MCAT Review - Kinematics & Dynamics
Distance
(Italicized d) - Distance traveled, considers the pathway taken and is a scalar quantity
How do you find the sum or resultant of two vectors A&B?
** when adding vectors, always add tip to tail ** One way to find the sum or resultant of two vectors A & B is to place the tail of B at the tip of A without changing, either the length or the direction of either arrow
Displacement
- A vector quality; therefore, it has both magnitude and direction. An object in motion may experience a change in its position in space, known as displacement (x or d). The displacement vector connects (in a straight line) the objects initial position and it's final position. - Displacement does not account for the actual pathway taken between the initial to the final positions - only the net change in position from initial the final.
Speed
- scalar quantity The rate of actual distance traveled in a given unit of time
Velocity
- vector quantity Its magnitude is measured as the rate of change of displacement in a given unit of time; SI units are m/s The direction of the velocity vector is necessarily the same as the direction of the displacement factor.
The inequality that describes the magnitude of static friction is
0 < fs < usN where us is the coefficient of static friction N is the magnitude of the normal force
To find the resultant of a vector using the compliment method, follow what four steps?
1) resolve the vectors to be added into their c- and y-components. 2) add the x-component of the resultant (Rx). At the y-components to get the y-component of the resultant (Ry). 3) find the magnitude of the resultant by using the Pythagorean theorem. If Rx and Ry are the components of the resultant then, R = sqrt Rx^2 + Ry^2a. 4) find the direction (@) of the resultant by using the relationship @tan = tan^-1 Ry/Rx
What are the two important distinction between the equation for kinetic friction and static friction?
1) the kinetic friction equation has an =; this means that kinetic friction will have a constant value for any given combination of a coefficient of kinetic friction in normal force 2) the two equations have a different coefficient of friction; the value of us is always larger than the value of uk—> the maximum value for static friction will always be greater than the constant value for kinetic friction (object will stick until they start moving and then will slide more easily over one another)
Provide a description of newtons laws of motion.
1: in the absence of any forces - or when the net force is zero - there will be no change in velocity. 2: acceleration results from the sum of the first factors. 3: for any to interacting objects, all forces acting on one object, have an equal and opposing force acting on the other object.
During a test crash, a 500 kg car is driven at a constant velocity of 50 mph until it hits a wall without braking. Apply all three of Newton's laws to this situation.
1: prior to the collision, the vehicle is traveling at a constant velocity, which indicates that there is no acceleration and no net force. 2: the collision with the wall, create a sudden deceleration. Because there is acceleration, there must be a net force. The value of the net force can be calculated by multiplying the mass of the car times acceleration. 3: when the car collides with the wall, the car exerts a force on the wall. Simultaneously, the wall exerts a force of equal magnitude in the opposite direction or the core.
Acceleration due to gravity
9.8 m/s^2
The angle of the resultant vector can also be calculated by knowing inverse trigonometric functions. What is the inverse tangent calculation?
@ = tan^-1 Y/X
Newton's first law of motion • define • equation
A body, either at rest or in motion with constant velocity will remain that way, unless I net force acts upon it Fnet = ma = 0 where Fnet is the net force m is the mass a is the acceleration
What will an object experiencing translational equilibrium have?
A constant velocity: both a constant speed and a constant direction
centripetal force
A force that always points ridially inward
Mass (m)
A measure of a bodies inertia - the amount of matter in the object - A scalar quantity, and, as such, has magnitude only - SI units: kilogram, which is depending on gravity
Weight (Fg)
A measure of gravitational force on an objects mass - A vector quantity, because weight is a force - SI units: Newton
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.
How is a scalar calculated from the product of two factors? How is a vector calculated?
A scaler is calculated from two vectors by using the dot product: A•B = |A| |B| cos@ A vector is calculated by using the cross product: AxB = |A| |B| sin@
Friction
A type of force that opposes the movement of objects - Unlike other kinds of forces, such as gravity, or electromagnetic force, which can use objects, either to speed up, or slow down, friction forces always oppose an objects motion, and cause it to slow down or to become stationary
acceleration due to gravity
Acceleration due to gravity, g, decreases with height above the Earth, and increases the closer one gets to the Earth center of mass. Near the Earth surface, use g = 10 m/s^2
Deceleration
Acceleration in the direction opposite the initial velocity
Gravity
An attractive force that is felt by all forms of matter - all objects, exert gravitational forces on each other; there is a small (but measurable) force of gravity between you and this book, the chair, you're sitting on, and all the objects around you. - Gravitational forces usually do not have much significance of a small scale because other forces tend to be much larger and magnitude.
