Introduction to Mechanics

Mechanics It has nothing to do with the people you call when your car needs to be repaired. It is the study of motion.

Historical Development of Mechanics Aristotle vs. Galileo

He said that we must first understand why objects move. Aristotle He said that we must first understand why objects move.

Aristotle Things move because they “desire” to do so. Light things “desire” to rise to the heavens. Heavy things “desire” to sink to earth. In short, objects have a natural tendency.

Early scientists like Aristotle were called natural philosophers.

Galileo Galileo said that we should first study how things move, and then we should describe why they move.

Stationary things react to pushes and pulls. Mechanics the study of motion Dynamics Kinematics Statics Why? cause How? Stationary things react to pushes and pulls.

Mechanics is the study of life. motion. work. systems. Question

T/F Aristotle believed that we should first determine why things move. Question

an artificial boundary used to isolate an object or objects System an artificial boundary used to isolate an object or objects

everything outside of the system Surroundings everything outside of the system

Systems Scientists are free to select any system as they study the motion of objects. Examples: you, your desk, the floor you and your desk

Frame of Reference When a car zooms by you, it is moving.

Frame of Reference But if you are in the car, it seems that the car is standing still and everything else is speeding past the windows.

your frame of reference What’s the difference? your frame of reference

you (How self-centered!) Frame of Reference What is THE frame of reference? you (How self-centered!) the earth the sun the galaxy

Frame of Reference There is no “THE frame of reference.” Choose the best frame of reference for the problem being solved.

Frame of Reference The frame of reference you choose Sun Earth North Pole The frame of reference you choose determines how the motion will be described.

Kinds of Reference Frames Fixed—the reference frame is stationary, but the system moves. Accelerated—the reference frame accelerates with the system. Rotational—the reference frame accelerates, but the system is stationary.

Coordinate Axis Number Line Zero is the origin. Negative numbers are to the left of the origin. Positive numbers are to the right of the origin.

non-physical continuum that orders the sequence of events Time non-physical continuum that orders the sequence of events

Time sometimes called the space-time continuum created by God Before time was, God is. “I AM.”

Time Any event that happens must occur within a span of time. The start of that time span is called the initial time (ti). The end of that time span is called the final time (tf).

Time The difference between the initial and final time is the time interval. It is called Δt (“delta tee”) and is found by subtracting the initial time from the final time.

Question What is another name for a coordinate axis? fulcrum space-time continuum number line reference frame Question

measurement that has a magnitude (amount) with no direction indicated Scalar measurement that has a magnitude (amount) with no direction indicated Examples: 13 m 47 km/h

This paper has a measurement of 215.7 mm. Scalar Since the smallest measurement is zero, scalars never have a negative magnitude. This paper has a measurement of 215.7 mm.

measurement that has both a magnitude and a direction Vectors measurement that has both a magnitude and a direction Examples: 13 m forward 47 km/h ENE

The magnitude part of a vector is considered to be a scalar. Vectors The magnitude part of a vector is considered to be a scalar.

Vectors Vectors are shown on the coordinate axis by an arrow. The length indicates the magnitude. The arrowhead indicates the direction. force (F) weight (w)

T/F Scalar measurements have magnitude and direction. Question

Kinematics: Describing Motion

Motion a change of position during a time interval It can be in one, two, or three dimensions. X1 = 0.5 cm X2 = 1.5 cm X3 = 2.5 cm x v 0 cm 1 cm 2 cm 3 cm t1 t2 t3

Distance a positive scalar quantity that indicates how far an object has traveled during a time interval

the overall change in position during a time interval Displacement the overall change in position during a time interval (how much it moved)

Displacement Displacement is a vector quantity. The distance is the magnitude of the displacement vector.

X X

Speed distance d speed (v) = = time Δt the rate at which an object changes position the distance traveled in a period of time As an equation: distance d speed (v) = = time Δt

using the speed triangle: distance speed = time

using the speed triangle: distance time = speed

using the speed triangle: distance = speed × time

Sample Problem 1 If a motorcycle travels 540 km in 2 hours, what is its speed? distance 540 km speed = = time 2 h = 270 km/h

What does speed equal? time / distance the rate at which an object changes time the amount of time traveled over a distance distance / time

If a car travels 400 km and the trip takes 5 hours, how fast is the car traveling? 405 km/h 395 km/h 2000 km/h 80 km/h

If an object is traveling at 100 km/h for 5 hours, how far does it travel? d = s × t d = 100 km/h × 5 h d = 500 km

rate of motion over a time interval Average Speed rate of motion over a time interval V1 + V2 2 V = This is only true in cases of uniformly accelerated motion, in the context of the one-dimensional motion discussed in Chapter 4.

