1. Dynamics: Newton’s Laws of Motion
Chapter 4
Dynamics: Newton’s Laws of Motion

2. Units of Chapter 4 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

3. Units of Chapter 4
Solving Problems with Newton’s Laws: Free-Body Diagrams
Applications Involving Friction, Inclines
Problem Solving – A General Approach

4. 4-1 Force
A force is a push or pull. An object at rest needs a force to get it moving; a moving object needs a force to change its velocity.
The spring scaled measures the amount of force it takes to pull the olive oil box across the table. If the object is at rest, to get it moving a force is required. So a force is required to get the box to move from zero velocity to nonzero velocity. If an object is already moving, then a force is required to change the velocity either in direction or magnitude.
A spring scale is used to find a weight of an object. In other words, the scale measures the force of gravity on the object. Force has direction and magnitude so it is a vector. We use an arrow to represent force on an illustration.
The magnitude of a force can be measured using a spring scale.

5. 4-2 Newton’s First Law of Motion
Newton’s first law is often 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.
So now we need to figure out if there is a relationship between force and motion. Aristotle believed that a force was required to keep an object moving along a plane. He believed the natural state was at rest years later, Galileo disagreed and believed that it was just as natural for an object to be in motion with a constant velocity. Think of pushing an object across a rough surface at a constant speed. As the surface gets smoother and smoother, it takes less force. If there was no friction, it will maintain a constant velocity once it starts moving. To push this book across the table, you need a force from you hand that balances out the friction. If you look at the signs, the two forces cancel each other out and the net forces is zero. The tendency of an object to maintain it’s state of rest or of uniform motion in a straight line is called inertia.

6. 4-2 Mass
Mass is the measure of inertia of an object. In the SI system, mass is measured in kilograms.
Mass is not weight:
Mass is a property of an object. Weight is the force exerted on that object by gravity.
If you go to the moon, whose gravitational acceleration is about 1/6 g, you will weigh much less. Your mass, however, will be the same.
Okay, let’s talk about the mass of something. It is not weight. Mass is the measure of the amount of matter in an object. It is also the amount of inertia of an object. The more the mass of an object, the greater the force needed to give a particular acceleration. It is harder to start it moving or to stop it. Think about a ford F250 and a baseball. The truck has more inertia than a baseball moving at the same speed and it requires a much greater force to change the velocity of the truck than the baseball. The mass of an object is the same anywhere in the universe. The weight of an object is the force exerted on that object by gravity and it will vary if the gravity is different. The ford f250 has the same amount of matter in no matter whether it is on the earth or on the moon. So if you don’t have friction, it will be just as hard to start a truck moving on the earth as on the moon or to stop it.

7. 4-3 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.
(4-1)
Newton’s first law say that if no force is acting on an object and it is at rest (not moving) then it will stay at rest and if it is moving, it will continue to move (frictionless). But if there is a net force on an object, then the velocity will change. If the force is in the direction of the motion then the velocity will increase. If the force is in the opposite direction, the velocity will decrease. If the force is at an angle to the direction of the movement, then the direction of the object will also change. Since the velocity changes then the acceleration changes.
Now let’s look at the relationship between acceleration and force. If you have a cart with almost no friction and you push it gently with a constant force over a specific period of time, the cart will accelerate from rest to a speed. Let’s say 3 km/h If you push with twice the force, it will reach 3 km/h in half the time. The acceleration is twice as great. If you triple the force then you will have three time the acceleration. But the acceleration also depends on the mass of the object. If you have a grocery cart empty and push it with the same force as a full one, the full one moves more slower. The greater the mass, the less the acceleration for the same force. The relationship is that the acceleration of an object is inversely proportional to the mass. The acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction o of the acceleration is in the direction of the net force acting on the object.

8. 4-3 Newton’s Second Law of Motion
Force is a vector, so is true along each coordinate axis.
The unit of force in the SI system is the newton (N).
Note that the pound is a unit of force, not of mass, and can therefore be equated to newtons but not to kilograms.
Force is a vector. So it has direction and magnitude. Force is an action capable of accelerating an object. We will use SI units in this class. Make sure you always get you mass into kg.

