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Chapter 4 Dynamics: Newton’s Laws of Motion

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Chapter 4

Dynamics: Newton’s Laws of Motion

• Force

• Newton’s First Law of Motion

• Mass

• Newton’s Second Law of Motion

• Newton’s Third Law of Motion

• Weight – the Force of Gravity; and the Normal Force

• Applications Involving Friction, Inclines

AssignmentAssignment 33

Textbook (Giancoli, 6 th edition). Due on Oct. 8 th.

http://ilc2.phys.uregina.ca/~barbi/academic/phys109 /2009/phys109.html

1. On page 68 of Giancoli, problem 36.

2. On page 68 of Giancoli, problem 41.

3. On page 68 of Giancoli, problem 47.

4. A car is skidding out of control on a horizontal icy (frictionless) road. It has a mass of 2000 kg and it is headed directly at Lois Lane at a speed of 45.0 m/s. When the car is 200 m from her, Superman begins to exert a steady force on the car, , relative to the horizontal with a magnitude of 1.20 × 104 N at a downward angleof 300. Was Superman correct? Was this force enough to stop the car before hitting Lois?

Recalling Recalling LastLast LecturesLecturesRecalling Recalling LastLast LecturesLectures

Relative VelocityRelative Velocity

Consider a person walking on the boat with velocity relative to the boat.His velocity relative to the shore will be given by:

using:

Then

(3.19)

Relative VelocityRelative Velocity

Notes:

In general, if is the velocity of A relative to B, then the velocity of B relative to A,will be in the same line as but in opposite direction:

If is the velocity of C relative to A, then , the velocity of C relative to B, will be given by:

(3.20)

be given by:

(3.21)

Relative VelocityRelative Velocity

Problem 5.50 (textbook) : An unmarked police car, traveling a constant 95 Km/h is passed by a speeder traveling 145 Km/h. Precisely 1.00 s after the speeder passes, the policeman steps on the accelerator. If the police car’s acceleration is 2.00 m/s2.

How much time elapses after the police car is passed until it overtakes the speeder (assumed moving at constant speed)?

Solution developed on the blackboard

Relative VelocityRelative Velocity

52. Take the origin to be the location at which the speeder passes the police car, in the reference frame of the unaccelerated police car.

The speeder is traveling at

relative to the ground, and the policeman is traveling at

1 m s145 km h 40.28 m s

3.6 km h=

relative to the ground.

Relative to the unaccelerated police car, the speeder is traveling at

and the police car is not moving.

1m s95 km h 26.39 m s

3.6 km h=

13.89 m s sv=

Relative VelocityRelative Velocity

Do all of the calculations in the frame of reference of the unaccelerated police car.

The position of the speeder in the chosen reference frame is given by

The position of the policeman in the chosen reference frame is given by

s sx v t∆ =

( )21 1 , 1x a t t∆ = − >

(Note that he policeman does not move relative to the reference frame until he starts accelerating his car).

The police car overtakes the speeder when these two distances are the same.; i.e.:

( )212

1 , 1p px a t t∆ = − >

s px x∆ = ∆

Relative VelocityRelative Velocity

Then,

( ) ( ) ( )( )2 2 2 21 12 2

22

1 13.89m s 2m s 2 1 2 1

15.89 15.89 415.89 1 0 0.0632 s , 15.83 s

s p s px x v t a t t t t t t

t t t

∆ = ∆ → = − → = − + = − +

± −− + = → = =

.

15.89 1 0 0.0632 s , 15.83 s2

t t t− + = → = =

1.00 st =

15.8 st =

Since the police car doesn’t accelerate until , the correct answer is

ForceForce

We have introduced the concept of motion and obtained a set of equations that describe the motion of an object given some initial conditions. We have also introduced the concept of vector and have shown that velocity and acceleration are vectors (have magnitude and direction).

But.. we have not discussed what sets an object into motion, or change its state of motion. This is what we will be learning in this chapter.

ForceForce

The area that studies the connection between motion and its cause is called dynamics .

The cause of the motion is force .

But what force is??

Force is the interaction between two objects.

For example: When you push a grocery cart which is initially at rest, you are exerting For example: When you push a grocery cart which is initially at rest, you are exerting a force on it.

The force you apply on the cart changesits state of motion

ForceForce

Force is the interaction between two objects.

Side note :

You may ask me: but how do two objects interact?

For example: You have seen that magnets attract some materials such as a piece of iron. But the magnet does not need to touch the iron to attract it. So, how do these two objects interact?

The answer to this question requires advanced physics. However, let’s say that there are some special particles that operates as messengers between the two objects. These messengers are the carriers of the force between these objects and therefore responsible for their interaction.

There are four fundamental kinds of forces:

� Gravitational � Responsible for having the Moon orbiting the Earth….� Electromagnetic � Responsible for the attraction between the magnet and iron.� Weak � Main responsible for the processes that make stars to shine� Strong � Responsible for the nuclear interaction

ForceForce

� Note that the direction of the grocery cart depends on the direction of the force you apply on it:

You can push it on a straight line, orto the left, or to the right, etc..

� The strength you push the cart will determine how fast the cart goes or how fast you will change its direction.how fast you will change its direction.

So, it is clear that you should assign a direction and a magnitude tothe force you apply on the cart. In otherwords,

And we will represent it using the standard vector representation:with the arrow representing the direction a force is applied on an object.

Force is a vector.

ForceForce

Forces being vectors can be added using the methods discussed in chapter 3.

Example : In the figure you and your mother are pushing the same grocery cart with forces represented by and , respectively. The total force applied onthe cart, , will be the vector addition of both forces:

You can then use, for example,You can then use, for example,the method of components forvector addition after having selected an appropriate reference frame and coordinate system.

