Position
Location of a particle in space
Represented as (x), (x,y), or (x,y,z)
Distance
The total length of the path traveled by an object.
Does not depend upon direction.
Displacement
The change in position of an object.
Depends only on the initial and final positions, not the path.
Includes direction.
Represented as (see image) x
Acceleration
A change in velocity.
Can be speeding up, slowing down, or turning.
SI unit is m/s^2
If the sign of the velocity is the same as the sign of the acceleration, the object speeds up.
If the sign of the velocity is different than the sign of the acceleration,
Uniformly Accelerated Motion
a(sub)avg = (change in)v / (change in)t
Average Speed
s(sub)avg = d / (change in)t
Average Velocity
v(sub)avg = (change in)x / (change in)t
Kinematic Equations
v = (vi) + at
x = (xi) + (vi)t + 1/2at^2
v^2 = (vi)^2 + 2a(change in)x
Free Fall
Occurs when an object falls unimpeded.
Gravity accelerates the object toward the earth.
g = 9.8 m/s^2 downward
a = -g if up is positive
Acceleration is down when ball is thrown up and/or everywhere is the balls flight.
Projectile Motion
2D motion
Object is fired, thrown, shot, hurled, etc. near the earth's surface
Horizontal velocity is constant
Vertical velocity is accelerated
Air resistance is ignored
Trajectory of Projectile
The path the object follows
Defined by a parabola
Range = distance traveled horizontally
Maximum Height = halfway through range (as long as projectile fired over level ground)
Acceleration points down at 9.8 m/s^2 for the entire trajectory
Velocity is tan
2D Motion Problems
Resolve vectors into components
Solve as 1D problems
Horizontal Component of Velocity
Not accelerated
Not influenced by gravity
Equation: x = (vix)t
Vertical Component of Velocity
Accelerated by gravity (9.8 m/s^2 down)
V(sub)y = (viy) - gt
y = (yi) + (viy)t - 1/2gt^2
Vy^2 = (viy)^2 - 2g(y-yi)
Force
A push or a pull on an object
Causes object to accelerate (speed up, slow down, change direction)
SI unit: Newton (N)
Newton's First Law
The Law of Inertia: A body in motion stays in straight-line motion at constant velocity and a body at rest stays at rest unless acted upon by an external, unbalanced force.
Periodic Motion
Repeats itself over a fixed and reproducible period of time
Mechanical devices that do this are called oscillators
Simple Harmonic Motion (SHM)
Periodic motion described by a sin or cos function
Springs and pendulums are common examples of Simple Harmonic Oscillators (SHOs)
All SHOs experience a "restoring force
Restoring Force
F = -kx (basic form of restoring force)
Restoring force is greatest at maximum displacement and is zero at equilibrium
Equilibrium
The midpoint of the oscillation of a SHO
Position of minimum potential energy and maximum kinetic energy
Amplitude
Represented as A
How far the oscillating mass is from equilibrium at its maximum displacement
Period
The length of time it takes for one cycle of periodic motion to complete itself
Frequency
How fast the oscillation is occurring
Frequency is inversely related to period
f = 1 / T
Units are Hz where 1 Hz = 1 s(sup)-1
Springs
A common SHO
F(sub)s = -kx (Hooke's Law)
Period of a Spring
T = 2pisqrt(m/k)
Potential Energy of a Spring
U(sub)s = 1/2kx^2
Pendulum
Can be thought of as an oscillator
Displacement needs to be small for it to work properly
Pendulum Forces: Gravity and Tension
Period of a Pendulum
T = 2pisqrt(1/g)
Potential Energy of a Pendulum
U(sub)g = mgh
Work
The bridge between force and energy
Work is a scalar
W = Fdcostheta
Measured in Joules (J)
Counterintuitive Results
There is no work if there is no displacement
Forces perpendicular to the displacement don't work
By doing positive work on an object, a force or collection of forces increases its mechanical energy in some way
The two forms of mechanical energy are potent
Kinetic Energy
Energy due to motion
K = 1/2mv^2
The Work-Energy Theorem
W(sub)net = (change in)K
Net work is used here. Net work = ALL forces acting on object
When net work is positive, the kinetic energy of the object will increase (object will speed up)
When net work is negative, the kinetic energy of the object will decrea
Work and Graphs
The area under the curve of a graph of force vs. displacement gives the work done by the force in performing the displacement. For springs, this is stretching and compressing
Power
The rate at which work is done
When we run upstairs, t is small so P is big
When we walk upstairs, t is big so P is small
P = W / t
P = FV
Measured in Watts where 1 Watt = 1 J/s
Force Types
Conservative Forces
Work in moving an object is path independent
Work in moving an object along a closed path is zero
Work done against conservative forces increases potential energy; work done by them decreases it.
