Physics 1

45-45-90 Triangle

*X and Y are the same* Use pythagorean theorem to solve for hypotenuse

Force

*any influence capable of causing a mass to accelerate* o Examples of Force: The force due to gravity, contact forces, electrostatic forces, torque (a force at a distance from a point of rotation), tension, magnetic forces, etc. o Forces are vectors and can thus sum to zero. This is called "no net force" and is the same as if no force existed at all. If one of those forces is suddenly changed, a net force is instantly created. o Force is measured in Newtons. Determine the units included in a Newton using the equation F=ma. m=kg a=m/s^2 F=(kg*m)/s^2

Conceptually define acceleration and relate it to gravity

-Acceleration is the change in velocity each second; or the change in m/s each second; or the "rate of change" of velocity. -Because we know how velocity changes for all bodies under the influence of earth's gravity (10 m/s each second; ignoring air resistance), we can predict velocity at any time period. -*If a ball is going up, for example, it loses 10 m/s of velocity each second. If a ball is falling, it gains velocity at 10m/s each second*

MCAT Question 3

45-45-90 triangle- solve for c^2

If force increases linearly

Acceleration increases linearly

Displacement

The shortest distance between point A and point B

Projectile Motion

Think of projectile motion as nothing more than falling body problems in two dimensions. -Immediately resolve the given vectors into components; then solve in your head as you would any other motion problem.

Use this equation when asked for final or initial vertical velocity, given drop height

V = √(2gh) Also use it as a stepping stone to find the final velocity can also be written V = √(2ax) -You can calculate either initial or final velocity using this equation because, when ignoring air resistance, initial velocity and final vertical velocity are the same for any projectile (due to the symmetry of projectile motion). *This is only for objects leaving the ground and returning to earth* -If the motion starts from rest, say at the top of a cliff or building, then the initial velocity would be zero. That would always be known, or easily inferred, and wouldn't need to be calculated. *Memorize for the purposes of manipulating equations and seeing relationships between variables only*

COMMON MISCONCEPTION #3

you CANNOT accelerate a ball horizontally across the room by throwing it. To accelerate, an object must be either 1) in contact with the object creating the force, or 2) be under the influence of a field force (e.g., gravitational or electrical) at that exact moment. Thus, the ball only accelerates horizontally during the brief time it is in contact with the object creating the horizontal force—in this case, your hand. In the vertical direction the ball is always accelerating (after it leaves your hand) because it is always under the influence of earth's gravitational field.

mass →liquid conversion

1 cm3 = 1 mL= 1g 1 m3 = 1000 L 1 L of water = 1 kg

Factors that affect the magnitude of air resistance:

1) Cross-sectional Area: greater cross-sectional area = more air resistance 2) Shape: less aerodynamic = more air resistance 3) Velocity: increased velocity = more air resistance.

When you see "Constant Velocity" or "Constant Speed" THINK:

1) No acceleration 2) No net force 3) All forces sum to zero (i.e., up forces = down forces, left forces = right forces, etc.) 4) No change in direction 5) The object is in equilibrium *TOP 10 MOST FREQUENTLY TESTED MCAT PRINCIPLE*

What is the tension in a rope being pulled from opposite ends with identical forces of 50 N?

50 N

Use Pascal's Law to explain the physics of a hydraulic lift

A hydraulic lift demonstrates Pascal's Law because it is the undiminished transmittal of pressure through the incompressible liquid that accounts for the input force being translated into a magnified output force. At the input cylinder, a force F1 is applied to a plunger with a cross- sectional area of A1. That creates a pressure equal to F1/A1 that is transmitted to the output cylinder. At the output cylinder, the pressure MUST be exactly the same according to Pascal's Law. However, the cross-sectional area of the output cylinder is usually many factors larger. According to P = F/A, in order for pressure to remain constant when area increases, force must increase.

Real life examples of Simple Harmonic Motion

A pendulum and a mass on a spring are the most common. -Almost any circular motion, when viewed from the side, approximates simple harmonic motion. -Waves sloshing back and forth in a container can approximate SHM. -Molecular vibrations approximate SHM. -any movement that oscillates about an equilibrium position, and shows the characteristic sinusoidal pattern, qualifies as SHM.

