internal forces Forces that originate within the object
itself
They cannot change the object’s velocity
Net Force
Net force is
the combination of all forces that change an object’s state of motion.
example: If you pull on a box with 10 N and a friend
pulls oppositely with 5 N, the net force is 5N in the direction you are pulling.
Net Force
Vector quantity
a quantity whose description requires both magnitude (how much) and direction (which way)
can be represented by arrows drawn to scale, called vectors
length of arrow represents magnitude and arrowhead shows direction
examples: force, velocity, acceleration
The Equilibrium Rule
The equilibrium rule
the vector sum of forces acting on a non-accelerating object equals zero
in equation form: F = 0
The Equilibrium Rule
example: a string holding up a bag of flour
two forces act on the bag of flour:
–tension force acts upward
–weight acts downward
equal in magnitude and opposite in
direction
when added, cancel to zero
bag of flour remains at rest
Inertia
Is the tendency of an object to continue in its original motion
Galileo (1564– 1642)
Inertia
Galileo’s Concept of Inertia
Italian scientist Galileo demolished Aristotle’s assertions in early 1500s.
Galileo’s discovery • objects of different weight fall to the ground
at the same time in the absence of air resistance
• a moving object needs no force to keep it moving in the absence of friction
Galileo’s Concept of Inertia
Force
is a push or a pull
Inertia
is a property of matter to resist changes in motion
depends on the amount of matter in an object (its mass)
The use of inclined planes for Galileo’s experiments helped him to discover the property called inertia. Intertia is a property of matter.
Galileo’s Inclinded Plane
Mass—A Measure of Inertia
Mass
a measure of the inertia of a material object
independent of gravity
greater inertia greater mass
unit of measurement is the kilogram (kg)
Weight
the force on an object due to gravity
scientific unit of force is the Newton (N)
unit is also the pound (lb)
Mass—A Measure of Inertia
Mass and weight in everyday conversation are interchangeable.
Mass, however, is different and more fundamental than weight.
Mass versus weight • on Moon and Earth
– weight of an object on Moon is
less than on Earth (1/6th)
– mass of an object is the same
in both locations
Mass—A Measure of Inertia
One Kilogram Weighs 9.8 Newtons
Relationship between kilograms and pounds
1 kg = 2.2 lb = 9.8 N at Earth’s surface
1 lb = 4.45 N
Mass
A measure of the resistance of an object to changes in its motion due to a force
Scalar quantity
SI units are kg
Sir Isaac Newton
1642 – 1727
Formulated basic concepts and laws of mechanics
Universal Gravitation
Calculus
Light and optics
Isaac my man
Newton’s First Law
An object moves with a velocity that is constant in magnitude and direction, unless acted on by a nonzero net force
The net force is defined as the vector sum of all the external forces exerted on the object
Newton’s Second Law
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
F and a are both vectors
Can also be applied three-dimensionally
Units of Force
SI unit of force is a Newton (N)
US Customary unit of force is a pound (lb)
1 N = 0.225 lb
2s
mkg1N1
Problem 4.1
A 6.0-kg object undergoes an acceleration of 2.0 m/s2. (a) What is the magnitude of the resultant force acting on it? (b) If this same force is applied to a 4.0-kg object, what acceleration is produced?
Problem 4.12
The force exerted by the wind on the sails of a sailboat is 390 N north. The water exerts a force of 180 N east. If the boat (including its crew) has a mass of 270 kg, what are the magnitude and direction of its acceleration?
Problem 4.19
A 2 000-kg car is slowed down uniformly from 20.0 m/s to 5.00 m/s in 4.00 s. (a) What average force acted on the car during that time, and (b) how far did the car travel during that time?
Gravitational Force
Mutual force of attraction between any two objects
Expressed by Newton’s Law of Universal Gravitation:
2
21g
r
mmGF
Weight
The magnitude of the gravitational force acting on an object of mass m near the Earth’s surface is called the weight w of the object w = m g is a special case of Newton’s
Second Law g is the acceleration due to gravity
g can also be found from the Law of Universal Gravitation
More about weight
Weight is not an inherent property of an object
mass is an inherent property
Weight depends upon location
Newton’s Third Law
If object 1 and object 2 interact, the force exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force exerted by object 2 on object 1.
Equivalent to saying a single isolated force cannot exist
12 21 F F
Newton’s Third Law cont.
