Download Chapter 4 Dynamics: Newton`s Laws of Motion

Chapter 4
Dynamics: Newton’s Laws of Motion
Dynamics
So far we have only discussed kinematics:
r = r0 + v0t + ½ at2
v = v0t + at
but ignored what causes the acceleration.
For now, we will limit our study to classical mechanics:
• Objects are macroscopic
• Objects move slowly compared to the speed of light
Dynamics
The central problem of classical mechanics:
• We are given a particle with known characteristics and initial v.
• The particle is placed in a completely described environment.
• Problem: What is the subsequent motion of the particle?
So what do we need to go from kinematics to dynamics?
- A relationship between the response of a particle (acceleration) and the
external environment acting on it.
- What properties of a particle affect the ability of the external
environment to affect it.
-How to predict the response of a given particle to a given environment?
Force
The outside influence called the force which acts on the particle.
• If multiple forces act, the particle responds to the net force.
Force is a vector quantity - has both magnitude and direction.
• SI units are Newtons (N)
Mass
Mass is the measure of inertia of an object.
• Also related to the quantity of matter in the object.
• SI units: kilogram (kg)
Inertia is the resistance of an object to a change in its state of motion.
Mass is not weight!!!
• Weight is the force exerted on that object by gravity.
• The imperial unit of weight and mass are the same (pound).
Newton’s Laws of Motion
First Law: “Every body persists in its state of rest or of uniform motion
in a straight line unless it is compelled to that state by forces impressed
upon it.”
- Does not hold in non-inertial reference frames (accelerated frames).
Second Law: Acceleration due to an applied force is proportional to
the force applied, and inversely proportional to the objects mass.
F=ma
- Force and acceleration are vector quantities.
Third Law: “To every action, there is an equal and opposite reaction.”
Newton’s First Law of Motion
also called the law of inertia:
Every object continues in its state of rest, or of uniform
velocity in a straight line, as long as no net force acts on
it.
Newton’s First Law of Motion
Inertial reference frames:
Newton’s first law does not hold in every reference frame,
(e.g. a reference frame that is accelerating or rotating).
An inertial reference frame is one in which Newton’s first
law is valid.
By definition: an inertial reference frame is one in which
the first law is valid.
Newton’s First Law of Motion
“Conceptual Question”:
A school bus comes to a sudden stop, and all of the
backpacks on the floor start to slide forward. What
force causes them to do that?
Newton’s Second Law of Motion
Newton’s second law is the relation between acceleration
and force.
Acceleration is proportional to force and inversely
proportional to mass.
Weight—the Force of Gravity; and the
Normal Force
Weight is the force exerted on an object by
gravity.
Close to the surface of the Earth, where the
gravitational force is nearly constant, the
weight of an object of mass m is:
where
Mass, Weight, and Gravity
•
Weight is the force of gravity on an object:
•
•
Mass doesn't depend on the presence or strength
of gravity.
• Weight depends on gravity, so varies with location:
• Weight is different on different planets.
• Near Earth's surface, has magnitude 9.8 m/s2
or 9.8 N/kg, and is directed downward.
All objects experience the same gravitational
acceleration, regardless of mass.
• Therefore objects in free fall—under the influence
of gravity alone—appear "weightless" because they
share a common accelerated motion.
• This effect is noticeable in orbiting spacecraft
• because the absence of air resistance means
gravity is the only force acting.
• because the apparent weightlessness continues
indefinitely, as the orbit never intersects Earth.
Weight—the Force of Gravity; and the
Normal Force
An object at rest must have no net force on it. If it is sitting
on a table, the force of gravity is still there; what other
force is there?
The force exerted perpendicular
to a surface is called the normal
force.
It is exactly as large as needed to
balance the force from the object.
Weight—the Force of Gravity; and the
Normal Force
Example: Weight, normal force, and a box.
A friend has given you a box of mass 25.0 kg. The
box is resting on the smooth (frictionless)
horizontal surface of a table.
(a) Determine the weight of the box and the
normal force exerted on it by the table.
(b) Now your friend pushes down on the box with a
force of 100.0 N. Again determine the normal
force exerted on the box by the table.
(c) If your friend pulls upward on the box with a
force of 400.0 N, what now is the normal force
exerted on the box by the table?
