Science Vault - Year 11 HSC Physics

8.4 - Moving About

8.4.2 - Forces and Acceleration

forces / freebody force diagrams / newton's laws / mass and weight

Forces

When we say that there is a force on an object we mean that there is a push or a pull on the object. Some outside agent is placing this push or pull on the object. Forces change the velocity of objects or cause deformation.

In the metric system forces are measured in units of Newtons (N). This unit is named after Isaac Newton.

Forces are vectors. That is, they are quantities that have both size and direction. Forces can be symbolized with arrows, as can all vectors. The arrow in the picture below is meant to show a force with the size of 15 Newtons directed toward the left.

More than one force can act on an object at once. For example, two people could push on a book at the same time. One person could push toward the left and the other could push toward the right. In this case the two forces would act against each other. If the force toward the left was greater than the force toward the right, then the prevailing force would be toward the left. The strength of this prevailing force would be the difference between the strengths of the two separate forces. In this prevailing force is called the net force. Basically, the word net means total.

For example, if one person pushed the book with a force of 20 Newtons toward the left, and the other person pushed the book with a force of 18 Newtons toward the right, then that the net force would be 2 Newtons in size and directed toward the left. As you can see, the net force is aimed toward the left. This object will move as if it had a single force of 2 Newtons pushing on it toward the left.

Actually, forces which are aimed to the right are usually called positive forces. And forces which are aimed to the left are usually said to be in a negative direction. So, to be more accurate, in the above diagram we should call F1 a negative force, that is, negative 20 Newtons, and we should call F2 a positive force, or positive 18 Newtons. This is shown in the following diagram. Notice that the net force is aimed to the left; so, it is a negative force.

Now, perhaps the two people pushing on the book each push with the same strength, again one person pushing toward the left and the other pushing toward the right. In this case the two forces would exactly cancel each other out. The net force on the book would be 0 Newtons. Here is a picture of that situation:

When two or more forces cancel each other out to create a net force of 0 as in the above diagram, we say that the forces on the object are balanced. We also say under these conditions that the forces on the object are in equilibrium. Under such conditions the object will not accelerate. That is, the velocity of the object will not change. Also, consider the two people to be pushing the book both in the same direction. If one person pushed to the right with a force of 8 Newtons, and the other person also pushed to the right with a force of 6 Newtons, then the net force on the book would be 14 Newtons toward the right. In this case the two forces would complement each other, rather than tending to cancel each other. Here is a picture of that:

This object will move as if it had a single force of 14 Newtons pushing on it toward the right. The above three situations have all been one-dimensional. When the only directions considered are to the left or to the right, then the problem can be thought of as operating on a horizontal number line, usually called the x-axis. Such situations occurring along one line are called one-dimensional. One-dimensional problems may also be vertical. In our context here we might consider only forces that act up or down. That would be a one-dimensional problem, also. It would be operating on a vertical number line, usually called the y-axis. However, if several forces act upon one object they need not act only along one line. For example, an object could be pushed toward the right by one force and upward by another. In this case the net force would be neither simply horizontally toward the right nor simply vertically upward. It would be aimed in a slanted direction upward and to the right. Nor would the size of this net force simply be the arithmetic sum of the sizes of the two other forces. In this situation one would use vector mathematics to calculate the net force.

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Free-Body Force Diagrams

Free-body diagrams are diagrams used to show the relative magnitude and direction of all forces acting upon an object in a given situation. The size of the arrow in a free-body diagram is reflective of the magnitude of the force. The direction of the arrow reveals the direction in which the force acts. Each force arrow in the diagram is labelled to indicate the type of force. It is customary in a free-body diagram to represent the object by a box and to draw the force arrow from the centre of the box outward in the direction in which the force is acting. One example of a free-body diagram is shown on the left. The free-body diagram on the left depicts four forces acting upon the object. Objects do not always have four forces acting upon them. There will be cases in which the number of forces depicted by a free-body diagram will be one, two, or three. There is no hard and fast rule about the number of forces which must be drawn in a free-body diagram. The only rule for drawing free-body diagrams is to depict all the forces which exist for that object in the given situation. Thus, to construct free-body diagrams, it is extremely important to know the types of forces. If given a description of a physical situation, begin by using your understanding of the force types to identify which forces are present. Then determine the direction in which each force is acting. Finally, draw a box and add arrows for each existing force in the appropriate direction; label each force arrow according to its type.

