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KEY CONCEPTS

NEWTON’S LAWS OF

1. ’S FIRST LAW - STATIC EQUILIBRIUM

Whereas describes motion without regard to the cause of that motion, dynamics specifically studies the relationship between motion and the that result in that motion. The study of dynamics deals with such concepts as why an object moves, and what makes it start or stop moving. The main concept we are concerned with in the study of dynamics is . A force is any kind of push or pull on an object. Thus, a force is required to push a crate or to throw a ball.

In 1687 in his great The Principia, Sir described three rules, known as Newton’s Laws of Motion, that relate the concepts of force and motion. Newton’s first law states:

Every body continues in its state of rest or of uniform in a straight line unless acted on by a nonzero net force.

In other words an object that is at rest remains at rest unless a force pushes or pulls on it. Likewise, an object that is moving will continue to move, without changing its speed or direction, unless a force is exerted upon it. This tendency for objects to resist a change in their motion is called . An object’s is a measure of its inertia. Thus, the more massive an object, the more difficult it is to move or alter the motion of that object. The SI unit of mass is the kilogram (kg). Mass is not the same thing as weight. Mass is an intrinsic property of the object, whereas weight is actually a measure of some force on the object. At first glance, Newton’s first law does not seem to apply to real world objects because they slow down without any apparent force acting on them. Actually, these objects are being acted upon by forces such as and air resistance, which are not readily visible. For simplicity, such forces are often ignored in dynamical studies.

2. NEWTONS SECOND LAW-DYNAMICS OF A SINGLE PARTICLE

Newton’s first law states that a force is required to change an object’s motion, but it doesn’t specifically indicate how that force is related to the object’s . This is provided by Newton’s second law, which states:

KEY CONCEPTS NEWTON’S LAWS OF MOTION The total or net force on an object is equal to its mass its acceleration.

Mathematically, it is given as:

Σ F = ma

This is probably the most important equation in Newtonian mechanics. Notice that both force and acceleration are vectors, whereas mass is a scalar. The net force is the vector sum of all forces acting on an object. The unit of Force is the newton (N). A newton is defined as:

kg⋅ m2 N = s2

In other words, if you push on a 1 kg mass with a force of 1 N, it will accelerate at a rate of 1 m/s2.

3. NEWTON’S THIRD LAW – SYSTEMS OF TWO OR MORE BODIES

Newton’s third law explains where forces come from, and how they are applied. It states:

For every there is an equal and opposite reaction.

In other words, any one object pushes on another, the other object pushes back with a force that is equal in magnitude, but opposite in direction. For example, if you push forward on a crate with a force of 100 N, the crate pushes backward on you with the same force of 100 N. You can never have one force without the other. A common misconception is that action-reaction pairs should cancel each other. In fact, the two forces never cancel out, because they are acting on different objects. If an action is a vehicle pushing a crate, then the reaction is the crate pushing back on the vehicle. The first force causes the crate to move, while the second makes it more difficult to move the crate.

4. EXAMPLES OF VARIOUS FORCES

The weight of an object on the Earth is a measure of the force of gravity on that object. The acceleration due to gravity near the Earth’s surface, denoted as g, is a

2 KEY CONCEPTS NEWTON’S LAWS OF MOTION vector pointing towards the Earth’s surface. Newton’s second law can be used to determine the weight of an object, which is just the force due to gravity:

F = mg g

Notice that the weight of an object depends on both the mass and the acceleration due to gravity. An object sitting on a floor has weight because gravity is pulling down on it. However, the object is not accelerating downward. That is because there is another force, known as the normal force, acting upon it to keep it from accelerating through the floor. This is called the normal force because it is always perpendicular to the surface. The normal force arises because the molecules of the floor and the object resist being pushed together. Its magnitude is always exactly what is required to prevent an object from moving through some surface. The normal force is not a reaction force to gravity. Both gravity and the normal force act on the same object, whereas action reaction pairs always act on different objects.

Friction is a force that resists the motion of an object sliding on a surface. It comes in two common varieties, static and kinetic. Static friction occurs when an object is at rest on the surface. It is the force necessary to prevent an object from sliding, and can vary from zero to some maximum value. The force of static friction is given by the equation:

F ≤ μ F fric s N

where μs is a material-dependent constant known as the static coefficient of friction. Kinetic friction, occurs when an the object is sliding on a surface, and is directed in the direction opposite that of the object’s motion. It is given by the equation:

F = μ F fric k N

where μk is a material-dependent constant known as the kinetic coefficient of friction.

3 KEY CONCEPTS NEWTON’S LAWS OF MOTION It is important to remember that forces are vector quantities and are not limited to a single dimension. Forces exerted within a two-dimensional plane can be broken down into forces acting in the x and y directions:

Σ Fx = max ΣF = ma y y

The x and y components can then be manipulated independently. It is generally also helpful to choose a coordinate system in such a way that an object’s acceleration is directed along one of the coordinate axes. Doing so can simplify calculations.

5. PROBLEM SOLVING STRATEGY

The following steps are useful for solving dynamics problems:

1. Identify all the forces acting on each independent object. If there is more than one object, remember action reaction pairs, but also remember that they act on different objects.

2. Chose coordinate axes that simplify the problem. Generally put the acceleration and as many unknown forces as possible on axis.

3. Break the forces into components and use Newton’s Second law to write equations in the x and y directions.

FNet, x = ∑Fx = max

FNet, y = ∑Fy = may

4. Insert any 0 values.

5. Solve the equations for the unknown quantities.

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