Specific Gravity

A ratio that describes how dense something is *compared to water*. SG = Dsubstance/DH2O

COMMON MISCONCEPTION #2

A related misconception is that a constant force applied to an object will cause it to travel at a constant velocity. *This is absolutely NOT true*. *A constant net force will always cause a constant acceleration—and therefore a changing velocity* Rehash these relationships until they are perfectly clear in your mind

Center of Mass Center of Gravity Center of Buoyancy

A weighted average of mass distribution. -The unique point where the weighted relative position of the distributed mass sums to zero or the point where if a force is applied causes it to move in direction of force without rotation. -To calculate the center of mass of multiple objects use the following formula, where r is the displacement vector between a reference point and each mass -Center of gravity and center of buoyancy also occur at the center of the object/mass Cmass = (r1m1 + r2m2 + r3m3 . . . ) /mtotal

Pendulums Potential Energy and Kinetic Energy

-Potential Energy (PE) is at a maximum at the maximum height of the bob, and is at a minimum at the bottom of the pendulum's arc. -Kinetic energy (KE) is at a maximum at the bottom of the pendulum's arc and is at a minimum at the maximum height of the bob. -Gravitational potential energy is usually assumed to be zero for a pendulum bob at the lowest point of its arc. In other words, at that point we assume that h = 0. - One cycle for a pendulum would be movement of the bob from one side to the other, and then back to the starting point ( Once clue to thinking of this correctly is the fact that the movement must be periodic, and therefore it must repeat. When the bob is swinging back, it is doing something it has not done before. Once it gets back to the starting point, however, it is repeating the same motion; )

Use the relationship Q = AV to explain how velocity varies as blood flows throughout the human circulatory system (i.e., aorta → arteries →arterioles → capillaries → venules → veins → vena cava).

-The cross-sectional area of individual vessels decreases as you go from the aorta to the capillaries. -However, it is the TOTAL cross-sectional area that we would apply to Q = AV. -Total cross-sectional area increases as you go from aorta to capillaries. -Because area is greatest at the capillaries, the velocity of blood is lowest at the capillaries. -The reverse is true on the return trip: cross-sectional area decreases as we go from capillaries back to the vena cava. Therefore, velocity increases.

Equation for simple harmonic motion with mass on a *pendulum*

T = 2π√(L/g) L=length

Equation for simple harmonic motion with mass on a *spring*

T = 2π√(m/k) [mass on a spring] -T=time it takes to complete a cycle (period) -m is the mass of the object attached to the spring -k is the spring constant of the spring. -The equation can be interpreted to mean that more massive objects will vibrate with a longer period. -Their greater inertia means that it takes more time to complete a cycle. -And springs with a greater spring constant (stiffer springs) have a smaller period; masses attached to these springs take less time to complete a cycle. -Their greater spring constant means they exert stronger restoring forces upon the attached mass. This greater force reduces the length of time to complete one cycle of vibration

If force increases exponentially

Acceleration increases exponentially

Acceleration

Acceleration is any change in velocity *velocity is a vector with both magnitude and direction and a change to either magnitude or direction constitutes a change—and therefore acceleration* Ex :When you walk around a corner at a constant speed you are still accelerating because there is a change in direction *ACCELERATING = CHANGE IN SPEED OR DIRECTION* *Acceleration describes how quickly the velocity is changing, -Not the magnitude of the velocity itself. Going from 0 mph to 60 mph is acceleration, but so is going from 0.00001 m/s to 0.000011 m/s. For that matter, going from 5 m/s to 4 m/s is also acceleration. m/s^2 If there is no net force, there can never be acceleration according to Newton's Second Law.

A 500 kg elevator is being accelerated upward by a cable with a tension of 6,000 N. What force does the elevator exert on the cable?

According to Newton's Third Law, if the elevator cable is pulling on the elevator with 6,000 N of force, the elevator must be pulling on the rope with a force of 6,000 N.

For any object floating in any liquid, the ratio of SGobject to SGliquid will exactly equal the fraction of the object submerged in the liquid.

Any object displaces an amount of fluid exactly equal to its own volume (if fully submerged), or to the volume of whatever fraction of the object is submerged (if floating). -The weight of the displaced fluid is exactly equal to the buoyant force pushing up on the object.

Simple Harmonic Motion (SHM)

Anything that oscillates back and forth, and can be represented by a sine wave graphically, constitutes Simple Harmonic Motion.

Turbulence:

At low velocities real fluids exhibit laminar flow. -*As velocity increases, and especially for non-viscous fluids, flow becomes turbulent*—meaning that although the net flow is still in one direction, there are random eddies, changes in direction, changes in velocity, and so forth.