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Newton's Laws

Newton’s First Law

This law is also called the Law of Inertia or Galileo's Principle.

'An object at rest will remain at rest unless acted on by an outside, external force.'

An object may be acted upon by many forces and maintain a constant velocity so long as these forces are balanced. For example, a rock resting upon the Earth keeps a constant velocity (in this case, zero) because the downward force of its weight balances out the upward force (called the normal force) that the Earth exerts upwardly on the rock. Only unbalanced forces induce acceleration, or a change in the velocity or an object. If you push someone, he or she will accelerate in the direction of the unbalanced force that you have provided (called the applied force). Likewise if you roll a ball along the floor, the unbalanced force of friction will decelerate the ball from some positive velocity to rest.

Before Galileo, people agreed with Aristotle that a body's natural state was at rest, and that movement needed a cause. This is understandable, since in everyday experience, moving objects eventually stop because of friction (except for celestial objects, which were deemed perfect). Moving from Aristotle's "A body's natural state is at rest" to Galileo's discovery was one of the most profound and important discoveries in physics.

There are no true examples of the law, as friction is usually present, and even in space gravity acts upon an object, but it serves as a basic axiom for Newton's mathematical model from which one could derive the motions of bodies from elementary causes: forces. Another way to put it is, "An object in motion tends to stay in motion and an object at rest tends to stay at rest until a force acts upon it".

Newton’s Second Law

‘The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction.’

Newton's second law as originally stated in terms of momentum, p, is

'An applied force is equal to the rate of change of momentum'.

The physical meaning of this equation is that objects interact by exchanging momentum, and they do this via a force. You will learn more about this in the next section on momentum.

For our purposes now, when the mass, m, of the object is constant, the relation p = mv gives another useful form of the second law:

F = ma

where

a = v-u / t is the acceleration of the object (i.e., the rate of change of its velocity) in metres per second squared (ms-2)
m = mass in kilgrams (kg)

For example, if a bowstring exerts a constant force of 100 N on an arrow having a mass of 0.10 kg, then the arrow's acceleration will be 1000 ms-2 until it leaves the bow (after which the arrow will stop speeding up).

In these equations, F is the net force, i.e., the sum of all the forces acting on the object. When the forces on the object all act along the same line, they can be added as positive and negative numbers, depending on their direction. When they do not all act along the same line, the total must be found by vector addition.

The quantity m, or mass, is a characteristic of the object. The greater the total force acting on an object, the greater the change in its acceleration will be. This equation, therefore, indirectly defines the concept of mass. In the equation, F = ma, a is directly measurable but F is not. The second law only has meaning if we are able to assert, in advance, the value of F. Rules for calculating force include Newton's law of universal gravitation, Coulomb's law, and other principles.

Said simply, an object needs a net force if it is to accelerate. The size of the acceleration is directly proportional to the force and inversely proportional to the mass.

Newton’s Third Law

‘All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.’

As shown in the diagram above, the skaters' forces on each other are equal in magnitude, and opposite in direction. Although the forces are equal, the accelerations are not: the less massive skater will have a greater acceleration due to Newton's second law. If a basketball hits the ground, the basketball's force on the Earth is the same as Earth's force on the basketball. However, due to the ball's much smaller mass, Newton's second law predicts that its acceleration will be much greater. Not only do planets accelerate toward stars; but, stars accelerate toward planets.

The two forces in Newton's third law are of the same type, e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing against the road.

