Ch1: Kinematics and Dynamics - Kaplan MCAT P+M

the metric system may be given in 2 ways:

1. MKS: meters, kilograms, seconds

2. CGS: centimeters, grams, seconds

SI unit for length

meter (m)

SI unit for mass (not weight)

kilogram (kg)

SI unit for time

second (s)

SI unit for current

ampere (coulomb/second): (A)

SI unit for amount of substance

mole (mol)

SI unit for temperature

kelvin (K)

SI unit for luminous intensity

candela (cd)

base units

standard units around which the measurement system itself is designed

derived units

created by associating base units with each other

SI unit of force

newton (N)

units: (kg*m)/(s²)

SI unit of work and energy

joule (J)

units: (kg*m²)(s²)

J=N*M

SI unit of power

watt(W)

units: (kg*m²)/(s²)

W=J/s

ångströms

1 Å = 10^-10 m

nanometers

1 nm = 10^-9 m

electron-volts

1 eV = 1.6×10^-19J

• represents amount of energy gained by an e- accelerating through a potential difference of 1 volt

vectors

numbers that have magnitude and direction

• includes: displacement, velocity, acceleration, and force

• may be represented by arrows: direction of arrow = direction of vector

• length of arrow is proportional to magnitude of vector quantity

scalars

numbers that have magnitude only and no direction

• includes: distance, speed, energy, pressure, mass

common notations for vector quantity

either arrow or boldface

(A or 𝐴⃗)

common notation for magnitude of displacement between two positions (vectors)

|𝐴⃗|, |A|, or A

common notation for scalar quantities

italics: the distance between two points could be represented by d

resultant

the sum or difference of two or more vectors

• can be found by tip-to-tail method or components

tip to tail method

place the tail of B at the tip of A without changing either the length or direction of either arrow

• lengths of the arrows must be proportional to the magnitudes of the vectors

components

breaking each vector into perpendicular components

• usually components are horizontal and vertical (x- and y- components)

• sometimes it makes more sense to do components parallel and perpendicular to some other surface

component equations

if θ is the angle between V and the x-component, then cosθ=X/V and sinθ=Y/V

• X=Vcosθ

• Y=Vsinθ

if you know X and Y, you can find the magnitude of V with pythagorean theorem

• X²+Y²=V²

angle of resultant vector

• θ=tan^-1(Y/X)

steps of finding the resultant vector using components

1. resolve the vectors to be added into their x and y components

2. add the x components to get the x component of the resultant (R_x). add the y components to get the y component of the resultant (R_y)

3. find the magnitude of the resultant using the pythagorean theorem. if R_x and R_y are the components of the resultant, then R=√(R_x²+R_y²)

4. find the direction (θ) of the resultant by using the relationship θ=tan^-1(R_y/R_x)

vector subtraction

subtracting one vector from another: add a vector with equal magnitude but opposite direction to the first vector

multiplying vectors by scalars

• when a vector is multiplied by a scalar, its magnitude will change

• its direction will be either parallel or antiparallel to its original direction

• if vector A is multiplied by the scalar value n, a new vector, B is created such that B=nA

• to find the magnitude of the new vector B, multiply the magnitude of A by |n|

• to find the direction of B, look at the sign on n

• → if n is positive, they are the same direction

• → if n is negative, they are opposite direction

dot product (A⋅B)

A⋅B = |A| |B| cosθ

to generate a scalar quantity like work, multiply the magnitudes of the two vectors of interest (force and displacement) and the cosine of the angle between the two vectors

cross product (AxB)

AxB = |A| |B| sinθ

• when generating a third vector like torque, need to determine its magnitude and direction

• multiply the magnitudes of the two vectors of interest (force and lever arm) and the sin of the angle between the two vectors

• once we have the magnitude, use the right hand rule to determine its direction

the resultant of a cross product will always be:

perpendicular to the plane created by the two vectors

• on the MCAT the vector of interest will usually be going into or out of the page

right hand rule where C=AxB

1. thumb points in direction of vector A

2. pointer finger in direction of vector B

3. palm is the plane between the two vectors. the direction your palm points (or middle finger) is the direction of resultant C

for cross products and the right hand rule, does order of operations matter?

