Kaplan MCAT Physics/Math chapter 1 Kinematics and Dynamics
multiplying vectors by other vectors: dot product
-generate a scalar quantity -multiply the magnitude of the two vectors and cosine of the angle between them *A · B = |A| |B|* cosθ -is commatative
multiplying vectors by other vectors: cross product
-generates a vector quantity -multiply magnitudes of two vectors and sine of angle between them *A × B = |A| |B|* sin θ -then use right hand rule to determine direction -not commutative
Multiplying Vectors by Scalars
-magnitude will change (use magnitude A*|n|) -direction will be either parallel or antiparallel to its original direction (direction of vector B depends on the sign of n; positive=same direction as A) -If a vector A is multiplied by the scalar value n, a new vector, B, is created such that B = nA.
torque (τ)
-moment of force -generates rotational motion -depends on magnitude of force and the length of the lever arm an the angle at which force is applied
sin 0°
0
sin 90°
1
right hand rule
1. start by pointing your thumb in the direction of vector *A* 2. extend your fingers in the direction of vector *B*. you may need to rotate your wrist to get the correct configuration of thumb and fingers 3. your palm establishes the plane between the two vectors. the direction your palm points is the direction of the resultant *C*
two distinctions between kinetic friction equation and static friction equation
1. the kinetic friction equation has an equals sign. kinetic friction will have a constant value for any given combination of a coefficient of kinetic friction and normal force. 2. the two equations have a different coefficient of friction. The value of μs is always larger than the value of μk. the maximum value for static friction will always be greater than the constant value for kinetic friction -it always requires more force to get an object to start sliding than it takes to keep an object sliding
centripetal force
A force that causes an object to move in a circle
Friction
A force that opposes the movement of objects
Can a moving object be in equilibrium? why or why not?
A moving object can be in either translational or rotational equilibrium (or both). Translational equilibrium only requires the net force on an object be zero—its velocity is constant. The corresponding condition in rotational equilibrium is that net torque equals zero—its angular velocity is constant.
force (*F*)
A push or pull exerted on an object -SI unit: N, or (kg x m)/s^2
How is a scalar calculated from the product of two vectors? How is a vector calculated?
A scalar is calculated from two vectors by using the dot product: *A* · *B* = *|A| |B|* cosθ. A vector is calculated by using the cross product: *A* × *B* = *|A| |B|* sinθ.
Newton's First Law
An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. -AKA law of inertia
Newton's Second Law equation
F=ma
Newton's Third Law equation
Fab = -Fba
True or false: the earth creates a larger force on you than you create on the earth
False. Forces are always reciprocal in nature. When the Earth creates a force on a person, the person also exerts a force of the same magnitude on the Earth (in the opposite direction). The difference in masses gives the Earth an apparent acceleration of zero.
True or False: If *C* = *A* × *B*, where *A* is directed toward the right side of the page and *B* is directed to the top of the page, then *C* is directed midway between *A* and *B* at a 45° angle.
False. This would be true of an addition problem in which both vectors have equal magnitude, but it is never true for vector multiplication. To find the direction of *C*, we must use the right-hand rule. If the thumb points in the direction of *A*, and the fingers point in the direction of *B*, then our palm, *C*, points out of the page.
Weight and mass equation
Fg= weight of the object m= mass g= acceleration due to gravity
Newton's First Law equation
Fnet = m a = 0
Newton's Third Law
For every action there is an equal and opposite reaction
gravitational force
G= 6.67 x 10^-11 (N x m^2)/kg^2 m1 and m2 are the masses of the two objects r is the distance between their center of mass
name two forces in addition to mechanical manipulation (pushing or pulling forces created by contact with an object):
Gravity and frictional forces were discussed in this chapter. Electrostatic, magnetic, elastic, weak nuclear, and strong nuclear forces are other examples of forces.
when no force is being applied, the velocity must be
If there is no net force acting on an object, then that object is not experiencing an acceleration and it has a constant velocity.
What is the relationship between instantaneous velocity and instantaneous speed? Between average velocity and average speed?
Instantaneous speed is the magnitude of the instantaneous velocity vector. Average speed and average velocity may be unrelated because speed does not depend on displacement, but is rather the total distance traveled divided by time.
linear motion
Movement in a straight line.
if you have an object three times as heavy as you can lift, how could a lever be used to lift the object? where would the fulcrum need to be placed?
One could place the fulcrum one quarter of the way across the lever, closer to the object. The ratio of the lever arms would then be 3:1, which means that only one-third of the original force is necessary. (Alternatively, the fulcrum could be placed at the end with the object one-third of the way across the lever. This would again result in a 3:1 ratio of lever arms, meaning that only one-third of the original force is necessary.)
gravity
The attractive force between two objects as a result of their masses
when calculating frictional forces, how is directionality assigned?
The direction of the frictional force always opposes movement. Once the instantaneous velocity vector is known (or net force, in the case of static friction), the frictional force must be in the opposite direction.
how do the forces acting in free fall and projectile motion differ?
The only force acting in both free fall and projectile motion is gravity. If the equation for centripetal force is and force is simply mass times acceleration (from Newton's second law), then 1.7 A moving object can be in either translational or rotational equilibrium (or both). Translational equilibrium only requires the net force on an object be zero—its velocity is constant. The corresponding condition in rotational equilibrium is that net torque equals zero—its angular velocity is constant. One could place the fulcrum one quarter of the way across the lever, closer to the object. The ratio of the lever arms would then be 3:1, which means that only one-third of the original force is necessary. (Alternatively, the fulcrum could be placed at the end with the " Excerpt From: Kaplan. "Kaplan MCAT Physics and Math Review: Created for MCAT 2015 (Kaplan Test Prep)." Apple Books.
lever arm
The perpendicular distance from the axis of rotation to a line drawn along the direction of the force
at what angle of launch is a projectile going to have the greatest horizontal displacement? what angle will result in the greatest vertical displacement, assuming a level surface?
