### Newton's Laws and Forces; Gravitation

The central problem of mechanics is this: if we have a particle whose characteristics we knoe (mass, shape, volume, charge, etc.) how can we predict that it will move? Newton answered this question with his laws of motion and his universal law of gravitation. Newton's laws introduced the concept of force and talked about force in terms of acceleration given to an object. The laws introduced the concept of mass which tells us how one body differs from another body and predicts how two bodies in the same environment can have different accelerations. The force laws give us ways to calculate the force acting on a body using the properties of the body and its environment.

Kinematics
the study of how objects move
Dynamics
the study of why objects move
Force
a push or a pull; symbol is F; SI unit is the Newton, or N
One Newton is the force necessary to cause a one kilogram mass to accelerate at the rate of 1 m/s2
1 N = 1 kg m/s2

Four basic forces:

1. Gravitational force   -an attractive force that exists between all objects with mass; an object with mass attracts another object with mass; the magnitude of the force is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between the two objects.
Stated mathematically:
Where G is the universal gravitational constant (meaning it has the same value throughout the universe), m1 and m2 are the masses of the objects in kilograms, and d is the distance between them in meters

G = 6.67 x 10-11 N m2 / kg2

Cavendish found the universal gravitation constant, allowing the earth to be "weighed."
2. Electromagnetic force    -an attractive or a repulsive force between charged particles; when charged particles are in motion, they produce magnetic forces on each other
3. Strong nuclear force    -an attractive force between the particles in the nucleus; it is the strongest of the four forces, but only acts over very small distances
4. Weak nuclear force   -a force involved in the radioactive decay of some nuclei (in the 1960’s, Weinberg theorized the existence of the electroweak force, combing the electromagnetic and the weak nuclear force)

Newton’s Laws of Motion:

• Newton’s First Law – an object in motion remains in motion at constant speed and moving in a straight line and an object at rest remains at rest unless acted upon by an outside force; called the law of inertia. You do not need a force to keep a body moving at a constant velocity. Also, a body at rest in one frame of reference could be moving at constant speed relative to another frame of reference.
• Equilibrium
an object is in equilibrium when its velocity is zero or its velocity is constant
Inertia
a measure of how an object resists changes in motion; it is a measure of an object’s mass

• Newton’s Second Law – An unbalanced force (or net force) causes an object to accelerate; this acceleration is directly proportional to the unbalanced force and inversely proportional to the object’s mass; called the law of acceleration

a = F / m or F = m a

Equilibrium
an object is in equilibrium when no unbalanced force (or net force) acts on it

• Newton’s Third Law – When one object exerts a force on another object, the second object exerts a force on the first object that is equal in magnitude, but opposite in direction; for every action, there is an equal, but opposite reaction; called the law of action-reaction

Applications of Newton's Laws

Free-body diagrams represent forces (their magnitudes and their directions). In a free-body diagram, all forces are represented using arrows. The direction of motion is considered positive (usually assigned the right direction); the direction opposite the motion is negative (typically assigned the left direction); up is positive and down is negative

If the sum of all the horizontal forces acting on an object is zero, then the object is in equilibrium horizontally. If the sum of all the vertical forces acting on an object is zero, then the object is in equilibrium vertically. If the sum of the forces in a direction is not zero, then the forces in that direction are not balanced. We say that an unbalanced force acts in that direction. An object can be in equilibrium in one direction and not in equilibrium in another direction.

Forces that act horizontally are independent of forces that act vertically.

To work free-body diagram problems:

1. Draw the free-body diagram labeling all forces (their magnitudes and directions). Remember to use the appropriate positive and negative signs.
2. Add all the forces in the vertical direction. If the sum is zero, the forces are balanced, and there is no acceleration. If the sum is not zero, the forces are unbalanced, and there is acceleration in the vertical direction. This sum equals the product of mass times acceleration.
3. Add all the forces in the horizontal direction. If the sum is zero, the forces are balanced, and there is no acceleration. If the sum is not zero, the forces are unbalanced, and there is acceleration in the horizontal direction. This sum equals the product of mass times acceleration.

Common types of forces involved in motion:

1. Weight – a measure of the gravitational force acting on an object; direction is down (toward the earth’s center); symbol is W
2. W = m g

Where W is the weight of the object in Newtons, m is the mass of the object in kilograms, and g is the acceleration due to gravity.

Letting the gravitational force equal represent the weight of an object yields,
W = (G m1 m2) / d2
or    m1 g = (G m1 m2) / d2
or     g = (G m2) / d2
where m2 is the mass of the earth

3. Normal force – the force exerted by a surface to support an object; symbol is FN; direction is always upward; when an object rests on a horizontal surface, the normal force equals the object’s weight; when an object is being pushed or pulled by a horizontal force, the normal force equals the object’s weight. When a push or a pull is something other than horizontal, a free-body diagram is used to determine the normal force (taking into account the vertical component of the push or the pull). A normal force is a "response" force. In other words, a surface responds to a weight resting on it by acting to oppose it to the extent necessary to "cancel out" these forces. For our purposes, normal forces only exist on a surface (if the object is in the air, there is no normal force).
4. Friction  – a force that opposes the motion of an object; symbol is Ff; direction is negative; friction is an electromagnetic force between surface atoms      Characteristics of friction:

• Friction acts parallel to the surfaces in contact
• Friction acts opposite the direction of the motion
• Friction depends upon the types of surfaces in contact (All surfaces are described by a coefficient of friction, m , which is a characteristic of that surface. m has no units.)
• Friction is independent of the surface area in contact.

Types of friction:

• Starting friction – opposes the beginning of motion of an object; is always greater than sliding friction
• Sliding friction  – opposes the motion of an object

Friction can be described mathematically:

Ff = m FN

5. Applied force – the push or pull that "you" use to move an object; symbol is Fapp
6. Unbalanced force (or net force ) – the sum of all the forces in a direction; it is what causes the acceleration of an object (I usually refer to it as UBF – this is not official, it is just me!)
7.      S F = m a

8. Tension  - a force usually associated with a rope or a cable; it is a "response" force. In other words, if one pulls on a rope, the rope "fights back" by resisting being stretched.

Forces on an Inclined Plane

The normal force exerted by the incline to support the weight, W. The parallel force is the part of the object's weight that tends to make it slide down the incline.
FN = W cos q
FP = W sin q

When you apply Newton's laws of motion to solve problems, you must think them out.
1. What is involved in the problem? Identify all forces involved.
2. What law of physics does this problem involve?
3. If Newton's Second Law is applied to this problem, to what particles is it applied?
4. Represent all forces as a free-body diagram.
5. Combine equations to solve problems.