Order the following units from smallest to largest: centimeters, Angstrom, inch, mile, foot.
Angstrom < cm < inch < foot < mile
Components
Another method for finding the resultant of several factors involved, breaking each factor into perpendicular components. In most cases, these components are horizontal and vertical (x- and y-components); however, in some instances, such as inclined planes, it makes more sense to find the components as parallel and perpendicular to some other surface.
Kinetic friction occurs
Any time to surfaces slide against each other Ex: a tire that is rolling along the road does not experience kinetic friction; however, attire, sliding on an icy patch does
Average speed versus average velocity
Average speed will not necessarily always be equal to the magnitude of the average velocity; this is because average velocity is the ratio of the displacement factor over the change in time (vector), whereas average speed is the ratio of total distance traveled (scalar) over the change in time. v = delta x / delta t where v is the average velocity Delta x is the change in position Delta t is the change in time
Dot Product of Vectors
A•B = |A| |B| cos@
base unit vs derived unit
Base unit - standard units around which the system itself is designed Derived unit - created by associating base units with each other
Given any vector V, how would you find the x- and y-components?
By drawing a right triangle with V as the hypotenuses
Contact points
Contact points are the places were friction occurs between two rough surfaces sliding past each other. If the normal load - the force that squeezes the two together - rises, the total area of contact increases. That increase, more than the surface is roughness, governs the degree of friction.
instantaneous acceleration
Defined as the average acceleration as a delta t approaches zero
Energy on the atomic scale can be expressed in
Electron-volts (1eV = 1.6 x 10^-19 J) - Represents the amount of energy gained by an electronic seller, and he through a potential difference of one bolt
Name two forces in addition to mechanical manipulation (pushing or pulling force, is created by contact with an object):
Electrostatic, magnetic, elastic, weak nuclear, and strong nuclear forces
projectile motion • define
Emotion that follows a path along to 2 dimensions
Translational Equilibrium
Exist only when the vector sum of all the forces acting on an object is zero
Rational equilibrium
Exist only when the vector sum of all the torques acting on the object is zero. Called the second condition of equilibrium
When are linear motion equations used most frequently?
Falling objects, exhibit linear motion with constant acceleration; this one dimensional motion can be fully described by the equation
True or false: the Earth create a larger force on you than you create on the Earth.
False. Forces are always reciprocal in nature. When the Earth exerts 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 massive skip the Earth an apparent acceleration of zero.
True or false: if C = AxB, 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 there's some points in the direction of A, in the fingers pointing the direction of B, then our palm, C, points out of the page.
Circular motion equation
Fc = mv^2/r Fc is the magnitude of the centripetal force m is the mass v is the speed r is the radius of the circular path
Magnitude of the gravitational force between two objects
Fg = Gm1m2/r^2 where G is the universal gravitational constant (6.67 x10^-11 N•m^s / kg^s) m1 and m2 are the masses of the two objects r is the distance between their centers of mass - equation is commonly tested in the context of proportionality
While mass and weight or not synonymous, they are related by the equation:
Fg = mg where Fg is the weight of the object m is its mass g is acceleration due to gravity (9.8 m/s^2)
What are the metric units for force, work and energy, and power?
Force - Newton {kg•m/s^2} Work & Energy - joule {kg•m^2/s^2} Power - watt {kg•m^2/s^3}
If the Newton is the product of kilograms in meters per second squared, what units comprise the pound?
Force will obey the same relationship with mass and acceleration, regardless of the unit system. Forces always the product of mass and acceleration, so 1 pound (lb) must be equal to one slug•ft/s^2.
centripetal acceleration
Generated by centripetal forces - It is this acceleration generated by the centripetal force that keeps an objects in its circular pathway; when the 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.
What is the equation for centripetal acceleration?
If the equation for centripetal force is Fc = mv^2/r And force is simply mass times acceleration (from Newton second law), then — ac = v2/r
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.
In uniform circular motion, what is the tangent of force?
In uniform circular motion, the tangential forces zero, because there's no changing the speed of the object.
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 he does not depend on displacement, but it's rather the total distance traveled divided by time.
air resistance
Like friction, air resistance, opposes the motion of an object. Its value increases as the speed of the object increases
What does torque depend on?
