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### Dynamics

Chapter 4

Expectations

After Chapter 4, students will:

- understand the concepts of force and inertia.
- use Newton’s laws of motion to analyze situations involving force, inertia, and acceleration.
- understand the meaning of dynamic equilibrium and identify objects or systems that are, or are not, in equilibrium.

Expectations

After Chapter 4, students will:

- recognize the properties of, and perform calculations involving, particular kinds of forces:
- gravitational
- normal
- frictional
- tension

Force and Inertia

Newton’s second law of motion: If an object experiences a net (unbalanced) force, it accelerates in the direction of the force. The magnitude of its acceleration is directly proportional to the magnitude of the force, and inversely proportional to its inertia (mass).

Force and Inertia

Newton’s second law of motion: If an object experiences a net (unbalanced) force, it accelerates in the direction of the force. The magnitude of its acceleration is directly proportional to the magnitude of the force, and inversely proportional to its inertia (mass).

In mathematical terms:

Force and Inertia

Sir Isaac Newton 1642 – 1727

Natural philosopher

(scientist / mathematician);

the original mathematical

physicist, and inventor of

the calculus. He was also

greatly interested in theology.

Force and Inertia

We need more definition of the terms “force” and “mass.”

Force: a push or a pull, which tends to produce acceleration.

Mass: the property of an object which resists acceleration by a force (inertia), and which is determined by the amount of matter that constitutes the object.

Force and Inertia

Mass: a scalar quantity

SI unit: the kilogram (kg) … a basic unit

Force: a vector quantity. We get the dimensions and units of force from Newton’s second law:

dimensions:

SI units:

Forces and Free-Body Diagrams

We said that an object accelerates as a result of the “net” or “unbalanced” force acting on it.

More than one force can act on a body at the same time.

Forces are vectors. We can add them together as vectors to determine the net or total force acting on an object. The net force is simply the vector sum of all the forces acting on an object.

Forces and Free-Body Diagrams

A convenient method for keeping track of the forces acting on an object is to draw a “free-body diagram” for it.

A free-body diagram:

- is drawn “of” a single object, and shows that object.
- shows every force that acts on an object.
- does not show any force that acts on anything besides the object.

Forces and Free-Body Diagrams

Example: a wooden crate is being dragged across a level floor by a rope that makes an angle of 30° with the horizontal.

Forces and Free-Body Diagrams

Resolve forces into x and y components (as needed):

Forces and Free-Body Diagrams

Add the x-direction forces, and also the y-direction ones. The resultant or net force is entirely horizontal, toward the left. The crate accelerates in that direction. If its mass is 17.9 kg:

Equilibrium: Newton’s First Law

Why did we discuss Newton’s second law before discussing his first law?

Because the first law is really a special case of the second.

In words: an object remains at rest, or moves in a straight line with constant velocity, unless it is acted on by a net force.

Equilibrium: Newton’s First Law

Expressed mathematically, the second law:

and the first law:

An object is in equilibrium if the sum of the forces (“net force”) acting on it is zero. Its velocity is constant. It does not accelerate.

Action - Reaction: Newton’s Third Law

If object A exerts a force on object B, Newton’s third law says that object B exerts a force, having the same magnitude and the opposite direction, on object A:

FAB and FBA are an action-reaction pair.

Kinds of Forces

Three fundamental categories of forces:

- Gravitational
- Weak (electromagnetic)
- Strong nuclear

They are listed here in ascending order of strength.

Gravitational Force

As with many other things, we have Isaac Newton to thank for the first mathematical description of the operation of the gravitational force.

Newton’s law of universal gravitation: any two objects exert equal attractive forces on each other. The magnitude of the forces is proportional to the product of the objects’ masses, and inversely proportional to the square of the distance separating them.

Gravitational Force

Newton’s law of universal gravitation (mathematically):

We can make this proportionality into an equation by introducing a constant of proportionality, G:

G is called the universal gravitational constant.

Gravitational Force

Do not confuse G, the gravitational constant, with g, the acceleration due to gravity!

If we use SI units, the value of G is 6.67×10-11 N·m2/kg2.

If one of the masses is the Earth’s mass, and r is the Earth’s radius, the gravitational force is called the weight of an object.