rate of motion at a specific time Instantaneous Speed rate of motion at a specific time

Sample Problem 2 It takes a fast cyclist 0.35 h (20.85 min) to cover the 19 km stage of a European biking race. What is his average speed in km/h? Known: time interval (Δt) = 0.35 h distance (d) = 19 km Unknown: speed (v)

Sample Problem 2 = = 54.2 km/h = 54 km/h 2 Sig Digs allowed d 19 km Known: time interval (Δt) = 0.35 h distance (d) = 19 km Unknown: speed (v) d 19 km v = = = 54.2 km/h Δt 0.35 h = 54 km/h 2 Sig Digs allowed

Velocity technically different than speed involves both speed and direction It is the rate of displacement. Example: The car is traveling east at 65 mph.

Velocity Velocity is called a vector measurement because it includes how fast and which direction. Speed is called a scalar measurement because it only involves how fast.

Scalars and Vectors Scalars Vectors Scalars Vectors distance (d) displacement (d) speed (v) velocity (v) Scalars Vectors distance (d) displacement (d) speed (v) velocity (v) Remember, vectors can be positive or negative, but scalars are only positive.

Scalars and Vectors Vfuel cell Vhybrid −45 m/s +30 m/s West (−) East (+)

Acceleration Acceleration is an increase in velocity in a given space of time (speeding up). Deceleration is a decrease in velocity in a period of time (slowing down).

Acceleration Formula = = Δv Acceleration Δt change in velocity change in time The Greek letter delta (Δ) stands for “change in.” Δv Acceleration = Δt

Acceleration Units The units for velocity are distance/time. Since acceleration is velocity/time, the units must be distance/time/time.

Acceleration Units This is rewritten distance/time2. Actual units could be miles/sec2. a v West (−) East (+)

Sample Problem 3 A car moving at +5.0 m/s smoothly accelerates to +20.0 m/s in 5.0 s. Calculate the car’s acceleration. North is positive. Known: car’s vi = +5.0 m/s car’s vf = +20.0 m/s time interval (Δt) = 5.0 s Unknown: acceleration (a)

Sample Problem 3 = = Known: car’s vi = +5.0 m/s car’s vf = +20.0 m/s time interval (Δt) = 5.0 s Unknown: acceleration (a) vf − vi (+20.0 m/s) − (+5.0 m/s) = = a Δt 5.0 s

Sample Problem 3 = = = = = vf − vi (+20.0 m/s) − (+5.0 m/s) a Δt 5.0 s +3.0 m/s/s a 5.0 s = 3.0 m/s2 north

Sample Problem 4 A car moving at +20.0 m/s smoothly slows to a stop (0 m/s) in 6.0 s. Calculate the car’s acceleration. East is positive. Known: car’s vi = +20.0 m/s car’s vf = 0.0 m/s time interval (Δt) = 6.0 s Unknown: acceleration (a)

Sample Problem 4 = = Known: car’s vi = +20.0 m/s car’s vf = 0.0 m/s time interval (Δt) = 6.0 s Unknown: acceleration (a) vf − vi (0.0 m/s) − (+20.0 m/s) = = a Δt 6.0 s

Sample Problem 4 = = = = = vf − vi (0.0 m/s) − (+20.0 m/s) a Δt 6.0 s -3.3 m/s/s a 6.0 s = -3.3 m/s2 west

What is acceleration? going a distance the direction you travel how fast you move a change in how fast you move

If a car takes 5 seconds to change speed by 40 m/s, what is its acceleration?

A car going 40 m/s takes 10 s to speed up to 140 m/s A car going 40 m/s takes 10 s to speed up to 140 m/s. What is its acceleration? 400 m/s2 10 m/s2 1,400 m/s2 4 m/s2

Which of the following are possible units of acceleration? seconds feet feet / second feet / second2

Two and Three Dimensional Motion These examples had motion in only one dimension. Two dimensional motion is common also. An example is a car rounding a corner.

Two and Three Dimensional Motion Three dimensional motion is not unusual. An example is a car rounding a corner on a hill. This type of motion uses “spatial” dimensions, so called because 3 dimensions enclose a volume or space.

Two and Three Dimensional Motion A car rounding a corner is changing its direction. Direction is part of velocity, so the car is accelerating even if its speed is constant.

Two and Three Dimensional Motion When a car repeatedly comes to a stop from the same speed—as in a busy downtown with red lights—its acceleration is closer to zero if it takes longer to stop.