Physics 20 - STA

Note Booklet

Unit 2 – Dynamics Chapter 3 – Forces

Name: ______________________________________________________________________________

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USE THE FORCE

Equation (F=ma) young padawan.

Table of Contents

3.1 – The Nature of Force (Dynamics, Types of Forces, and Free-Body Diagrams

3.2 – Newton’s First and Second Law

3.3 - Multi-Force Analysis

3.4 – The Force of Friction

3.5 – Newton’s Third Law

3.6 – Application Problems (Elevators, Pulleys, and Inclined Planes)

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3.1 – The Nature of Force

Previously, we have studied motion without regard to what caused the motion. We have already studied kinematics. Dynamics, on the other hand, is the study of motion as we pay attention to what causes the motion. Dynamics is the study of forces.

A force is defined as a push or a pull. Force is a vector, as it has a direction associated with it. The typical SI unit used to describe force is the Newton (N). A force acting on an object can have one or both of the following effects on the object:

1. accelerate the object (cause the object to speed up, slow down, or change direction)2. deform the object

Before we continue, it is important to distinguish force from two other quantities: mass, and volume. Mass is defined as the amount of matter possessed by something. Mass is a scalar quantity. The typical SI unit used to describe mass is the kilogram (kg). Volume is the amount of space that something takes up. Volume is a scalar quantity. The typical SI unit used to describe volume is the Litre (L).

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Types of Forces

There are different types of forces and scientists distinguish among them by giving these forces special names. When an object is in contact with another, the objects will have a common surface of contact, and the two objects will exert a normal force on each other. The normal force, FN, is a force that is perpendicular to this common surface. Depending on the situation, another force called friction, Ff, may be present, and this force acts parallel to the common surface.

The adjective “normal” simply means perpendicular. Figure 3.9 (a) shows a book at rest on a level table. The normal force exerted by the table on the book is represented by the vector directed upward. If thetable top were slanted and smooth as in Figure 3.9 (b), the normal force acting on the book would not be directed vertically upward. Instead, it would be slanted, but always perpendicular to the contact surface.

A free-body diagram is a powerful tool that can be used to analyze situations involving forces. This diagram is a sketch that shows the object by itself, isolated from all others with which it may be interacting. Only the force vectors exerted on the object are included and, in this physics course, the vectors are drawn with their tails meeting at the centre.

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A stationary object may experience an applied force, Fapp, if, say, a person pushes against the object (Figure 3.10). In this case, the force of friction acting on the object will oppose the direction of impending motion.

Since force is a vector quanity, its direction is important. This is especially important if two or more forces are acting on an object.

The net force may be found using vector additionFnet = F1 + F2 + F3+…..

(Directions must be accounted for .. i.e. + and – signs)

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ExampleA car with a weight, Fg, of 10 000 N [down] is coasting on a level road. The car experiences a normal force, FN, of 10 000 N [up], a force of air resistance, Fair, of 2500 N [backward], and a force of friction, Ff, exerted by the road on the tires of 500 N [backward]. Draw a free-body diagram for this situation.

ExampleDraw a free body diagram of a 20 kg object that is sliding along a horizontal surface, as it is pushed with an applied force of 30 N and is resisted by a frictional force of 10 N.

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3.2 - Newton’s First and Second Law

First Law

Inertia is defined as the tendency of an object to stay in motion. Every object that is moving has a tendency to stay in motion, in a straight line, at a constant speed. Every object that is at rest has a tendency to remain at rest. In order to change an object’s tendency, an unbalanced force must be applied.

The Law of Inertia (Newton’s First Law): An object at rest will remain at rest will remain at rest until acted upon by an unbalanced force, and an object in motion will continue in motion in a straight line at a constant speed until acted upon by an unbalanced force.

This of course implies that an object that remains at rest must have zero net, or total, force acting on it. It is said to be in static equilibrium. If there were a net force, it would begin to move. Because force is a vector, two forces of the same magnitude acting in opposite directions can balance each other to produce zero net force. An object at rest therefore does not necessarily have zero force acting on it, just no net force acting on it.

Similarly, an object is moving at a constant velocity must have zero net force acting on it. It is said to be in translational equilibrium. If there were a net force, it would speed up, slow down, change direction, or deform. Again, two forces of the same magnitude acting in opposite directions can balance each other to produce zero net force. An object moving at a constant velocity therefore does not necessarily have zero force acting on it, just no net force acting on it.