F12 may be called the action force and F21 the reaction force
Actually, either force can be the action or the reaction force
The action and reaction forces act on different objects
Some Action-Reaction Pairs
is the normal force, the force the table exerts on the TV
is always perpendicular to the surface
is the reaction – the TV on the table
'andn n
' n n
n
n
'n
More Action-Reaction pairs
is the force the Earth exerts on the object
is the force the object exerts on the earth
'
g gandF F
gF
'
gF
'
g g F F
Forces Acting on an Object
Newton’s Law uses the forces acting on an object
are acting on the object
are acting on other objects
gandn F
'' gandn F
Applications of Newton’s Laws
Assumptions
Objects behave as particles
can ignore rotational motion (for now)
Masses of strings or ropes are negligible
Interested only in the forces acting on the object
can neglect reaction forces
Free Body Diagram
Must identify all the forces acting on the object of interest
Choose an appropriate coordinate system
If the free body diagram is incorrect, the solution will likely be incorrect
Free Body Diagram, Example
The force is the tension acting on the box The tension is the same
at all points along the rope
are the forces exerted by the earth and the ground
gandn F
Free Body Diagram, final
Only forces acting directly on the object are included in the free body diagram
Reaction forces act on other objects and so are not included
The reaction forces do not directly influence the object’s motion
Solving Newton’s Second Law Problems
Read the problem at least once
Draw a picture of the system
Identify the object of primary interest
Indicate forces with arrows
Label each force
Use labels that bring to mind the physical quantity involved
Solving Newton’s Second Law Problems
Draw a free body diagram If additional objects are involved, draw
separate free body diagrams for each object
Choose a convenient coordinate system for each object
Apply Newton’s Second Law The x- and y-components should be taken
from the vector equation and written separately
Solve for the unknown(s)
Equilibrium
An object either at rest or moving with a constant velocity is said to be in equilibrium
The net force acting on the object is zero (since the acceleration is zero)
0F
Equilibrium cont.
Easier to work with the equation in terms of its components:
This could be extended to three dimensions
0 0x yF and F
Equilibrium Example – Free Body Diagrams
Problem 4.45
(a) What is the resultant force exerted by the two cables supporting the traffic light in Figure P4.45? (b) What is the weight of the light?
Problem 4.26 (a) An elevator of mass m moving upward has two forces
acting on it: the upward force of tension in the cable and the downward force due to gravity. When the elevator is accelerating upward, which is greater, T or w? (b) When the elevator is moving at a constant velocity upward, which is greater, T or w? (c) When the elevator is moving upward, but the acceleration is downward, which is greater, T or w? (d) Let the elevator have a mass of 1 500 kg and an upward acceleration of 2.5 m/s2. Find T. Is your answer consistent with the answer to part (a)? (e) The elevator of part (d) now moves with a constant upward velocity of 10 m/s. Find T. Is your answer consistent with your answer to part (b)? (f) Having initially moved upward with a constant velocity, the elevator begins to accelerate downward at 1.50 m/s2. Find T. Is your answer consistent with your answer to part (c)?
Inclined Planes
Choose the coordinate system with x along the incline and y perpendicular to the incline
Replace the force of gravity with its components
Multiple Objects – Example
When you have more than one object, the problem-solving strategy is applied to each object
Draw free body diagrams for each object
Apply Newton’s Laws to each object
Solve the equations
Multiple Objects – Example, cont.
Problem 4.20
Two packing crates of masses 10.0 kg and 5.00 kg are connected by a light string that passes over a frictionless pulley as in Figure P4.20. The 5.00-kg crate lies on a smooth incline of angle 40.0°. Find the acceleration of the 5.00-kg crate and the tension in the string.
Connected Objects
Apply Newton’s Laws separately to each object
The magnitude of the acceleration of both objects will be the same
The tension is the same in each diagram
Solve the simultaneous equations
More About Connected Objects
Treating the system as one object allows an alternative method or a check
Use only external forces
Not the tension – it’s internal
The mass is the mass of the system
Doesn’t tell you anything about any internal forces
Problem 4.52
Three objects are connected by light strings as shown in Figure P4.52. The string connecting the 4.00-kg object and the 5.00-kg object passes over a light frictionless pulley. Determine (a) the acceleration of each object and (b) the tension in the two strings.
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 is resistance is called friction
More About Friction
Friction is proportional to the normal force
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
The direction of the frictional force is opposite the direction of motion
The coefficients of friction are nearly independent of the area of contact
Static friction acts to keep the object from moving
If F increases, so does ƒs
If F decreases, so does ƒs
ƒs µ n
Static Friction, ƒs
Kinetic Friction, ƒk
The force of kinetic friction acts when the object is in motion
ƒk = µ n Variations of the
coefficient with speed will be ignored
Block on a Ramp, Example
Axes are rotated as usual on an incline
The direction of impending motion would be down the plane
Friction acts up the plane Opposes the motion
Apply Newton’s Laws and solve equations
Problem 4.41
Find the acceleration reached by each of the two objects shown in Figure P4.41 if the coefficient of kinetic friction between the 7.00-kg object and the plane is 0.250.
