Mechanics

Mechanics: The analysis of the action of forces on matter.

All of PH201 deals with mechanics.

Mechanics is generally subdivided into:

Kinematics

- The “how” of motion.

- Measurement and/or description of motion

Dynamics

- The “why” of motion.

- The study of the relationship between motion

and the forces that effect that motion.

We’ll start with analysis of 1D (straight line) motion and then broaden our scope to include 2D motion and, later still, circular motion.

Distance versus Displacement

Distance - a scalar quantity

- length

- ex: 425 m

Displacement - a vector quantity

- distance in a given direction.

- distance plus a direction.

- a measure of a change in position

- straight line distance from starting point.

- ex: 425 m to the left

Common usage: Don’t be confused with the water displacement that occurs when you lower yourself into a tub of water.

If all of the motion is 1-D, and in one direction, the values of distance and displacement will be exactly the same.

Displacement Definition:

D x = x - x_o

Note subscripts…

Speed versus Velocity

speed - a scalar quantity

- distance traveled divided by elapsed time.

- =slope on a distance versus time graph.

- ex: 32 m/s

velocity - a vector quantity

- displacement or change in position divided by

elapsed time.

- speed plus a direction.

- =slope on a displacement versus time graph.

- ex: 32 m/s due south

Common usage: These two words are interchangeable.

Physics usage: The difference between the two words (and concepts) is crucial to make.

If all of the motion is 1-D, and in one direction, the values of speed and velocity will be exactly the same.

average versus instantaneous

- Instantaneous values can be computed from a

limiting process. (calculus)

- If there is no change in speed over some time

interval, then, avg. = inst. in that interval.

Acceleration

acceleration - a vector quantity

- change in velocity divided by elapsed

time.

- dimensions of distance over time squared.

- =slope on a velocity versus time graph.

- ex: 9.8 m/s/s downward

Common usage: Acceleration is commonly used to mean only speeding up. Deceleration is used to mean slowing down.

Physics usage: Acceleration refers to ANY change in velocity, thus, a car that is speeding up is accelerating, a car that is slowing down is accelerating, and a car that is turning (changing direction) is accelerating.

average versus instantaneous

- We will make little/no distinction.

- All accelerations will be average, and, we’ll

assume, constant over the time interval. Thus,

avg. = inst. for our problems.

Speed, Velocity, Acceleration

Average Speed Definition:

v = distance / elapsed time

Average Velocity Definition:

v = displacement / elapsed time

v = (x - x_o) / (t - t_o) = D x / D t

Average Acceleration Definition:

a = change in velocity / elapsed time

a = (v - v_o) / (t - t_o) = D v / D t

Average Velocity (Speed)

The arithmetic average of any evenly incremented set of values can be found by: (first + last) / 2.

Any object undergoing straight line, uniform, constant acceleration (either speeding up or slowing down) is passing through an evenly incremented set of velocity values. The average velocity is thus:

v_avg = (v_o + v) / 2 = ½ (v_o + v)

Distance traveled is always = v_avg times t, therefore:

x = ½ (v_o + v) t

gives us the distance traveled during uniform acceleration.

Note that this also works fine (of course) for an acceleration of zero where v_o = v = v_avg .

1-Dimensional Kinematics Equations

x = ½ (v_o + v) t

x = v_o t + ½ at²

v = v_o + at

v² = v_o² + 2ax

These equations can be applied to the motion of an object only when:

t_o = 0

x_o = 0

a is constant.

Acceleration (continued)

1-D Speeding up and slowing down

If the acceleration vector points in the same direction as the velocity vector, the object is speeding up.

If the acceleration vector points in the opposite direction of the velocity vector, the object is slowing down.

velocity acceleration result

+ + speeding up

+ - slowing down

- + slowing down

- - speeding up

Negative acceleration does not (necessarily) mean deceleration (or slowing.)