Fnet = 0 (If the object is stationary or moving at a constant rate)

When you are in a moving car, you can feel the effects of your own inertia. If the car accelerates forward, you feel as if your body is being pushed back against the seat, because your body resists the increase in speed. If the car turns a corner, you feel as if your body is being pushed against the door, because your body resists the change in the direction of motion. If the car stops suddenly, you feel as if your body is being pushed forward, because your body resists the decrease in speed.

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Second Law

Recall that the acceleration of an object is defined as the rate of change of an object’s velocity. This rate of change depends on two things: the net force acting on the object, and the mass of the object. The larger the net force acting on an object, the greater the acceleration. The larger the mass of the object, the smaller the acceleration.

Newton’s Second Law: The acceleration of an object is directly proportional to, and in the same direction as, the net force acting on the object. The acceleration of an object is inversely proportional to the mass of the object.

Newton’s second laws can be stated mathematically as a= F

m , although it is typically rearranged to take the form F=m a .

ExampleA 100 kg car accelerates at a rate of 2.0 m/s2. What is the net force required to do this?

ExampleA 100 kg car moving at 72 km/h slows to a stop in 10 s. How much force was required to stop the car?

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Multi-Force ProblemsRecall that the net force acting on object is the total force acting on the object (the vector sum of all forces acting on the object). When the net force results from only one force, F=m acan simply be used to analyze the problem. When the net force results from two or more forces, however, a slightly more in-depth analysis must be performed. The following are the steps of that analysis:

1. Draw a FBD. Cross off all forces that balance each other. If the only forces remaining are all in one dimension, this method will work. If they are in two difference dimensions, another method to be described later will have to be used.

2. Fnet is the sum of the forces, therefore Fnet=F A+F B+. . . (all of the forces that are not crossed off).

3. Fnet is also equal to the product of an object’s mass and acceleration (Newton’s second law),

therefore m a=F A+FB+. ..4. Plug in the values for the known variables, and solve. Remember that the accleration and forces

are vectors, therefore accelerations and forces that act to the left (or down, South, etc.) will have a negative value and accelerations and forces that act to the right (or up, North, etc.) will have a positive value.

Example

A 50 kg box is pushed to the right with a force of 150 N by one person. It is pushed to the left with a force of 320 N by a second person. A 30 N force of friction acts on the box as well. Calculate the acceleration of the box.

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3.3 - Multi-Force Analysis

Mass vs. Weight

To understand the differences we need to compare a few points:

Mass is a measurement of the amount of matter something contains, while Weight is the measurement of the pull of gravity on an object.

The Mass of an object doesn't change when an object's location changes. Weight, on the other hand does change with location. 

To calculate the weight : Fg = ma or Fg = mg

Example

Calculate the weight of a 40.0 kg thumbnail.

Multi-Force Problems Involving Vector Components

If a force contributes to the motion of an object but the acceleration is not entirely in the direction of the force, the component of the force in the direction of the acceleration must be found. The problem can then be treated as any other multi-force problem.

To find the component of the force in the direction of the acceleration, follow the steps outlined in your kinematics notes for finding vector components.

ExampleA 500kg wagon is pulled by a force of 20N North and by 30N East.

a) What is the net force (magnitude and direction)?

b) If the force of friction is 8.0N, calculate the acceleration.

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Example

If a 10kg ball is being thrown up with an applied force of 5000N:

a) Draw a FBD of what is happening

b) Write an Fnet equation.

c) Calculate the acceleration of the ball being thrown.

Example

If a 20kg box is moving 2.0 m/s to the right at a constant rate, what is the applied force if the force of friction is 13.4 N?

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3.4- The Force of FrictionThe force of friction is a force that opposes the motion between two surfaces that are in contactThere are two types

STATIC friction: opposes the start of motion between two surfaces that are not in relative motion.

KINETIC friction: the force between two surfaces that are in relative motion. It always acts in the opposite direction to the motion of the object.

Static friction is ALWAYS bigger than kinetic friction

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Friction is directly proportional to the weight of an object (FN)

The coefficient of friction is a number (no units) that depends on the two surfaces in contact. The larger the number is, the more “sticky” the situation. So, a very slippery surface (ice) would have a low coefficient of friction.

ExampleA 12 kg piece of wood is placed on a floor. There is 35 N of maximum static friction measured between them. Determine the coefficient of friction between the piece of wood and the floor.

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“mew”

0.06 – 0.20 0.8 – 1.2 1.0 – 1.6

Example

What frictional force must be overcome to start a 50 kg object sliding across a surface with a static coefficient of friction equal to 0.35?

