SPH 3UI Forces

Lesson 1 – Dynamics & Force

Up until now, we have mainly observed motion and the way it happens. We’ve successfully described position, velocity and acceleration. But we never asked why motion happens in the first place. If kinematics explains the how of motion, dynamics certainly explains the why.

During this unit, we are going to pay attention to reasons behind motion and consider a discussion of forces. Nearly all of what we will study is based upon the work of a young scientist, who at the time he developed his laws of motion was not much older than you. The World According to Newton Isaac Newton was just 18 when he started at Cambridge University. While a student, he postulated three laws of motion: Law 1: Law 2: Law 3: We’ll investigate each of these laws in turn, but we are actually getting ahead of ourselves. Consider what the word force means to you. Using words or a picture, explain your meaning.

SPH 3UI Forces

Our textbook defines force as: There are many types of forces:

According to your definition of force, and that of the textbook, determine if there is a force acting on the objects specified. By adding to the picture or by using words, explain your answers.

The person is trying to move a stalled car but the car is not moving. Is there a force on the car?

The bicycle and rider are slowing down as they coast along a flat road. There is no pedaling or braking. Is there a force on the bicycle?

A golf ball flies through the air after being struck by the club. Is there a force on the golf ball?

Good to Know: Forces can be classified as CONTACT forces or NON-CONTACT forces. A book in direct contact with a table is experiencing a contact force because of gravity. Two magnets attract each other with distance between them, no contact is involved.

SPH 3UI Forces

A policeman is moving a demonstrator. Is there a force on the demonstrator?

Newton’s First Law

All objects will remain in a state of rest or continue to move with a constant velocity unless acted upon by an unbalanced force.

This law describes the two possible conditions of an object if it is left alone: either it stays put (at rest) or it continues moving with a constant velocity. The property this law refers to is inertia, which is the ability of an object to resist changing its motion. The object’s mass is a measure of this property. Some examples of Newton’s 1st Law:

Homework:

1. Read Pg 122 Force and Motion and answer # 2, 3. 2. Pg 143 # 1, 2 and 3

SPH 3UI Forces

Lesson 2 – Newton’s 2nd Law

The acceleration of an object depends inversely on its mass and directly on the unbalanced force applied to it.

Mathematically: And we arrive at our equation for Newton’s 2nd Law, Here lies the connection between force and motion. This law tells us that an unbalanced force on an object will cause it to accelerate. The magnitude of the acceleration, however, depends not only on the force applied, but also on the mass of the object. Consider the following: Case 1 – normal baseball Case 2 – lead baseball

SPH 3UI Forces

In order for both balls to have the same acceleration, you would have to apply a far greater force to the lead ball, because of it much larger mass.

The units of force are kg ⋅ms2

. This quantity has been simplified and given a

derived SI unit called a Newton.

1N ≡1kg ⋅ms2

Example 1 Calculate the net force acting on a 20 kg object if the acceleration is 0.28 m/s2.

Example 2 A skydiver, accelerating at 9.8 m/s2, experiences a force of 1000 N. What is the skydiver’s mass?

Example 3 A bicycle of mass 14.6 kg and a rider of mass 50 kg generate a force of 12 N. How fast are they going after 2.0 s?

SPH 3UI Forces

Lesson 3 – Free Body Diagrams A Free Body Diagram (FBD) is a powerful tool for analyzing force problems. The FBD isolates the object under inspection from its environment, and provides an analysis of the forces acting on it. In the situation below two horses are pulling carriage. A FBD of the carriage would looks as follows. From Newton’s 2nd Law we know that

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F net = m

a . If there is more than one

force acting on the object in the FBD (which is usually the case),

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F net =

F acting on object∑

Where the symbol “ Σ ” means “sum of” and is used to indicate all of the forces in the diagram. In this example, we should note that the forces pulling in opposite directions were unequal. Had the forces been the same, they would have ‘canceled out’ and the toy would not have moved. Since the forces are unbalanced, they cause the toy to accelerate.

Example 1 Two children pull on a toy, one with a force of 2.0 N [E], the other with a force of 1.5 N [W]. What is the net force acting on the toy? Draw a FBD.

Example 2 Find the acceleration of the toy from Example 1, if it has a mass of 50 g.

SPH 3UI Forces

Example 3 Find the net force for each FBD shown below. a) b)

Example 4 A race car can accelerate from rest to 97 m/s in 6.2 s. If the car has a mass of 800 kg and its engine generates a driving force of 16 000 N [forward], find the force of friction on the car.

3 N

2 N

2 N

6 N 2 N

1 N

1 N

2 N

5 N

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FBD Worksheet Find the NET force in each case. 1. 2. 3. 4. 5. Answers: 1. 2 N [E] 2. 3 N [S] 3. 4.2 N [NE] 4. 7.1 N [E 8.1° N] 5. 9.8 N [S 66° E]

3 N

3 N

4 N 2 N

10 N

10 N

5 N

5 N 3 N

3 N

2 N

3 N

2 N

9 N 2 N

4 N

3 N

6 N

6 N

6 N

6 N

3 N

7 N

18 N 9 N

SPH 3UI Forces

For every action force, there is an equal and opposite reaction force.

Lesson 4 – Newton’s 3rd Law

The third law of motion, unlike the second law, has no direct correlation to mathematics. There isn’t an equation to describe it specifically. However, understanding the third law is of great importance in order to continue to study dynamics. Consider some action-reaction pairs:

Consider this: “if action forces cause equal and opposite reaction forces, then why don’t all forces cancel out and why does anything happen at all?” This is true, of course, for the pair. But the answer lies in the free body diagrams for each object. Consider the example of a hockey player pushing against the boards.

