MiniTest 1 Lecture Summary

• discussed the meaning of "sound," including looking in dictionary
• Basic skills for working with numbers and physical measurements
  • Numbers
    • reviewed significant digits
    • reviewed scientific notation

• Physical measurements and Units
  • reviewed units and metric prefixes: know M-, k-, m-, and µ-, plus cm
  • units conversion with hours, minutes, seconds as example
  • Units conversion: in compound units, how to handle parts that are raised to a power

Sound as something that travels

• timed how long sound takes to travel quad
• for sound, distance traveled is proportional to time taken
• Proportions
  • how to work with proportional quantities by creating ratio equations
  • a ratio IS a fraction
  • why the proportionality makes it convenient to define speed
• standard speed of sound is 340 m/s
• Word problems involving traveling sound
  • the usefulness of algebra, define speed algebraically for steady motion

d∝Δt

s=d/Δt

s_sound=340 m/s

• did several examples of working out algebraic problems using the speed of sound
• note that echos introduce factors of 2 in equations
• Listed some systematic steps to help structure an algebraic problem

Motions that produce sounds: vibrations

• Question: why does smooth oscillatory motion produce "nice" (i.e., not harsh) sounds
• Some kinds of motion are made up of repeating units, called cycles..
• For repeating motions, period is the time required to do one cycle
• For fast vibrations, it's sometimes easier to think about how many cycles are done in one second. That is frequency. Think of frequency as "cycles per second" and think of period as "seconds per cycle."
  • Units of frequency are 1/s. This is defined as Hertz (Hz).  Note that this means that 1/ms = kHz.
• A key idea is that number of vibrations is proportional to time.
• Describing vibrational motion in detail
  • Tracking the vibration
    • defined distance and displacement.
    • displacement is generally a vector to indicate direction, but for this class we'll just need displacement to be signed to indicate left or right
    • position is displacement from an origin
  • Extend speed definition to average speed for non-uniform motion
  • closely related to speed is velocity, defined as a displacement divided by the time taken for that displacement to occur
  • velocity is signed (can be positive or negative), indicating direction of motion.
  • to deal with changing velocities, GRAPHS!
  • looked at position versus time graphs
  • velocity is the slope of a position versus time graph

f = 1/T

f = N /Δt

s_avg=d/Δt

Δx=x_final-x_start

v_avg=Δx/Δt

• Tracking the velocity
  • practiced interpreting velocity versus time graphs
  • get velocity graph from position graph by the rule "SLOPE TO VALUE ". Useful steps for that are:
    1. Where is the slope 0?
    2. Where is the slope positive or negative?
    3. Where is the slope steepest or shallowest?
  • "Knematics" is about the description of motion. "Dynamics" is about explaining the reasons for motion.
  • Forces cause changes in velocitiy.
• Tracking the acceleration
  • Define average acceleration as the rate of change of velocity (see equation).  Acceleration units are m/s².
  • Note: sign of acceleration can be confusing. Sometimes helpful to consider "which way is it trying to go?"
  • As can been seen by comparing defining equations for v and a, acceleration graph is obtained from velocity graph by "slope to value".
  • Near surface of earth, all unsupported objects accelerate down at g=9.81 m/s² (if air has little effect). That's called "free fall."
  • Constant acceleration can only result in one direction reversal. So for vibrations, even acceleration must not be constant.  One could continue by defining "rate of change of acceleration," but luckily we'll see that we don't have to.
  • Completed example of position graph → velocity graph → acceleration graph

a_avg=Δv/Δt

g = 9.81 m/s²

F = m a

F = –k Δx

• After two slope-to-value transformations, the acceleration graph can be pretty sloppy. Clean it up with other information, such as "free-fall implies constant acceleration of 9.8m/s² down."
• Force: the why of motion
  • To quantify force, two overall options:
    1. Measure by the acceleration which the force can produce. This leads to Newton's second law. Mass is how hard it is to accelerate something.
    2. Measure by the amount of deformation the force can produce. In particular, how for can it stretch a spring. This leads to Hooke's Law. k is the "spring stiffness constant", and describes how hard it is to deform something.
  • These two turn out to be compatible with each other.
  • Minus sign in Hooke's law expresses direction of force.
  • Units:
    1. Mass is fundamental. Base unit is gram (g). Common unit is kilogram (kg)
    2. By Newton's 2nd law, this means force is kg m/s². This is given the name Newton (N), which is about equal to a 1/4 pound of force..
    3. Therefore, Hooke's law tells us that the units of k are N/m.

Simple Harmonic Motion: a special vibration

• Hooke's Law has brought us full circle back to displacement. This means that the motion of a mass bouncing on a spring has to be very special!
• We tried a position vs. time graph consisting of straight line motion and turn-arounds. This did not satisfy the "circle of physics".

Notice that s, s_avg, and v_avg are extremely similar equations, with only slight differences in the quantities involved.

• s=d/Δt
• s_sound=340 m/s
• f = 1/T
• Δ(anything)=(thing)_final-(thing)_start
• s_avg=d/Δt
• v_avg=Δx/Δt
• a_avg=Δv/Δt
• F = m a

• π = 3.1415
• g = 9.81 m/s²
• f = N /Δt
• F = –k Δx