Mechanics Outline
Mechanics terms:
Velocity = (change in displacement) / time
Speed = (distance traveled) / time {no direction}
Acceleration = (change in velocity) / time
Deceleration = acceleration opposed to velocity
Distance = shortest distance from start to finish. Displacement = distance & direction
Distance Traveled = measured distance along actual path
Position = coordinates of object’s location
Mass = amount of matter, measure of inertia
Force = push or pull
Inertia = resistance to acceleration
Momentum = quantity of motion = mass x velocity
Work = energy transferred by a force = (force)(distance force is applied)
Energy: Potential = stored energy
Kinetic = energy of motion = ½ mass (velocity)2
Modeling tools and tips:
Rate of change = (Change in quantity) / (time elapsed)
Direction and sign (+ or -); Choice of direction for + sign is arbitrary, opposite direction is then - .
Choose one axis along direction of acceleration
Choice of origin: Again, choice of 0 value is arbitrary.
Split problem into parts with constant acceleration.
Split motion into motion from place to place, rotation and vibration; handle separately.
Motion from place to place and vibration may each be split into perpendicular
components, each handled separately.
Choose a system for balancing external forces.
Draw a force diagram of the chosen system. Only forces from direct contact (plus
gravity, for now) are included, with the proper direction.
Geometry and angles for distances will be the same for the forces in the problem.
Patterns and results of demos and experiments:
Distance fallen during succeeding time intervals increases in the ratio of the odd integers
Total distance fallen during a given time = ½ g t2
Systems.
A group of objects which is being studied can be considered a system.
Limits and logic of relations:
What COULD BE TRUE and what MUST BE TRUE?
What conditions imply zero acceleration?
Zero acceleration guarantees what else to be true?
Zero velocity guarantees what else to be true?
What conditions imply zero average velocity?
What is the difference between speed and velocity?
If two objects are at the same place, how do their velocities and accelerations compare?
If the sum of the forces in the horizontal direction is zero and so is the sum of the forces in the
vertical direction, can they produce any change in the motion of the object?
If a large object is pushing a small object, how do the forces they exert on each other compare?
How much do those forces affect the motion of each object?
Applying the model
Split the problem into one dimensional and rotational parts (vibration part later).
For each linear dimension:
First, is velocity involved in the problem?
If it is, is the velocity constant over time?
Yes, then the object is in equilibrium, which means
1. acceleration = 0,
2. sum of forces = 0,
3. the object is moving in a straight line at constant speed.
4. the object’s momentum is constant.
If not, then the acceleration is not zero and
1. the acceleration = (change in velocity) over (time for the change)
2. the acceleration = net force over the mass.
3. The net force = sum of forces is not zero.
4. the direction of motion may change.
5. the inertia of an object resists acceleration.
6. the momentum of the object is changing.
7. from rest, an accelerating object travels farther each second.
8. the distance traveled in equal times are the odd numbers times the
distance traveled in the first period.
Forces:
A force causes an object to accelerate in the same direction as the force.
A force is a push or a pull, nothing else is a force.
The net force is the sum of forces acting on one object.
An object not traveling in a straight line has a force acting on it.
Forces transfer momentum and energy from one object to another.
friction: parallel to surface, opposes motion,
equals opposing force up to a maximum value.
contact or support force : perpendicular to surface of contact.
tension : pulls toward center of rope or object under tension;
same magnitude at both ends.
Basic equations:
average velocity = (total displacement) over (total time)
displacement = initial displacement + (initial velocity)time +
1/2 acceleration(time squared)
velocity = initial velocity + acceleration times time.
acceleration = (change in velocity) over time
= net force over mass
net force = mass times acceleration = sum of forces.
Two dimensional problems are solved by combining the 2 one-dimensional parts.
Rotation:
Any object in circular motion has some force acting on it.
The net force is directed toward the center of the circular path,
so it is called centripetal.
The magnitude of this force is
mass times speed squared divided by the radius of the path.
When the centripetal force ceases, the object flies off in
a straight line due to inertia.
There are rotational counterparts for the linear motion variables:
Inertia :
Moment of inertia = factor times mass times radius squared
radius is measured from the axis of rotation.
Momentum:
Angular momentum = momentum times radius
Force :
Torque = force times radius
Net force :
Net torque
Every force one object exerts on the other has an equal and opposite
reaction the second object exerts on the first.
 you cannot touch without being touched.
 equal force does not mean equal effect; (masses matter)
 Forces always come in pairs.
 Both forces act for the same length of time.
 Both forces act over the same distance, in opposite direction.
Equilibrium :
1. net force = 0
2. net torque = 0
3. center of mass is over (or below) the base of support.
Forces
A Force is a push or pull. Anything which is not a push or pull is not a force.
Forces have a strength and direction.
Forces acting on one object add together to give the net force.