Ex: Gravity, springs
Nonconservative Fo
Potential Energy
Energy an object possesses by virtue of its position or configuration
Ex: Gravitational Potential Energy and Spring Potential Energy
Potential Energy is Related to Work Done by Conservative Forces ONLY
(change in)PEg = -W (gravity)
(change in)PEsp = -W (spring)
Gravitational Potential Energy Close to Earth's Surface
W(gravity) = -mgh (close to earth's surface)
(change in)U = -W(gravity) = mgh
Spring Potential Energy
U(sub)sp = 1/2kx^2
U(sub)sp is zero when a spring is in its preferred, or equilibrium position where the spring is neither compressed or extended.
Law of Conservation of Mechanical Energy
U + K = Constant
(change in)U + (change in)K = 0
(change in)U = -(change in)K
Conservation of Energy and Dissipative Forces
Dissipative forces cause loss of mechanical energy by producing heat
W = (change in)U + (change in)K
Newton's Second Law
(sum of)F = ma
General Second Law Problems Solving Procedure
Draw the problem
Free Body Diagram
Set up equations (F = ma, Fx = max, Fy = may)
Substitute givens from problem
Solve
Newton's Third Law
For every action there exists an equal and opposite reaction
If A exerts a F on B then B exerts a -F on A
Weight
W = mg
Normal Force
Force that prevents objects from penetrating each other
Reaction to other forces
Ex: Commonly a reaction to gravity
Friction
The force that opposes a sliding motion
Static friction exists before sliding occurs
Kinetic friction exists after sliding occurs
In general, kinetic friciton < static friciton (It takes more force/work to get something moving than to keep something in mo
Calculating Static Friction
F(sub)f(sub)s < or = (mew)(sub)sF(sub)n
Static friction increases as the force trying to push an object increases until the force overcomes the friction and it becomes nonexistent.
Calculating Kinetic Friction
F(sub)f(sub)k = (mew)(sub)kF(sub)n
Uniform Circular Motion
An object moves at uniform speed in a circle of constant radius
Acceleration in Uniform Circular Motion
Turns object, doesn't speed it up or slow it down.
Acceleration points toward center of circle.
AKA centripetal acceleration
Centripetal Acceleration
a(sub)c = v^2 / r
Force in Uniform Circular Motion (Centripetal Force)
Any force responsible for uniform circular motion is referred to as a centripetal force.
Centripetal force can arise from one force, or from a combination of sources.
F(sub)c = (sum of)F = ma(sub)c
F(sub)c = (sum of)F = mv^2 / r
Centripetal force(s) alway
Universal Law of Gravity
F(sub)g = -Gm1m2 / r^2
Negative sign means the force is attractive
Most orbit problems can be solved by setting the gravitational force equal to the centripetal force
Gm1m2 / r^2 = m1v^2 / r
Torque
Torque is a twist as opposed to a push or pull
t = Frsin(theta)
Units: mN or Nm
Rotational Equilibrium
If counterclockwise torques equal the clockwise torques, the system is balanced and no rotation occurs.
(sum of)tcw = (sum of)tccw
Momentum
How hard it is to stop a moving object
Related to both mass and velocity
For one particle: p = mv
For multiple particles: P = (sum of)p1 = (sum of)m1v1
Units: Ns or kgm/s
Impulse
Represented as J
The product of an external force and time, which results in a change in momentum
J = Ft
J = (change in)P
Units: Ns or kgm/s
Law of Conservation of Momentum
If the resultant external force on a system is zero, then the momentum of the system will remain constant.
The sum of the momentums before a collision is equal to the sum of the momentums after a collision
(sum of)P(sub)b = (sum of)P(sub)a
Collisions
Follow Newton's Third Law which tells us that the force from body A to body B in a collision is equal and opposite to the force from body B to body a
During a collision, external forces are ignored
The time frame of the collision is very short
The forces
Collision Types
Elastic: P is conserved, K is conserved
Inelastic: P is conserved, K is not conserved
Perfectly inelastic means the bodies stick together
Explosion
Mathematically, handled just like an ordinary perfectly inelastic collision
Momentum is conserved, kinetic energy is not