9. Example
What average net force is required to bring a 1500 – kg car to rest from a speed of 100km/h within a distance of 55 m.

10. Solution What do we know: vf = 0 vo = 100 km/h = 28 m/s x – xo = 55m
v2 = vo2 + 2a(x-xo)
a = v2 – vo2 = 0 – (28 m/s)2 = -7.1 m/s2
2 (x – xo) (55m)
F = ma = (1500 kg)(-7.1m/s2) = -1.1x104 N

11. 4-4 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:
Whenever one object exerts a force on a second object, the second exerts an equal force in the opposite direction on the first.
Some people also say for every action there is an equal and opposite reaction. But it is important to realizes that the action force and the reaction force are on different objects. Everybody push your hand against the edge of the desk. You hands shape is distorted which is evidence that there is a force begin exerted on your hand. You can see the edge of the desk pressing into your hand. You can even feel the force. The harder you push against the desk the harder the desk pushes back on your hand.

12. 4-4 Newton’s Third Law of Motion
A key to the correct application of the third law is that the forces are exerted on different objects. Make sure you don’t use them as if they were acting on the same object.
Let’s look at an ice skater. There is very little friction between her sakes and the ices so she moves freely if a force is exerted on her. If she pushes against he wall, then she starts to move backward. The force she exerts on the wall cannot make her start moving because that force acts on the wall. Something had to exert a force on her to start her moving tend that force could only have been exerted by the wall. The force with which the wall pushes on her is by Newton's third law, equal and opposite to the force she exerts on the wall. If you were in a boat sitting still in the water, and threw out a package, the boat will move in the opposite direction. The person exerts a force on the package and the package exerts an equal an opposite force on the person which propels the person and the boat slightly backward.

13. 4-4 Newton’s Third Law of Motion
Rocket propulsion can also be explained using Newton’s third law: hot gases from combustion spew out of the tail of the rocket at high speeds. The reaction force is what propels the rocket.
Note that the rocket does not need anything to “push” against.
Lots of people think that the rocket accelerates because of the gas rushing out the engine pushing against the ground or the atmosphere. Not true. The rocket exerts a strong force on the gases, expelling them and the gases exert an equal and opposite force on the rocket. It is the latter force that propels the rocket forward. The force exerted on the rocket by the gases. That is why a space vehicle can maneuver in empty space by firing it’s rockets in the direction opposite to that in which it wants to accelerate.

14. 4-5 Newton’s Third Law of Motion
Helpful notation: the first subscript is the object that the force is being exerted on; the second is the source.
This need not be done indefinitely, but is a good idea until you get used to dealing with these forces.
(4-2)
Let’s look at how we walk. A person begins walking by pushing with the foot backward against the ground. The ground exerts an equal and opposite force forward on the person and it is this force not the person that moves the person forward. If you doubt this try walking on ice where there is almost not friction. Birds fly forward by exerting a backward force on the air but it is the air pushing forward on the birds wings that propels the bird forward.

15. What exerts the force on a car?
The engine makes the wheels go forward but it is the friction between the tires and the ground that makes the car go forward. Friction is the force. The ground pushes on the tires in the opposite direction (think about slick ice and tires).

16. 4-5 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 is:
On the earth, we have a force called gravity or g that exerts a pull on objects. The pull is vertical downward and negative. Weight is the magnitude of the force of gravity on an object. Sop that means that the weight of an object on the moon will be much less than on the earth since the moons gravity is 1/6 that of the earth.

17. 4-5 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 (if the required force gets too big, something breaks!)
Now we are going to start to look at how forces act on objects. First, let’s look at objects that are at rest. If you look at Newton's second law (f=ma) the net force on an object at rest is zero. So there must be a force on the object to balance the gravitational force. The table is slightly compressed beneath the object and due to its elasticity, it pushes up on the object. Some people call this the contact force or the normal force if it is perpendicular. So FG = FN F’N is the force exerted on the table by the statue. It is purple because it is acting on the table and not on the statue. The reaction to FG is not shown The pink Forces are not Newton’s third law which acts on different objects. These are acting on the same object. The pink and purple ones in b are the ones from Newton’s third law.