Walking

Apple on a branch

Swimming

Firing a rifle

Pushing the boards on skates

Space Shuttle lift off

SPH 3UI Forces

For the boards, the forces are balanced, and thus, there is no motion. But for the player, the push from the boards is the only external force and the motion of the player is in the same direction as the applied force. In essence, action-reaction forces can never cancel each other out since they act on different objects.

Example 1 A student sits on a chair with another student pressing down on her shoulders. Draw FBDs for the person on the chair, the chair and the ground. For each FBD, list the forces on the diagram and explain what is happening according to Newton’s 3rd law.

SPH 3UI Forces

Lesson 5 – Gravity Likely you have heard the story of Newton sitting under a tree and an apple falling from above, hitting him on the head. The story goes that this event started Newton thinking about why the apple fell, and lead to his derivation of the law. According to Newton, Gravity is one of the fundamental forces of the universe. It acts through the property mass and it is what binds the universe together. Mass versus Weight Mass Weight When considering problems where the force of gravity is included, we will employ the useful formula,

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F g = m

g

But where does this equation come from and how do we calculate

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g ? In

addition to Newton’s 3 laws of motion, he also developed the universal law of gravity, which explains planetary orbit, and provides reasoning for why we are stuck to the Earth and countless other phenomenon. Factors Affecting the Force of Gravity Lets say we were to pick up and move to the moon. And we took along our bathroom scale just because. Would the scale measure a different value on the moon than it did on Earth?

SPH 3UI Forces

This tells us that the _______________ of both objects affects the force of gravity. Now, what happens to the force of gravity as you increase your distance from the Earth?

So if we collect these ideas, we will arrive with the following statements:

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Fg ∝m1 Fg ∝m2 Fg ∝1

r2

Putting these three factors together: We have made two assumptions to arrive at this proportionality statement, (1) the objects masses are spherically distributed and (2) the distance between the objects is measured from their centres. In order to turn a proportionality statement into an equation, we need a constant. In this case, the constant is called the universal gravitation constant, and is given the symbol G. The value of G provides a measure of

SPH 3UI Forces

the strength of the force of gravity between any two objects anywhere in the universe. It’s value has been found to be

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6.67×10−11 Nm2 / kg2.

Example 1 Two nervous students are standing at opposite ends of the dance floor, 26.0 m apart. If one has a mass of 75.5 kg and the other, a mass of 66.0 kg, what is the force of gravitational attraction between them?

Example 2 Calculate the force of attraction between a 1.0 kg medicine ball and Earth of mass

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5.98×1024 kg if the object is sitting on the surface. The radius of the Earth is

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6.38×106 m.

SPH 3UI Forces

Lesson 6 – The Normal Force As we saw in lesson 4, Newton’s third law deals with situations involving two bodies. The initiating body creates the action force and the receiving body creates the reaction force. We create two FBDs, one for each object. When the body receiving the action force is a surface, the reaction force of the surface pressing back is called the normal force. As the term “normal” implies, the force is always perpendicular to the surface.

Example 1 For a person of mass 40 kg sitting on a box, what is the value of the normal force?

Example 2 The same person from example 1 is now harnessed to a long rope with a friend who pulls from above with a force of 92 N. What is the value of the normal force now?

SPH 3UI Forces

Lesson 7 – Friction Nearly every mechanical movement involves some type of friction. As such, it is important to study the force of friction if we wish to accurately describe and predict motion. To understand what friction is, we should note that, in general, there is a force of attraction between any given materials. This attraction occurs at an atomic level and gives rise to the amount of stickiness experienced between surfaces. Even smooth or polished materials are rough at an atomic level.

The magnitude of the frictional force is determined by the types of materials in contact and by the normal force exerted by one object on the other.

Combining the two components that make up the frictional force, we obtain the following equation:

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Ff = µ Fn

The Greek letter µ (mu) is called the coefficient of friction. It depends only on the kinds of surfaces in contact. This value has no units because it is a ratio of two forces. You have probably noticed that it takes a greater force to move a stationary object than it does to keep an object in motion. This is because of the fact that there are two types of friction:

static friction

kinetic friction

SPH 3UI Forces

The theory of frictional forces tells us that an object at rest experiences more friction than an object in motion. This is because the atomic attraction between materials is decreased when the object is in motion due to decreased contact time. The two types of friction lead to two different coefficients of friction:

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µs, the coefficient of static friction and

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µk , the coefficient of kinetic friction.

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µs > µk

Both of these coefficients are used in exactly the same way; you substitute them into the equation for friction, but they have physically different meanings and so must be calculated separately and used in their appropriate situations.

Example 1 85 kg of sand is added to the bed of a truck to reduce slipping on ice in the winter. Calculate the increased force of static friction that results from the added sand. The coefficient of static friction between rubber and concrete is 0.70.

Example 2 A hard-working physics student pushes a lawnmower with a mass of 12 kg. She is pushing with a force of 150 N horizontally and 40 N down. The coefficient of kinetic friction between the wheels and grass is 0.9. Find the force of friction acting on the lawnmower.

SPH 3UI Forces

Example 3 Find the acceleration of the lawnmower in the previous problem.

Example 4 You and a friend are pushing horizontally on a crate of mass 50 kg. Your friend is pushing from the right with a force of 50 N, and you are pushing from the left with a force of 80 N. The coefficient of kinetic friction is 0.3. Find the acceleration of the crate.

Example 5 A 0.5 kg block is being slid up a chalkboard with an applied force of 6.0 N upward and 2.0 N inward towards the board. If the coefficient of kinetic friction is 0.4, find the acceleration of the block.