ForceForce

In general we can state that if a system of two or more forces are applied on a object,the net force on this object will be given by:

Where the subscript “ i ” in denotes a particular force acting on the object; “i “ can be any number between 1 and N; N is the total number of forces applied on the object. The Greek symbol Σ denotes a summation over all possible forces from 1

(4.1)

the object. The Greek symbol Σ denotes a summation over all possible forces from 1to N.

Example: Determine the net force acting on an object subject to five different forces: ; ; ; ; and

x

y

Newton’s Laws of MotionNewton’s Laws of Motion

Relationship between force and motion

We have seen that given a reference frame, an object can be found in any of the following states of motion:

• at rest• moving with constant velocity• accelerated

But an applied force on an object can change its state of motion: But an applied force on an object can change its state of motion:

• The object can starting moving from rest• Can change velocity from a state of constant velocity• Can change direction (and therefore the vector velocity), and may be the

magnitude of its velocity.

It is clear that if NO forces are applied on an object, it will tend to remain on its original state of motion.

Newton’s Laws of MotionNewton’s Laws of Motion

Newton’s First Law of Motion

We can generalize the previous statement to say that:

If the NET FORCE of all forces applied on a object is ZERO, then the object will tend to remain in its state of rest, or constant velocity .

This follows from the fact that a non-zero net force will change the state of motion of any object, which implies in changing its velocity either in direction or magnitude or both. But, by definition, change in velocity implies acceleration . So, if no or both. But, by definition, change in velocity implies acceleration . So, if no acceleration is present, the object will remain in its original state of motion.

We can then state the Newton’s first law of motion as:

“ Every Object continues in its state of rest, or uni form velocity in a straight line, as long as no net force acts on it ”

Newton’s Laws of MotionNewton’s Laws of Motion

Example: An object initially at rest is suddenly subjected to two forces, and , of same magnitude and opposite direction. What is the net force acting on this object? Will it move?

Answer:

Let be the net force, then:

But , then

Since the net force is zero, the object will NOT move.

Newton’s Laws of MotionNewton’s Laws of Motion

Note 1:

You can always represent the forces on the previous diagram as

instead of

This can be generalized to any system of forces.

Newton’s Laws of MotionNewton’s Laws of Motion

Note 2:

Newton’s first law ONLY holds in inertial frames .

Inertial frames are frames FIXED on a system which is either at rest or moving with constant velocity (no changes in direct ion and magnitude).

Or, in other words, inertial frames are those where the Newton’s first law is valid.

Frames other than the inertial frames are called non-inertial frames (for obvious reasons).

An example of non-inertial frame is that of a frame fixed on an a ccelerated system (example: a car).

Newton’s Laws of MotionNewton’s Laws of Motion

Example :

Assume you are in a truck and have fixed the reference frame on a system at rest relative to the truck while moving at constant speed.

You then decide to accelerate the truck to increase its speed from a state of constant velocity.

You will then notice that anything free on the passenger sit or anywhere on the truck will tend to move. If you crash your truck on a wall, you will also tend to keep moving will tend to move. If you crash your truck on a wall, you will also tend to keep moving at the same velocity as before.

The reason is that the car is being accelerated due to aforce applied to it (but not to you.)

For a person at rest relative to the ground, the box will seem to continue moving with constant velocity while the car is being accelerated (ignore effects due to friction).

Newton’s Laws of MotionNewton’s Laws of Motion

The Concept of Mass:

Mass measures that quantity of matter in a body (or object). It is given in units of Kg.

But then, you may ask me: What is matter????

The answer is even more complicate than that to understand how objects interact. We will not discuss about it here.

A alternative definition can be given as:A alternative definition can be given as:

� Mass is a measure of the inertia of an object.

Inertia is the tendency an object has to continue i n its state of motion.

You may have experienced the fact that it gets harder to move a box of some material as you add more of the same material to the box. So you are adding mass. This means that adding mass in the box makes it more difficult to change its state of inertia. So, mass is related to inertia:

� The larger is the mass of an object, the larger is its inertia.

Newton’s Laws of MotionNewton’s Laws of Motion

Newton’s Laws of MotionNewton’s Laws of Motion

Newton’s Second Law of Motion:

We have seen that force and acceleration are related � In fact, acceleration is the product of an applied force.

We have also noticed that force and mass are also related � more mass implies that a stronger force is needed to change the state of motion of a body.

But how are these quantities related???

From the equation of motion we know that:

If you triple (or double, or etc) your acceleration a, you expect to triple the velocity added to your motion, v - v0.

It is also observed that if you triple the applied force on an object, its change in velocity will be triple.It is then clear that force and acceleration are directly proportional. We say:

The symbol means “proportional to”

Newton’s Laws of MotionNewton’s Laws of Motion

Newton’s Second Law of Motion:

It also has been observed that for the same applied force, the change in velocity will depend on the mass of the object:

� the greater the mass,� the smaller is the change in velocity,� the smaller is the acceleration.

So, here we can state the mass and acceleration are inversely proportional.So, here we can state the mass and acceleration are inversely proportional.

This, together with led Newton to state his second law of motion:

“ The acceleration of an object is directly proportio nal to the net force acting on it, and is inversely proportional to its mass. T he direction of the acceleration is in the direction of the net force a cting on the object ”

Newton’s Laws of MotionNewton’s Laws of Motion

Newton’s Second Law of Motion:

Newton’s second law can be summarized with the following equation:

where I have omitted the subscript “ i ” and the limits of summation: 1 and N (they are implicitly assumed to be present).

This equation can be re-written to yield the well known relationship between force, mass and acceleration:

Note that you can also write:

, where

(4.2)