18. 4-7 Solving Problems with Newton’s Laws – Free-Body Diagrams
Draw a sketch.
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.
Resolve vectors into components.
Apply Newton’s second law to each component.
Solve.
Fax = FAcos45.0 = (40N)(0.707) = 28.3 N
Fay = FAsin45.0 = (40N)(0.707) = 28.3 N

19. 4-7 Solving Problems with Newton’s Laws – Free-Body Diagrams
When a cord or rope pulls on an object, it is said to be under tension, and the force it exerts is called a tension force.
When a flexible cord pulls on an object, the cord is said to be under tension. The force it exerts on the object is the tension Ft. If the cord is really light in mass, then the force is transmitted to each adjacent piece of cord along the entire length. If m= 0 then the sum of F is 0 so Ft =-Ft.

20. Two boxes, A and B, are connected by a lightweight cord and are resting on a smooth (frictionless) table. The boxes have masses of 12.0 kg and 10.0kg. A horizontal force FP of 40.0N is applied to the 10.0-kg box, as shown. Find (a) the acceleration of each box, and (b) the tension in the cord connecting the boxes.

21. 4-7 Applications Involving Friction, Inclines
On a microscopic scale, most surfaces are rough. The exact details are not yet known, but the force can be modeled in a simple way.
For kinetic – sliding – friction, we write:
So far we have ignored friction but it must be taken into account for any engineering problem and is actually really important. Friction exists between everything even the smoothest surfaces. Exactly what happens at the microscopic level is not exactly understood. It may the attractive forces between atoms that make a tiny weld. Sliding friction is called kinetic friction. This force acts in the opposite direction from the velocity. The size of the force depends on the roughness of the two surfaces and is proportional to the normal force that either object exerts on the other. The friction forces on hard surfaces doesn’t depend very much on the total surface area of contact (i.e. the friction force on the book is almost the same on the flat part or the side) So we consider a simple model that is independent of area. This is not a vector equation.
is the coefficient of kinetic friction, and is different for every pair of surfaces.

22. 4-7 Applications Involving Friction, Inclines
The coefficients of friction are independent of speed and area of contact.

23. 4-7 Applications Involving Friction, Inclines
Static friction is the frictional force between two surfaces that are not moving along each other. Static friction keeps objects on inclines from sliding, and keeps objects from moving when a force is first applied.
This force is parallel to the two surfaces and arise when they are not sliding. Lets say I try to move this desk. It isn’t moving because there is a force exerted that keeps it from moving. This force it hes static friction exerted by the floor on the desk. If I push hard enough the desk will start to move and the kinetic friction takes over. You may have noticed that it is easier to keep something moving that starting which is why us is generally greater that uk.

24. 4-7 Applications Involving Friction, Inclines
The static frictional force increases as the applied force increases, until it reaches its maximum. Then the object starts to move, and the kinetic frictional force takes over.
This is a graph of the force of friction applied to an object initially at rest. At first, as the applied force is increased, the force of static friction increases linearly to just match it until the applied force equals usFN. If the applied force increase more, the object begins to move and the friction force drops to a roughly constant value of kinetic friction.

25. 4-7 Applications Involving Friction, Inclines
An object sliding down an incline has three forces acting on it: the normal force, gravity, and the frictional force.
The normal force is always perpendicular to the surface.
The friction force is parallel to it.
The gravitational force points down.
If the object is at rest, the forces are the same except that we use the static frictional force, and the sum of the forces is zero.
Last, now let’s look at objects sliding down an incline. Now gravity is involved as an acclerating force.

26. The skier below has just begun descending the 30 degree slope
The skier below has just begun descending the 30 degree slope. Assuming a coefficient of kinetic friction is 0.10, calculate (a) her acceleration and (b) the speed she will reach after 4.0 s.

27. 4-9 Problem Solving – A General Approach
Read the problem carefully; then read it again.
Draw a sketch, and then a free-body diagram.
Choose a convenient coordinate system.
List the known and unknown quantities; find relationships between the knowns and the unknowns.
Estimate the answer.
Solve the problem without putting in any numbers (algebraically); once you are satisfied, put the numbers in.
Keep track of dimensions.
Make sure your answer is reasonable.

28. 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.
The frictional force can be written:
(kinetic friction) or (static friction)
Free-body diagrams are essential for problem-solving