Example

A 10 kg box is dragged over a horizontal surface by a force of 40 N. If the box moves with a constant speed of 0.5 m/s, what is the coefficient of kinetic friction for the surface?

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3.5 Newton’s Third Law

Consider any two objects, A and B, interacting with each other:

When object A exerts a force on object B, object B always exerts an equal but opposite force on object A.

The above statement is known as Newton’s Third Law.

It is important to note that the net force is not necessarily zero here. The forces do not balance or cancel each other out because they are each acting on different objects. A FBD of object A would look like the following:

Object A FBA

While a FBD of object B would look like the following:

FAB Object B

It is entirely normal for one of the two objects to experience a greater acceleration than the other. Remember that the forces acting on each one are equal, but the masses are not necessarily equal.

Therefore, since a= F

m , the heavier object will not accelerate at as great of a rate as the lighter object. It is also possible that one of the two objects is attached to the ground or some other fixed object. In this case, the attached object would not move unless the Newton III force acting on it was large enough to overcome the attachment force (or force of static friction).

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Example

A 75 kg man and a 65 kg man push against each other as they stand on a horizontal, frictionless surface. If the 75 kg man accelerates at a rate of 1.5 m/s2, at what rate does the 65 kg man accelerate?

Example

Two people, one of mass 50 kg and the other of mass 60 kg, are pushing against each other on a horizontal, frictionless surface. If the first person accelerates from rest to 3.0 m/s in a time of 4.0 seconds, what is the acceleration of the second person?

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3.6 Applications of Newton’s Laws

Elevators

An elevator problem is just like any other multi-force problem. In an elevator problem, there will always be the same two forces drawn on the FBD: the force of gravity (acting down) and the normal force (acting up). The issue that sometimes makes elevator problems more difficult for students to solve is knowing which variable to solve for. There are several things that can be asked for in an elevator problem, but they all boil down to a few common variables:

Asked For: Solve For:actual weight force of gravity (Fg)normal force normal force (FN)apparent weight normal force (FN)force of floor normal force (FN)scale reading normal force (FN)

Example

A 73 kg man stands in an elevator that accelerates downward at a rate of 2.0 m/s2.

a. With what force does the elevator floor push him up?

b. What is the man’s actual weight?

c. What is the normal force acting on the man?

d. What is the man’s apparent weight?

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Pulley Problems

Sometimes two or mare masses are joined to each other by a rope that goes around a pulley. The analysis of a pulley problem is very much like the method previously learned for analyzing multi-force problems, with one addition: the system should be “straightened-out” before a FBD is drawn. This means that all forces should be treated as if they are acting left or right.

For example, the following:

m1

m2

Fg1

Fg2

would become:

m1 m2

Fg1 Fg2

Once the system is straightened out and the FBD is drawn, the problem is like any other multi-force problem. Note that in the above example, Fg1 is equal to m1g while Fg2 is equal to m2g. The mass that is used for the net force is the entire mass of the system (m1 + m2 in this case) because the entire mass is being accelerated. Also note that in the above example, Fg1 is negative because in the FBD it is drawn to the left, while Fg2 is positive because in the FBD it is drawn to the right.

Example

m= 2.0 kg

FF= 20 N

What is the acceleration of the system? m= 10 kg

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Inclined Planes

An object on a hill experiences at least two forces (the normal force and a force of gravity) that do not cancel each other out. A FBD must therefore be drawn. The problem can then be “straightened-out” and analyzed as a multi-force problem.

To “straighten-out” the problem, the force acting parallel to the hill must be found. Call this force FH. This is the force that acts to pull the object down the hill, and is really just a combination of the normal force and the force of gravity acting on the object.

FH FN

Fg

FH

The angle of incline of the hill () will always be the same as the angle between the force of gravity and the normal force acting on the object. This can be justified by using trigonometry. The resultant vector can be found using the angle, , and a trigonometric function (typically the sine function). The resultant vector is the force that acts on the object parallel to the hill, FH.

Now redraw the object horizontally with FH instead of Fg and FN. The diagram above “straightened-out” horizontally looks like the following:

FH

Don’t forget to include any other forces that act on the object, such as an applied force or force of friction. Now solve the problem as any other multi-force problem.

Example

If a 5.0 N force of friction acts to oppose the motion of the block in the diagram below, what is the acceleration of the block?

F= 20 N

m= 2.0 kg

= 300

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