Gravitational Force

We can use this expression for the gravitational force to calculate the value of g, the acceleration due to gravity at the Earth’s surface.

Ordinarily, we express an object’s weight as:

Normal Force

The atoms of which objects are made are surrounded by electron clouds. Due to the electromagnetic (“weak”) force, these electron clouds resist being brought close together. Their resistance to being actually merged is extremely powerful.

When objects are in macroscopic contact, each exerts a force on the other to prevent their occupying (macroscopically) the same space at the same time. These forces are called normal forces.

Normal Force

If objects A and B are in contact:

- They exert equal and opposite normal forces on each other. , in accordance with Newton’s third law.
- The directions of the forces are perpendicular to the contact surface. (“Normal” is another way of saying “perpendicular.”)
- The magnitude of the forces is as large as necessary to prevent the objects from merging.

Normal Force

Example: a box rests (equilibrium) on a table.

Since the box is in equilibrium

(does not accelerate), the sum of

the forces acting on it must be zero.

The magnitude of the normal force

on the box is equal to its weight.

Apparent Weight (Nonequilibrium)

Frame of reference: another name for a coordinate system

A coordinate system that does not accelerate is also called an inertial reference frame.

What happens in a frame of reference that does accelerate (a non-inertial frame)?

Apparent Weight (Nonequilibrium)

Consider an elevator car that can accelerate up or down … or can move with constant velocity or be at rest.

When it accelerates, it is a non-inertial frame of reference. When it does not accelerate, it is an inertial frame.

We place a scale inside the elevator, and an object on the scale. A scale reports the normal force that it exerts on an object (and that the object exerts on it).

Apparent Weight (Nonequilibrium)

When the elevator moves with

constant velocity and is an inertial

frame …

… the normal force exerted

(and reported) by the scale

is equal to the weight of the

object:

Apparent Weight (Nonequilibrium)

When the elevator accelerates upward …

… the normal force exerted

and reported by the scale

(the object’s “apparent weight”)

increases:

Apparent Weight (Nonequilibrium)

When the elevator accelerates downward …

… the normal force exerted

and reported by the scale

(the object’s “apparent weight”)

decreases:

Frictional Forces

Another force that objects in contact exert on each other is the frictional force.

Two kinds of frictional force:

- Static: when the objects are stationary with respect to each other.
- Kinetic: when the objects move relative to one another.

Frictional Forces

Where do frictional forces come from?

Even objects whose surfaces are macroscopically smooth have microscopic textures that tend to interlock.

Frictional Forces

Frictional forces are exerted parallel to the contact area between objects.

Their direction is such as to oppose motion between the objects.

The static frictional force is as large in magnitude as necessary to prevent motion … but not larger than a calculable maximum magnitude.

Frictional Forces

Once motion between the contacting objects takes place, the frictional force become a kinetic one. The kinetic frictional force between two objects is smaller than the maximum static frictional force.

The magnitude of the frictional force depends on the normal force, and on a coefficient of friction which is determined by the materials in the objects, as well as the condition / texture of their contacting surfaces:

Frictional Forces

Rearranging as a defining equation:

We see that the coefficient of friction is a dimensionless ratio of the magnitudes of two forces.

There are two coefficients of friction for any combination of surfaces, yielding two frictional forces:

Tension Forces

“Tension” means both the forces applied to the ends of a rope, string, belt, ribbon, wire, chain, cable, thread, etc. … and the force that the rope-like object (R-LO) exerts to resist being pulled apart.

The reluctance of R-LOs to be pulled apart means that when you pull on one end, the other end pulls on whatever it may be attached to. We can think of R-LOs as being transmitters of pulling forces.

Tension Forces

If the R-LO has mass in a non-equilibrium situation, the tension force is diminished in transmission:

Important to Remember

If an object or system does not accelerate (a = 0), it is in equilibrium, and

If the object or system does accelerate (a ≠ 0), it is not in equilibrium, and

In either case, Newton’s second law applies. In equilibrium, the right-hand side becomes zero, and the second law becomes the first law.

Important to Remember

Velocity and acceleration are vector quantities. So, Newton’s laws apply separately in the X and Y directions:

Often (as in the case of a projectile), an object is in equilibrium in one direction and not in equilibrium in the other. Both the dynamic and kinematic situations must be considered separately in each axis.

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