YES

• unlike scalar multiplication, which is commutative, vector multiplication is not cumulative (AxB ≠ BxA)

• ORDER MATTERS

displacement (x or d)

an object in motion may experience a change in its position in space

• this is a vector quantity (has direction and magnitude)

• the displacement vector connects the initial and final positions in a straight line

• does not account for pathway taken, only the net change in position

distance (d)

• considers the pathway taken

• scalar quantity

velocity (v)

• vector

• magnitude is rate of change of displacement in a given unit of time

• SI units: m/s

• direction of velocity vector is necessarily the same as direction of displacement vector

speed (v)

the rate of actual distance traveled in a given unit of time

instantaneous speed

will always be equal to the magnitude of the object’s instantaneous velocity

• scalar number

instantaneous velocity

measure of the average velocity as the change in time (Δt) approaches 0:

v= lim_Δt→0 (Δx/Δt)

average velocity

ratio of the displacement vector over the change in time

• is a vector

• does not account for actual distance traveled

• measure of the displacement of an object over a given period of time

v̄=Δx/Δt

average speed

measure of distance traveled in a given period of time

every change in velocity is motivated by what?

a push or a pull - a force

force (F)

a vector quantity that is experienced as a pushing or pulling on objects

• SI units: Newton

• F=ma

newton (N)

N=(kg*m)/s²

F=ma

acceleration due to gravity (g)

• decreases with height above the earth

• increases the closer you get to earth’s center of mass

• near the earth’s surface, use g = 10 m/s²

gravity

• attractive force that is felt by all forms of matter

• all objects exert gravitational forces on each other

magnitude of the gravitational force between two objects

F_g = (Gm₁m₂) / r²

• G: universal gravitational constant (6.67×10^-11 N*m²/kg²)

• m₁ and m₂: masses of the two objects

• r: distance between the two objects centers of mass

how is the magnitude of the gravitational force related to the square of the distance?

the magnitude of the gravitational force is inversely related to the square of the distance

• if r is halved, then F_g will quadruple

how is the magnitude of the gravitational force related to the masses of the objects?

the magnitude of the gravitational force is directly related to the masses of the objects

• if m₁ is tripled, then F_g will triple

friction

type of force that opposes the movement of object

• friction forces always oppose an object’s motion and cause it to slow down or become stationary

types of friction

static and kinetic

static friction (f_s)

static friction (f_s) exists between a stationary object and the surface upon which it rests

magnitude of static friction: 0 ≤ f_s ≤ μ_sN

• μ_s: coefficient of static friction

• N: magnitude of the normal force

• min is 0

• max is μ_sN

μ_s: coefficient of static friction

unitless quantity that is dependent on the two materials in contact

normal force

component of the force between two objects in contact that is perpendicular to the plane of contact between the object and the surface upon which it rests

contact points

places where friction occurs between two rough surfaces sliding past each other

• if the normal load rises, the total area of contact increases

• that increase, more than the surface’s roughness, governs the degree of friction

normal load

the force that squeezes two rough surfaces sliding past each other together

kinetic friction (f_k)

kinetic friction (f_k) exists between a sliding object and the surface over which the object slides

• any time two surfaces slide against each other (like ice), kinetic friction will be present

• f_k=μ_kN

• μ_k = coefficient of kinetic friction

• kinetic friction will have a constant value for any given combination of a coefficient of kinetic friction and normal force (doesnt matter how much contact or velocity of the object)

μ_k vs μ_s

• the value of μ_s is always larger than the value of μ_k

• max value for static friction will always be > the constant value for kinetic friction

• objects will “stick” until they start moving, and then will slide more easily over one another

• it always requires more force to get an object to start sliding more than it takes to keep an object sliding

mass (m)

measure of a body’s inertia - the amount of matter in the object

• scalar quantity (magnitude only)

• SI units: kg

• independent of gravity

weight (F_g)

measure of gravitational force (usually that of the earth) on an object’s mass

• weight is a force, so it is a vector quantity

• units: Newtons

equation that relates weight and mass

F_g=mg

• F_g: weight of the object

• m: mass

• g: acceleration due to gravity (10 m/s²)

center of mass or gravity

the weight of an object can be thought of as being applied at a single point in that object called center of mass or gravity

• center of mass of a uniform, homogeneous body (symmetrical shape and uniform density) is at the geometric center of the object

acceleration (a)

rate of change of velocity that an object experiences as a result of some applied force

• vector quantity

• SI units: m/s²

deceleration

acceleration in the direction opposite the initial velocity

average acceleration (ā)

ā=Δv/Δt

instantaneous acceleration

the average acceleration as Δt approaches zero

a=lim _Δt→0 Δv/Δt

• on a graph of velocity vs time, the tangent to the graph at any time t which corresponds to the slope of the graph at that time, indicates instantaneous acceleration

on a graph of velocity vs time, the tangent to the graph at any time t which corresponds to the slope of the graph at that time, indicates instantaneous acceleration.

if slope is positive, what does it mean?

the acceleration is positive and in the direction of the velocity

on a graph of velocity vs time, the tangent to the graph at any time t which corresponds to the slope of the graph at that time, indicates instantaneous acceleration.

if slope is negative, what does it mean?

the acceleration is negative (deceleration) and is in the direction opposite of the velocity

Newton’s first law

F_net = ma = 0

a body either at rest or in motion with constant velocity will remain that way unless a net force acts upon it

• aka law of inertia

• special case of Newton’s second law

Newton’s second law

F_net = ma

an object of mass m will accelerate when the vector sum of the forces results in some nonzero resultant force vector