Magnitude of the force The direction in which it acts The point at which it is applied on the object
Incline planes and force • describe • working with inclines planes • equations
Motion in 2 dimensions When working with an inclined plane, it is often best to divide force vectors into components that are parallel and perpendicular to the plane. Fg, || = mg sin @ Fg, (perpendicular line) = mg cos @ where Fg, || is the component of gravity parallel to the plane (oriented down the plane) Fg, (perpendicular line) is the component of gravity perpendicular to the plane (oriented into the plane) m is the mass g is acceleration due to gravity @ is the angle of the incline
How do you generate a scalar quality like work?
Multiply the magnitude of the two factors of interest (force and displacement) and the cosine of the angle between the two vectors.
SI unit of Force (F)
Newton kg•m/s^2
Vectors are
Numbers that have magnitude and direction Their qualities include displacement, velocity, acceleration, and force.
Scalars
Numbers that have magnitude only and no direction. Qualities include distance, speed, energy, pressure, and mass
Objects in projectile motion experience force, and acceleration only in what direction?
Objects in projectile motion on earth, such as cannonballs, baseball, or footballs, experience, the force and acceleration of gravity, only in the vertical direction (along the Y axis). This means that vy will change at the rate of g, but vx will remain constant
rotational motion
Occurs when forces are applied against an object in such a way as to cause the object to rotate around a fix pivot point
circular motion
Occurs when forces cause an object to move in a circular pathway Upon completion of one cycle, the placement of the object is zero
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 closer across the the lever, closer to the object. The ratio of the lever arms would then be 3:1, which means only 1/3 of the original force would be needed.
The Newton
SI unit of force that is derived from kilograms, meters, and seconds: 1 N = 1 kg•m / s^2
There are two types of friction, cold static and kinetic. Describe.
Static friction (fs) exist between a stationary object in the surface upon which it rests. Kinetic friction exists between a sliding object in the surface over which the object slides.
vector subtraction
Subtracting one vector from another can be accomplished by adding a vector with equal magnitude - but opposite direction - to the first vector. A - B = A + (-B), where -B represents a vector with the same magnitude as B, but pointing in the opposite direction
If we know X and Y, we can find V by
The Pythagorean theorem X^2 + Y^2 = V^2 or V = sqrt X^2 + Y^2
normal force
The component of the force between two objects in contact that is perpendicular to the plane of the contact between the object in the surface, upon which it rests
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.
lever arm
The distance between the applied force in the fulcrum
fulcrum
The fixed point around which a lever pivots
uniform circular motion
The instantaneous velocity vector that is always change it to the circular path This means that the object moving in the circular path has a tendency (inertia) to break out of it, circular path way, and moving a linear direction along the tangent
instantaneous velocity
The measure of the average velocity as the change in time approaches zero
linear motion
The objects velocity and acceleration are along a line of motion, so the pathway of the moving object continues along a straight line
How did the forces acting in free fall and projectile motion differ?
The only force acting in both free fall and projectile motion is gravity.
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 launch vertically will experience the greatest vertical distance.
Acceleration (a)
The rate of change of velocity of an object experiences, as a result of some applied force - Vector quantity - SI units: meters per second squared
Dynamics
The study of forces and torque
Inertia
The tendency of an object to resist a change in motion
Describe the velocities and acceleration of projectile motion.
The velocity and acceleration's in the two directions (usually horizontal and vertical) are independent of each other, and must be analyzed separately
center of mass or gravity
The weight of an object can be thought of as being applied at a single point in that object - the center of mass of a uniform object is at the geometric center of the object
drag force
Therefore, an object in freefall will experience a growing drag force as the magnitude of its velocity increases. Eventually, this drag force will be equal in magnitude to the weight of the object, and the object will fall with a constant velocity, according to newtons first law.
Newtons third law of motion • define • physical contact • equation
To every action, there is always in a post 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 - physical contact is not necessary for newtons third law; the mutual gravitational pull between the Earth and the moon, traverses hundreds of thousands of kilometers of space Fab = -Fba
How do you determine the magnitude and direction of the new vector B?
To find the magnitude of the new vector, B, simply multiply the magnitude of A by |n| (absolute value). To determine the direction of the vector, B, we must look at the sign of n - if n is a positive number, then B & A are in the same direction. However, if n is a negative number, then B & A point in opposite directions.
Application of force at some distance from the fulcrum generates what?
Torque, or the moment of force
How do you determine if a torque is positive or negative?
Torques that generate clockwise rotation are considered negative, while torques that generate counterclockwise rotation are positive.
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.