The net force on an object produces an acceleration, Force=mass x acceleration
Units: Newton; 1 N = 1 kg m/s2 Force = mass [kg] x acceleration [m/s2]
Types of forces:
• Gravity: pull between one mass and another,
 the weight of an object = force of gravity on the object
= mass x “g” “g” = 10 m/s2 on Earth
• Friction: resistance to motion when 2 surfaces “rub” together
 equal to applied force up to maximum possible while still,
 increases with normal force between surfaces.
 less but constant while moving.
 parallel to surface of contact, opposes motion
• Normal support or contact: replaces “the floor is in the way”
 always perpendicular to the surface of contact.
• Tension: the pull a rope exerts at both ends,
 pulls toward center
 tension is the same at any point along the rope, including both ends,
as long as the rope’s mass is negligible
 Electrical or Coulomb:
 opposite charges attract
 proportional to the product of the electric charges
 inversely proportional to the square of the distance between charges
 Magnetic:
 opposite poles attract
 proportional to the product of the magnet strengths (dipoles)
 inversely proportional to the square of the distance between poles
 Elastic
 proportional to the change in length
 opposite in direction to change in length
Systems.
A group of objects which is being studied can be considered a system.
Forces exchanged among the objects in a system are internal forces.
These forces do not affect the motion of the system as a whole.
Example: All the collisions and interactions on Earth do not affect
the motion of the Earth through space.
Forces from outside the system are external forces.
External forces do affect the motion of the system as a whole.
Example: The Sun’s gravity has a strong effect on the motion
of the Earth through space.
Forms of energy:
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Kinetic energy = ½ mass (velocity 2); energy of motion.
Gravitational potential = mass • “g”• height
Chemical potential; energy stored in fuels
Heat (thermal energy), random motion energy of matter produced by friction,
collision
Other potential; electrical, magnetic, mass, elastic (spring)
Sound
Work = force • distance = energy transferred by force
Unit : Joule [J] = [kg m2/s2]
Power
Power is the rate of doing work. The unit for power is the Watt.
1 Watt = 1 Joule/Sec [W] = [J/s]
Formulas:
momentum = mass times velocity
change in momentum = force times time
work = change in energy = force times distance
kinetic energy = ½ mass times (velocity squared)
Potential energy :
gravitational = mass times height times “g”
elastic = ½ spring constant times (stretch squared)
p= mv
Ft = Δp
W=F•s
KE = ½ m v2
PE = mg•h
PE = ½ k (Δx)2
Conservation laws.
System : the collection of objects you wish to observe. Everything else is referred to as
the surroundings.
External force : a force on a system from the surroundings.
If the net external force is zero, then
1. the total momentum is conserved;
2. the center of mass moves with constant velocity;
3. the angular momentum is conserved;
4. as mass moves inward, the rotational speed increases;
5. energy is conserved, but some may be converted to heat.
In an elastic collision, objects bounce apart and kinetic energy is conserved.
In an inelastic collision, objects stick together and some kinetic energy is converted to
heat.
Conservation of Momentum
In the absence of external forces, the momentum of a system is conserved.
This means that, if a collision occurs between two objects in the system, the total
momentum before the collision equals the total momentum after collision.
Some momentum is just transferred from one object to another.
Conservation of Energy
Energy is never created or destroyed, it is transformed from one form to another
or transferred from one object to another, but the total is always the same.
An object in motion may have different amounts of potential energy and kinetic energy at
various times and friction may convert some of this to heat and sound, but the sum of all
these will remain a constant.
Energy conservation problems basically involve adding energies and the complexities of
the situation need not be considered!!!!
Graphical representations
At right, the top graph shows the
momentum during the collision of
two equal mass carts, one moving
(green momentum and purple
kinetic energy) hitting the second
stationary cart (blue momentum
and red kinetic energy). The total
momentum (orange) and kinetic
energy (green) are also shown.
Nature of Science
All information in science comes from observation and experimental data.
Relations describing patterns which are always seen to be true are called Laws of Science.
Scientific laws
 do not explain patterns, they simply state them.
 Are confirmed by experiment.
A possible new relationship, based on knowledge of phenomena, is called an hypothesis.
Hypotheses are
 educated guesses at an answer to the questions “what happens?” or “what causes that to
occur?”.
 Experimentally testable.
 Based on some data and knowledge of other scientific laws.
A theory is
 an explanation of what causes phenomena, often explaining several laws.
 tested and confirmed in its predictions.
 Often involves a model of the physical causes.
Theories do not become laws, they are already larger than laws and already tested experimentally.
All laws and theories of science are tentative, they are subject to modification based on new
information. This is part of why science has been so successful at describing the world.
Authority in science is based on a scientist’s success at predicting and explaining phenomena.
Authority does not mean their statements are correct, the only basis for deciding whether a
statement is correct or not is its agreement with data.
In order to be accepted as correct, an hypothesis must be confirmed by independent