• no acceleration will occur when the vector sum of the forces results in a cancellation of those forces

• net force and acceleration vectors necessarily point in the same direction

Newton’s third law

F_AB = -F_BA

law of action and reaction: to every action, there is always an opposed but equal reaction

• for every force exerted by object A on object B, there is an equal but opposite force exerted by object B on object A

objects can only undergo two types of motion

• that which is constant (no acceleration)

• that which is changing (with acceleration)

if an object’s motion is changing (as indicated by a change in velocity)…

then the object is experiencing acceleration

• the acceleration may be constant or changing

linear motion

the object’s velocity and acceleration are along the line of motion, so the pathway of the moving object continues along a straight line

• does not need to be limited to vertical or horizontal paths (could be a ramp)

• on the MCAT this will usually be an object being dropped to the ground from some starting height

falling objects exhibit linear motion with constant acceleration (one dimensional motion). what equations describe this?

v = v₀ + at

x = v₀t + at²/2

v² = v₀² + 2ax

x= v̄t

free fall with negligible air resistance

• air resistance is negligible, so the only force acting on the object would be the gravitational force causing it to fall

• constant acceleration (the acceleration due to gravity, g=9.8 m/s²

• would not reach terminal velocity

air resistance

opposes the motion of an object

• its value increases as the speed of the object increases

drag force

an object in free fall will experience a growing drag force as the magnitude of its velocity increases

free fall with air resistance that is not negligible

• air resistance increases as speed of object increases

• as magnitude of velocity increases, drag force increases

• eventually the drag force will be equal in magnitude to the weight of the object, and the object will fall with constant velocity according to Newton’s first law

terminal velocity

• for an object in freefall with non-negligible air resistance, the drag force will eventually become equal in magnitude to the weight of the object

• then the object will fall with constant velocity according to Newton’s first law

the amount of time that an object takes to get to its maximum height is the same as:

same time it takes for the object to fall back down to the starting height (assuming air resistance is negligible)

how can you solve for the time to reach maximum height?

setting final velocity to zero

how can you get the total time in flight?

• set final velocity to zero

• multiply answer by 2 to get total time in flight

• this works as long as the object ends at the same height at which it started

• velocity in the x direction will remain constant

how can you find the horizontal distance traveled?

by multiplying the time by the velocity in the x direction

projectile motion

motion that follows a path along two dimensions

• velocities and accelerations in the two directions (usually horizontal and vertical) are independent of each other and must be analyzed separately

• objects in projectile motion on earth experience force and acceleration of gravity only in the vertical direction (y-axis)

• v_y will change at the rate of g, v_x will remain constant

inclined planes

• motion in two dimensions

• best to divide force vectors into components that are parallel and perpendicular to the plane

• F_g∥=mgsinθ

• F_g⊥=mgcosθ

circular motion

occurs when forces cause an object to move in a circular pathway

• upon completion of one cycle, the displacement of the object is 0

• uniform and nonuniform circular motion

uniform circular motion

• the instantaneous velocity vector is always tangent to the circular path

• the object moving in the circular path has a tendency (inertia) to break out of its circular pathway and move in a linear direction along the tangent - but it is kept from doing so by centripetal force

• the tangential force is 0 bc there is no change in the speed of the object

centripetal force

• in uniform circular motion, the object moving in the circular path has a tendency (inertia) to break out of its circular pathway and move in a linear direction along the tangent - but it is kept from doing so by centripetal force

• centripetal force always points radially inward

• centripetal force generates centripetal acceleration

• can be caused by tension, gravity, electrostatic forces, or other forces

• when the centripetal force is no longer acting on the object, it will simply exit the circular pathway and assume a path tangential to the circle at that point

• F_c = mv²/r

centripetal acceleration

acceleration generated by the centripetal force that keeps an object in its circular pathway

dynamics

the study of forces and torques

translational motion

occurs when forces cause an object to move without any rotation

translational equilibrium

• exists only when the vector sum of all of the forces acting on an object is 0

• this is the first condition of equilibrium

• an object experiencing translational equilibrium will have a constant velocity: constant speed (zero or nonzero) and constant direction

when the resultant force upon an object is 0, what does this mean?

the object will not accelerate

• may mean the object is stationary

• or it could mean the object is moving with a constant nonzero velocity

rotational motion

occurs when forces are applied against an object in such a way as to cause the object to rotate around the fulcrum

fulcrum

a fixed pivot point

torque (𝜏) or moment of force

application of force at some distance from the fulcrum generates torque

• it is the torque that generates rotational motion, not the mere application of the force itself

• torque depends on magnitude of force, length of the lever arm, and angle at which force is applied

• 𝜏 = rxF = rFsinθ

lever arm

the distance between the applied force and the fulcrum

sin 90° =

1

torque is greatest when?

when the force applied is 90° (perpendicular) to the lever arm