Linear motion equations
V = V0 + at x = V0t + 1/2at^2 V^2 = V0^2 + 2ax where x, v, and a are the displacement, velocity, and acceleration vector's v0 is the initial velocity v is the average velocity t it's time
What is the magnitude of a vector with the following components? X = 3 m/s Y = 4 m/s
V = sqrt 3^2 + 4^2 = sqrt 25 = 5 m/s
When calculating the sum of vectors A and B (A+B), we put the tale 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.
Multiplying Vectors by Scalars
When a vector is multiplied by scaler, it's magnitude will change. It's direction will be either parallel or anti-parallel to its original direction. If a vector A is multiplied by the scaler value n, a new vector, B, is created such that: B = nA
How do you generate a third vector?
When generating a third vector, like torque, we need to determine both it's magnitude and direction. To do so, we multiply the magnitude of the two vectors of interest (force and lever arm) and the sine of the angle between the two vectors. Once we have the magnitude, we use the right hand rule to determine its direction. This is called the cross product: AxB = |A| |B| sin@
How are vectors represented?
With arrows - the length shows the magnitude and the direction of the arrow shows the direction of the vector. Common notations for a vector quantity, or either an arrow or bold face
How are scalar quantities generally represented?
With italic type: the distance between two points could be represented by an italic d.
Find the x- and y-components of the following vector: V = 10 m/s @ = 30°
X = Vcos@ = 10cos (30°) = 10 x sqrt 3/2 = 5 sqrt 3 m/s Y = Vsin@ = 10sin (30°) = 10 x 1/2 = 5 m/s
Average acceleration is defined as
a = delta v / delta t where a is the average acceleration?l delta v is the change in velocity delta t is the change in time
velocity-versus-time graph
a line graph that places time on the x-axis and velocity on the y-axis; the slope at any point will be the instantaneous acceleration at that point and the area under the curve will be the displacement for that time interval If the slope is positive, comment in the acceleration is positive, and in the same direction is the velocity If the slope is negative, then the acceleration is negative, and in the opposite direction of the velocity (deceleration).
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 the length of the arrows must be proportional to the magnitude of the vectors. the vector sum of A + B is the vector joining the tail of A to the tip of B and pointing toward the tip of B.
Free Body Diagram (FBD)
a simple drawing of an object showing all the forces that are acting on it
coefficient of static friction
a unitless quantity that is dependent on the two materials in contact
Force (F)
a vector quantity that is experienced as pushing or pulling on objects - can exist between objects that are not even touching; while it is common for forces to be exerted by one object pushing on another, there are even more instances in which forces exist between objects, nowhere near each other, such as gravity, or electrostatic forces
instantaneous acceleration equation
a= lim as t→0 ∆v/∆t
First condition of equilibrium
an object is in translational equilibrium when the sum of forces pushing it one direction is counterbalanced by the sum of forces acting in the opposite direction —> a reiteration of newtons first law When the resultant force upon an object is zero, the object will not accelerate; this may mean that the object is stationary, but it could just as well mean that the object is moving with a constant nonzero velocity.
Newton's second law of motion • define • relationship with first law • equation
an object of mass (m) will accelerate when the vector sum of the forces resultant some 90 resultant force vector What newton second law state is actually a corollary of the first: no acceleration will occur when the vector, some of the forces result in a cancellation of those forces Fnet = ma where Fnet is the net force m is the mass a is the acceleration
How do you measure the magnitude of kinetic friction?
fk = ukN where uk is the coefficient of kinetic friction? N is the normal force
Every change in velocity is motivated by _________
force (push or pull)
The resultant of a cross product will always be
perpendicular to the plane created by the two vectors
free fall acceleration
the acceleration of an object due only to the effect of gravity
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
instantaneous speed
the speed of an object at one instant of time - as a measure of speed, instantaneous speed is a scalar number.
resultant
the sum of two or more vectors
equation of torque
torque = r x F = RF sin @ where r is the length of the lever arm F is the magnitude of the force @ is the angle between the lever arm and force vectors
What generates rotational motion?
torque, not the mirror application of the force itself
instantaneous velocity equation
v= lim as t→0 ∆x/∆t where v is the instantaneous velocity, Delta x is the change in position, and delta T is the change in time
When does translational motion occur?
when forces cause an object to move without any rotation
component vectors equations x= y=
x = v cos@ y = v sin@
For each quantity listed, named the unit and symbol, according to the SI unit/MKS system. • length • mass (not weight) • time • current • amount of substance • temperature • luminous intensity
• length - meter (m) • mass (not weight) - kilogram (kg) • time - second (s) • current - ampere (A) • amount of substance - mole (mol) • temperature - kelvin (K) • luminous intensity - candela (cd)