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Mass and Weight

"Weight" is often used as a synonym for mass. For instance, when we buy or sell goods "by weight", we are in principle interested in the amount of goods exchanged, not how hard it presses down on the table. Similarly, in measurements of body weight we are primarily interested in the amount of tissue (fat, muscle, etc.) present. Under most circumstances, this ambiguity is not a problem, because the weight of an object is directly proportional to its mass, and the constant of proportionality - the strength of the gravitational field -- is approximately constant everywhere on the surface of the Earth (around 9.8 m s-2). Correspondingly, weight is often given in kilograms and other units of mass.

In the physical sciences, people are usually more careful about the distinction between weight and mass. For instance, a body will have a smaller weight if it is located on the Moon than if it is on the Earth, since the gravitational field of the Moon is weaker; its mass, on the other hand, does not depend on position. Although terms such as "atomic weight", "molecular weight", and "formula weight" may still be encountered, such usage is often discouraged; terms like atomic mass are used instead.

Weight as a Force

The SI unit for weight is the newton (N), or kilogram metres per second squared (kg m s-2).

The weight force that we sense is actually the normal force exerted by the surface we stand on, which prevents us from being pulled to the centre of the Earth and not, the weight itself. This normal force, that we can call the apparent weight is the one that is measured by a weighing scale, not the weight itself. A good evidence of this is given by the fact that a person moving up and down on his toes does see the indicator moving, telling that the measured force is changing while his weight, that depends only on his mass, the Earth mass and the distance between his centre of mass and the centre of Earth obviously do not change.

In contrast, in free-fall, there is no apparent weight because we are not in contact with any surface to provide such a normal force. The experience of having no apparent weight is known as weightlessness or microgravity.

Comparative Weights on Bodies of the Solar System

The following is a list of the weights of a mass on some of the bodies in the solar system, relative to its weight on Earth:

Mercury = 0.378
Venus = 0.907
Earth = 1
Moon = 0.166
Mars = 0.377
Jupiter = 2.364
Saturn = 1.064
Uranus = 0.889
Neptune = 1.125
Pluto = 0.067

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download worksheet 8.4.2.B - forces

download worksheet 8.4.2.C - newton's laws

download worksheet 8.4.2.D - miscellaneous force problems

download worksheet 8.4.2.E - weight

download experiment 4 - newton's second law

 
References

Internet References

  • Under constructtion.

Syllabus References

These references relate to the content covered on this page and can be found in Section 8.4.2 of the syllabus.

1. An analysis of the external forces on vehicles helps to understand the effects of acceleration and deceleration.

Students learn to:

  • Describe the motion of one body relative to another.

  • Identify the usefulness of using vector diagrams to assist solving problems.

  • Explain the need for a net external force to act in order to change the velocity of an object.

  • Describe the actions that must be taken for a vehicle to change direction, speed up and slow down.

  • Describe the typical effects of external forces on bodies including:

    – friction between surfaces
    – air resistance.

  • Define average acceleration as:



    therefore

  • Define the terms ‘mass’ and ‘weight’ with reference to the effects of gravity.

  • Outline the forces involved in causing a change in the velocity of a vehicle when:

    – coasting with no pressure on the accelerator.
    – pressing on the accelerator.
    – pressing on the brakes.
    – passing over an icy patch on the road.
    – climbing and descending hills.
    – following a curve in the road.

  • Interpret Newton’s Second Law of Motion and relate it to the equation:

  • Identify the net force in a wide variety of situations involving modes of transport and explain the consequences of the application of that net force in terms of Newton’s Second Law of Motion.

Students:

  • Analyse the effects of external forces operating on a vehicle.

  • Gather first-hand information about different situations where acceleration is positive or negative.

  • Solve problems using vector diagrams to determine resultant velocity, acceleration and force.

  • Plan, choose equipment or resources and perform first-hand investigations to gather data and use available evidence to show the relationship between force, mass and acceleration using suitable apparatus.

  • Solve problems and analyse information using:



    for a range of situations involving modes of transport.

Textbook References

Taken from:

Heffernan, D., Parker, A., Pinniger, G. & Harding, J. (2002) Physics Contexts 1, Pearson Education, Melbourne

  • Sections x and x on pp. x - x