Problem 4.48
(a) What is the minimum force of friction required to hold the system of Figure P4.48 in equilibrium? (b) What coefficient of static friction between the 100-N block and the table ensures equilibrium? (c) If the coefficient of kinetic friction between the 100-N block and the table is 0.250, what hanging weight should replace the 50.0-N weight to allow the system to move at a constant speed once it is set in motion?
Friction of Air
When acceleration of fall is less than g—non-free fall • occurs when air resistance is non-negligible
• depends on two things: speed and frontal surface area
Friction of Air
example: A skydiver jumps from plane.
Weight is the only force until air resistance acts.
As falling speed increases, air resistance on diver builds up, net force is reduced, and acceleration becomes less.
When air resistance equals the diver’s weight, net force is zero and acceleration terminates.
Diver reaches terminal velocity, then continues the fall at constant speed.
Friction of Air
Terminal speed
occurs when acceleration terminates (when air resistance equals weight and net force is zero)
Terminal velocity
same as terminal speed, with direction implied or specified
When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is
A. less than g.
B. more than g.
C. g.
D. terminated.
Newton’s Second Law of Motion CHECK YOUR NEIGHBOR
When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is
A. less than g.
B. more than g.
C. g.
D. terminated.
Comment:
Acceleration of a nonfree-fall is always less than g. Acceleration will actually be (20 N – 5 N)/2 kg = 7.5 m/s2.
Newton’s Second Law of Motion CHECK YOUR ANSWER
If a 50-N person is to fall at terminal speed, the air resistance needed is
A. less than 50 N.
B. 50 N.
C. more than 50 N.
D. none of the above
Newton’s Second Law of Motion CHECK YOUR NEIGHBOR
If a 50-N person is to fall at terminal speed, the air resistance needed is
A. less than 50 N.
B. 50 N.
C. more than 50 N.
D. none of the above
Explanation:
Then, F = 0 and acceleration = 0.
Newton’s Second Law of Motion CHECK YOUR ANSWER
As the skydiver falls faster and faster through the air, air resistance
A. increases.
B. decreases.
C. remains the same.
D. not enough information
Newton’s Second Law of Motion CHECK YOUR NEIGHBOR
As the skydiver falls faster and faster through the air, air resistance
A. increases.
B. decreases.
C. remains the same.
D. not enough information
Newton’s Second Law of Motion CHECK YOUR ANSWER
As the skydiver continues to fall faster and faster through the air, net force
A. increases.
B. decreases.
C. remains the same.
D. not enough information
Newton’s Second Law of Motion CHECK YOUR NEIGHBOR
As the skydiver continues to fall faster and faster through the air, net force
A. increases.
B. decreases.
C. remains the same.
D. not enough information
Newton’s Second Law of Motion CHECK YOUR ANSWER
As the skydiver continues to fall faster and faster through the air, her acceleration
A. increases.
B. decreases.
C. remains the same.
D. not enough information
Newton’s Second Law of Motion CHECK YOUR NEIGHBOR
As the skydiver continues to fall faster and faster through the air, her acceleration
A. increases.
B. decreases.
C. remains the same.
D. not enough information
Newton’s Second Law of Motion CHECK YOUR ANSWER
Newton’s Second Law of Motion A situation to ponder…
Consider a heavy and light person jumping together with the same-size parachutes from the same altitude.
Who will reach the ground first?
A. the light person
B. the heavy person
C. both will reach at the same time
D. not enough information
A situation to ponder… CHECK YOUR NEIGHBOR
Who will reach the ground first?
A. the light person
B. the heavy person
C. both will reach at the same time
D. not enough information
Explanation:
The heavier person has a greater terminal velocity. Do you know why?
A situation to ponder… CHECK YOUR ANSWER
Newton’s Second Law of Motion
Coin and feather fall
feather reaches terminal velocity very quickly and falls slowly at constant speed, reaching the bottom after the coin does
coin falls very quickly and air resistance doesn’t build up to its weight over short-falling distances, which is why the coin hits the bottom much sooner than the falling feather
How to be a flying squirrel: Jeb Corliss "Grinding the Crack"
Air Resistance
Felix Baumgartner: Record high sky dive
Air Resistance
24 miles high 725 mph
Backup
Problem 4.42
A 2.00-kg block is held in equilibrium on an incline of angle θ = 60.0° by a horizontal force applied in the direction shown in Figure P4.42. If the coefficient of static friction between block and incline is μs = 0.300, determine (a) the minimum value of and (b) the normal force exerted by the incline on the block.
Forces on an Elevator
Tension T
Force of gravity
Force of elevator acceleration
mg mg
T ma T
ma
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