Topic 4a: Translational Motion

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translational motion: movement of an object from one position to another, but without reference to a fixed point

includes motion in a straight line and projectile motion

rotational motion: motion of an object in refence to an axis or fixed point

dimensions: physical quantities described by physical measurements; some quantities can be described by combinations of the below (ex: force, a combination of mass x length/time squared)

length

mass

time

temperature

amount of a substance

electric current

luminous intensity

Units: the amount of a physical quantity

meter

kg

second

Conversion factors are used to convert units consistently in terms of dimension. Example: (L/T) à (km/h or m/s).

Conversion techniques can also be used to estimate quantities, however these estimations are subject to assumptions.

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scalar quantities: require only magnitude

example: time, mass, area, volume, distance, speed, work, energy.

Vector quantities: quantities that require both the magnitude and direction

example: displacement, velocity, acceleration, force, momentum, torque.

Direction is often given as an angle.

methods of vector addition:

head to tail method:

one vector is strong with its tail positioned at the origin.

The tail of the second vector is John at the head of the first vector.

The resultant/vector sum is the magnitude and direction of the arrow drawn from the tail of the first vector to the head of the second vector

If the vectors are perpendicular to one another, then the resultant vector will be the hypotenuse of the right triangle formed by the two perpendicular vectors.

The magnitude of the resultant vector would be found using the pythagorean theorem.

The direction of the resultant vector would be found by calculating the angle using an inverse tangent function.

θ = tan−1 (opp/adj) = tan−1 (vector B/vector A)

component method: used for two dimensional vectors/when a vector quantity is oriented at an angle, requires that the X&Y components of each vector be determined.

Horizontal component: cos θ = (adj/hyp) = Ax/A

Vertical component: sin θ = (opp/hyp) = Ay/A

Exceptions: angles θ = 90◦, 180◦, 270◦, and 360◦ are special case

at 90◦

x component= 0

y component= +|A|

at 180◦

x-component= −|A|

y component= 0

at 270◦

x-component= 0

y-component= −|A|

at 360◦

x-component= +|A|

y-component= 0

The sum of the X components composes the X component of the resultant vector

the sum of the Y components composes the Y component of the resultant vector

the magnitude of the resultant vector is determined by plugging in the X&Y components of the resultant vector into the pythagorean theorem

the inverse tangent function is used to determine the direction of the resultant vector.

θ = tan−1 (opp/adj) = tan−1 (Ry/Rx)

Vector subtraction: the direction of 1 vector is reversed and it is added to the remaining vectors; This does not affect the magnitude of the vector.

Mathematical operations involving a vector and a scalar always result in a vector

Mathematical operations involving a scalar and a scalar always result in a scalar

Mathematical operations involving a vector and a vector DO NOT always result in a vector; ex: kinetic energy formula

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Displacement: measure of length required to get from point A to point B, irrespective of the path taken; the difference between final position and initial position

negative displacement implies that the direction of displacement is opposite to the direction considered positive

distance: the length required to get from the starting point to the ending point in respect to the path taken

Speed: rate of change of distance overtime, expressed in dimensions of length per time

average speed: DISTANCE traveled over the total time of travel

velocity: rate of change of displacement overtime; The speed of an object in a given direction, expressed in dimensions of length per time

average velocity: DISPLACEMENT over a given time interval

instantaneous velocity: velocity at a given instant of time, example the velocity of a car indicated by its speedometer

calculated by determining the slope of the line tangent to the point of interest

Acceleration: the rate at which velocity changes; The change of velocity in a given direction over a defined interval of time, expressed in dimensions of length per unit time squared

average acceleration: average velocity over a time interval

instantaneous acceleration: acceleration at a given instant of time

the direction of acceleration will be the direction of the change in velocity; this is NOT always the direction of motion

acceleration graphs can be derived from velocity or displacement graphs

Graphical representation of motion:

time typically represented along the X axis, and other dimensions are represented along the Y axis.

Calculations of slope:

Slope on a graph of displacement vs time = rise/run = (y2 − y1)/(x2 − x1) = displacement (meters)/time (seconds) = velocity (meters/second)

Slope on a graph of displacement vs time = rise/run = (y2 − y1)/ (x2 − x1) = velocity (meters/second)/time (seconds) = acceleration (meters/second2)

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Equations of motion:

applicable to uniformly accelerated motion

this includes free fall motion: describes motion with constant acceleration along the Y axis caused by gravity

example: during a ball in the air, dropping a pencil

to apply the equations to free fall, acceleration will be= g= 9.8 m/s2

exception: if an object is falling to the ground, initial velocity will equal 0 and the equations can be simplified

Displacement given both velocities = 1/2 *(initial velocity + final velocity)*(time interval)

Displacement given only initial velocity = (initial velocity)*(time interval) + 1/2 *(acceleration)*(time interval)2

Final velocity given time = initial velocity + (acceleration)*(time interval)

Final velocity given displacement: (Final velocity)2 = (initial velocity)2 + 2 *(acceleration)*displacement)