Newton’s Second Law of Motion
Force is a vector, so the second law is true along each
coordinate axis.
Newton’s Second Law of Motion
Example 4-2: Force to accelerate a fast car.
Estimate the net force needed to accelerate (a) a
1000-kg car at ½ g; (b) a 200-g apple at the same
rate.
Example 4-3: Force to stop a car.
What average net force is required to bring a
1500-kg car to rest from a speed of 100 km/h
within a distance of 55 m?
Solving Problems with Newton's Second Law
Example
• A 740-kg elevator accelerates upward at 1.1 m/s 2, pulled by a cable of
negligible mass. Find the tension force in the cable.
• INTERPRET The object of interest is the elevator; the forces are
gravity and the cable tension.
• DEVELOP Newton's law reads
•
EVALUATE In a coordinate system
with y-axis upward, Newton's law reads
Solving gives
• ASSESS Makes sense; look at some
special cases.
• When a = 0, T = mg and the cable tension balances gravity.
• When T = 0, a = –g, and the elevator falls freely.
The elevator revisited:
In this system, we have an elevator [A] with
a mass of 1100kg and a counterweight [B] with
mass 1000 kg. A driving mechanism [C]
is used to run the elevator in service. The
elevator accelerates upward at 2.0 m/s2.
Assume that the pulleys and cable
have no mass, and that there is no friction.
A) What is the tension T1?
B) What is the tension T2?
C) What force is exerted on the cable by the
driving mechanism?
The elevator revisited:
In this system, we have an elevator [A] with
a mass of 1100kg and a counterweight [B] with
mass 1000 kg. A driving mechanism [C]
is used to run the elevator in service. The
elevator accelerates upward at 2.0 m/s2.
Assume that the pulleys and cable
have no mass, and that there is no friction.
A) What is the tension T1? 1.3 x 104 N
B) What is the tension T2? 0.78 x 104 N
C) What force is exerted on the cable by the
driving mechanism?
5.2 x 103 N toward the counterweight
Announcements (9 Feb 2017):
- Exam 1 – next Thursday, February 16. Covers Chapters 1 through 3.
- Read Chapter 5 for next week.
- Mastering Physics and Written Homework for Chapter 4 posted today/Fri.
- Office hours next week: Monday 2-330p and Tuesday 4-5 pm in the OSL.
If an object moves with constant velocity in a straight
line, which of the following statements is true?
a)
b)
c)
d)
There are no forces acting on the object.
The net force on the object is zero.
There is a constant force in the direction of motion.
There is a constant force in the direction opposite of
motion.
If an object moves with constant velocity in a straight
line, which of the following statements is true?
a)
b)
c)
d)
There are no forces acting on the object.
The net force on the object is zero.
There is a constant force in the direction of motion.
There is a constant force in the direction opposite of
motion.
Consider a mass suspended from a spring. After being
released, while the mass is accelerating downward, the
magnitude of the force the mass exerts on the spring is
________ the weight of the mass.
a) larger than
b) equal to
c) smaller than
Consider a mass suspended from a spring. After being
released, while the mass is accelerating downward, the
magnitude of the force the mass exerts on the spring is
________ the weight of the mass.
a) larger than
b) equal to
c) smaller than
If a 5.0 kg box is pulled simultaneously by a 10.0 N
force and a 5.0 N force, then its acceleration must be:
A) 3.0 m/s2.
B) 2.2 m/s2.
C) 1.0 m/s2.
D) We cannot tell from the information given.
If a 5.0 kg box is pulled simultaneously by a 10.0 N
force and a 5.0 N force, then its acceleration must be:
A) 3.0 m/s2.
B) 2.2 m/s2.
C) 1.0 m/s2.
D) We cannot tell from the information given.
Newton’s Third Law of Motion
Any time a force is exerted on an object, that force is
caused by another object.
Newton’s third law:
To every action there is always opposed an equal
reaction. (For every action there is an equal but
opposite reaction).
Newton's Third Law
• Forces come in pairs.
• If object A exerts a force on object B,
then object B exerts an oppositely
directed force of equal magnitude on A.
• Obsolete language: "For every action
there is an equal but opposite reaction."