The product of sine and cosine is maximized when the angle is 45°. Because horizontal displacement relies on both measurements, the maximum horizontal displacement will also be achieved at this angle. Vertical displacement will always be zero as the object returns to the starting point. Objects launched vertically will experience the greatest vertical distance.
True or False: Total distance traveled can never be less than the total displacement.
True. Displacement considers the most direct route between two points. Distance will always be equal or larger in magnitude than displacement.
When calculating the sum of vectors *A* and *B* (*A* + *B*) we put the tail of *B* at the tip of *A*. What would be the effect of reversing this order (*B* + *A*)?
Vector addition, unlike vector multiplication, is a commutative function. The resultant of *A* + *B* is the same as *B* + *A*, so there would be no difference between the two resultants.
When calculating the difference of vectors *A* and *B* (*A* - *B*) we invert *B* and put the tail of this new vector at the tip of *A*. What would be the effect of reversing this order (*B* - *A*)?
Vector subtraction, like vector multiplication, is not a commutative function. The resultant of *A* - *B* has the same magnitude as *B* - *A*, but is oriented in the opposite direction.
Provide a definition for displacement or velocity in terms of the other variable.
Velocity is the rate of the change of the displacement of an object. Displacement is a function of velocity acting over a period of time.
Tip to tail method
a method for finding a resultant vector in which the tail of one vector is drawn from the tip of another vector
Acceleration equation
a= average acceleration delta v= change in velocity delta t= change in time
Deceleration
acceleration in the direction opposite to the initial velocity
centripetal acceleration
acceleration toward the center of a curved or circular path
Newton's Second Law
an object of mass m will accelerate when the vector sum of the forces results in some nonzero resultant force vector
Vectors
are numbers that have magnitude and direction -include displacement, velocity, acceleration, and force -may be represented by arrows; direction of arrow=direction of vector, length of arrow=magnitude of vector -often written in bold
Center of Mass of a uniform object is...
at the geometric center of the object
derived units
created by associating base units with each other (i.e. Newtons)
instantaneous acceleration
defined as the average acceleration as delta t approaches zero -on a graph of v vs. t, the tangent to the graph at any t, which corresponds to the slope of the graph at that time, indicates the instantaneous acceleration
electron-volts
energy on the atomic scale (1eV=1.6 X 10^-19 J)
free body diagrams
example:
kinetic friction (fk)
exists between a sliding object and the surface over which the object slides
static friction (fs)
exists between a stationary object and the surface upon which it rests
kinetic friction equation
f(k)=kinetic friction u(k)=coefficient of kinetic friction N= magnitude of the normal force
vector subtraction
flip the negative one and add head to tail; resultant vector is head to head A-B= A+ (-B)
air resistance
force that opposes the motion of objects that move through the air -value increases as the speed of object increases
if the newton is the product of kilograms and meters/secend^2, what units comprise the pound?
force will obey the same relationship mass and acceleration, regardless of the unit system. Force is always the product of mass and acceleration, so one pound (lb) must be equal to one slug*ft/s^2
Angstroms
length (1 Å = 10^-10 m)
instantaneous velocity
measure of average velocity as the change in time approaches zero
weight (Fg)
measure of gravitational force on an object's mass -vector -SI units: N
SI Units
meters, kilograms, seconds
Translational Equilibrium
net force acting on a body is zero -first condition of equilibrium=Newton's first law
scalars
numbers that have magnitude only and no direction -include distance, speed, energy, pressure, and mass -often written in italics
circular motion
occurs when a force causes an object to curve in a circular pathway -upon completion of one cycle, displacement=0
rotational equilibrium
occurs when an object's net torque is zero -clockwise rotation= negative
rotational motion
occurs when forces are applied against an object in such a way as to cause the object to rotate around a fixed pivot point (fulcrum)
Torque equation
r=length of the lever arm F=magnitude of the force Ѳ= angel between the lever arm and force vectors
distance (d)
scalar quantity of the length of a path between two points (the path traveled)
base units
standard units around which the system itself is designed
terminal velocity
the constant velocity of a falling object when the force of air resistance is equal in magnitude and opposite in direction to the force of gravity
projectile motion
the curved path that an object follows when thrown, launched, or otherwise projected near the surface of Earth
mass (m)
the measure of a body's inertia-the amount of matter in the object -scalar -SI unit: kg
uniform circular motion
the movement of an object at a constant speed around a circle with a fixed radius
speed (v)
the rate of actual distance traveled in a given unit of time
Acceleration (a)
the rate of change of velocity that an object experiences as a result of some applied force -vector -SI units: m/s^2
dynamics
the study of forces and torques
resultant
the sum of two or more vectors
velocity (*v*)
the vector representation of the change in displacement with respect to time SI units: m/s
Displacement (*x* or *d*)
the vector representation of the change in position of an object in space
inclined planes
to solve, most often gravity must be split into two components
static friction equation
where f(s)=static friction u(s)= coefficient of static friction N= magnitude of the normal force
order the following units from smallest to largest: centimeter, angstrom, inch, mile, foot
ångström < centimeter < inch < foot < mile
Average velocity equation
∆x=change in position ∆t= change in time