• Important point: The two forces
always act on different objects;
therefore they can't cancel each other.
• Example:
• Push on book of mass m1 with force
• Note third-law pair
• Third law is necessary for a consistent
description of motion in Newtonian physics.
Example: A 6.00-kg block is in contact with a 4.00-kg block on
a horizontal frictionless surface as shown in the figure. The 6.00kg block is being pushed by a horizontal 20.0-N force as shown.
What is the magnitude of the force that the 6.00-kg block exerts
on the 4.00-kg block?
Example: A series of weights connected by very light cords
are given an upward acceleration of 4.00 m/s2 by a pull P, as
shown in the figure. A, B, and C are the tensions in the
connecting cords. Calculate P and A.
Solving Problems with Newton’s Laws:Free-Body
Diagrams
1. For one object, draw a free-body
diagram, showing all the forces acting on
the object. Make the magnitudes and
directions as accurate as you can. Label
each force. If there are multiple objects,
draw a separate diagram for each one.
2. Resolve vectors into components.
3. Apply Newton’s second law to each
component.
4. Draw a sketch.
5. Solve.
Solving Problems with Newton’s Laws: Free-Body
Diagrams
Book example: Pulling the mystery box.
Suppose a friend asks to examine the 10.0-kg box
you were given previously, hoping to guess
what is inside; and you respond, “Sure, pull the
box over to you.” She then pulls the box by the
attached cord along the smooth surface of the
table. The magnitude of the force exerted by
the person is FP = 40.0 N, and it is exerted at a
30.0° angle as shown.
Calculate (1) the acceleration of the box, (2) the
magnitude of the upward force FN exerted by
the table on the box, and (3) what is the
maximum force she can pull on the box
without lifting it?
Solving Problems with Newton’s Laws:
Diagrams
Free-Body
Example 4-12: Two boxes connected by a
cord.
Two boxes, A and B, are connected by a
lightweight cord and are resting on a smooth
table. The boxes have masses of 12.0 kg and
10.0 kg. A horizontal force of 40.0 N is
applied to the 10.0-kg box. Find (a) the
acceleration of each box, and (b) the tension
in the cord connecting the boxes.
Solving Problems with Newton’s Laws: Free-Body
Diagrams
A
B
C
Clicker Question: The hockey puck.
A hockey puck is sliding at constant velocity across a flat
horizontal ice surface that is assumed to be frictionless.
Which of these sketches is the correct free-body diagram
for this puck?
Solving Problems with Newton’s Laws: Free-Body
Diagrams
A
B
C
Clicker Question: The hockey puck.
A hockey puck is sliding at constant velocity across a flat
horizontal ice surface that is assumed to be frictionless.
Which of these sketches is the correct free-body diagram
for this puck? B is the correct figure. No net force acts on
the puck.
Solving Problems with Newton’s Laws: Free-Body
Diagrams
Example: The advantage of a
pulley.
A mover is trying to lift a
piano (slowly) up to a secondstory apartment. He is using a
rope looped over two pulleys
as shown. What force must he
exert on the rope to slowly lift
the piano’s 2000-N weight?
Solving Problems with Newton’s Laws: Free-Body
Diagrams
Example: Accelerometer.
A small mass m hangs from a thin
string and can swing like a
pendulum. You attach it above the
window of your car as shown. What
angle does the string make
(a) when the car accelerates at a
constant a = 1.20 m/s2, and
(b) when the car moves at constant
velocity, v = 90 km/h?
Solving Problems with Newton’s Laws:Free-Body
Diagrams
Example: Box slides down an incline.
A box of mass m is placed on a smooth
incline that makes an angle θ with the
horizontal.
(a) Determine the normal force on the
box.
(b) Determine the box’s acceleration.
(c) Evaluate for a mass m = 10 kg and
an incline of θ = 30°.
Summary of Chapter 4
• Newton’s first law: If the net force on an object is
zero, it will remain either at rest or moving in a
straight line at constant speed.
• Newton’s second law:
• Newton’s third law:
• Weight is the gravitational force on an object.
• Free-body diagrams are essential for problemsolving. Do one object at a time, make sure you have
all the forces, pick a coordinate system and find the
force components, and apply Newton’s second law
along each axis.