Terminal Velocity

At terminal velocity, the object has stopped accelerating; the forces of gravity and air resistance are now balanced. -initially the downward force of gravity is greater -Air resistance increases as velocity increases until both are equal (no net force) -velocity is maintained, acceleration=0 *At terminal velocity, mg = Fair*-

MCAT QUESTION

B; Whenever you are given the fraction of one object that floats and asked for the fraction of another object that will float, you should first try to determine the density of the liquid. In this case, the ball is 1⁄4 submerged and the SG of that object is 2.5. This means the liquid must be exactly 4 times as dense as the object, and therefore has a SG of 10. Now we know from our basic buoyancy principles that if the SG of the second ball is 2.0, it is 1/5 as dense as the liquid and will float 1/5 submerged.

What is considered a fluid?

BOTH GAS AND LIQUID

MCAT QUESTION

C; In order to answer these apparent weight/apparent mass questions, you must be 100% conceptually sound on Archimedes' principle and buoyancy. The most important thing to remember is that the BUOYANT FORCE is exactly equal to an object's apparent weight loss in Newtons. If the weight loss is given in grams or kg, then we can say that the weight loss in g/kg is EXACTLY the mass of that much volume of the fluid. In this case, the block had a mass of 10 kg out of water and 5 kg in water. This means a "block's-worth" of whatever the block is made of is two times as massive as a block's-worth of the fluid. You should know the density of water to be 1.0 g/cm3. Multiply this by two to get 2.0 g/cm3, or Answer C.

State which of these two forces is greater in each of the following: beaded water droplets; a concave-up meniscus in a burette; a concave-down meniscus in a burette; mercury in a thermometer; water flowing up a capillary tube.

Cohesive (C) vs. Adhesive (A): Beaded water droplets = C > A ; concave-up meniscus (water): A > C ; concave-down meniscus (mercury): C > A ; Mercury in a thermometer: C > A ; water moving up a capillary tube: A > C.

Cohesive vs. Adhesive forces (Intermolecular forces of *liquid*)

Cohesive forces are intermolecular forces BETWEEN the molecules of a liquid, binding the molecules to one another -Adhesive forces are intermolecular forces between the molecules of the liquid and the molecules of the container. -This is well illustrated by the meniscus formed when water is in a burette. The adhesive forces are greater than the cohesive forces, and therefore the water appears to "stick" to the sides of the burette (i.e., concave-up meniscus). -Water droplets on a freshly waxed car are a good example of cohesive forces exceeding adhesive forces. There are intermolecular forces between the water and the surface of the car, but the cohesive forces between water molecules are much greater, causing them to coalesce into a sphere-like droplet.

Normal Force

Contact force

The Buoyant Force

Fbuoyant = ρvg ; -Where v is the *volume of fluid displaced*, -ρ is the *density of the fluid* -the buoyant force is always exactly equal to the weight of the amount of fluid displaced by the object., -a characteristic of liquids (as compared to solids) that liquids can permanently resist a compressive force applied normal to their surface, but not parallel to their surface. Solids permanently resist both (up to the elastic limit of the solid). - This gives us a rather general sense for the fact that liquids want to push objects up toward their surfaces -does *NOT* change with depth (h is not in the equation) -does *NOT* change with mass (m is not in the equation!)

Kinetic Friction Equations

Ff= μk x F(normal) or Ff= μk x mg x cosθ -θ is the angle between the force ,mg ,and a line perpendicular to the sliding surface (i.e., the angle between the mg vector and a line normal to the surface)

Force due to gravity

Fgravity = mg

If a projectile has an initial velocity of 60 m/s at an angle of 30 degrees from the horizontal, for how many seconds will the ball be in the air?

First, find the vertical component, which is all we care about when determining time in the air. The y-component is given by 60(sin30) = 60*0.5 = 30 m/s. This projectile will take 3 seconds to reach max height and 3 more seconds to reach the ground, for a total of 6 seconds.

Laminar Flow

Fluid flows in pipes in concentric sheets, each with different velocities. -The fastest flow is at the exact center of the pipe and the slowest is at the interface with the wall of the pipe.

Newton's 2nd Law

Fnet = ma, (where Fnet is the vector sum of all forces acting on the object) *The MCAT will try to make problems sound difficult that can really be easily solved using Newton's Second Law. Whenever force, mass, or acceleration are asked for, first try to solve using Fnet = ma.*

Density ratio of objects in floating liquids

For objects floating in liquids, the fraction of the object submerged = the ratio of the density of the object to the density of the liquid.

Calculating frequency for SHM

Frequency is the inverse of period, so the above equations can also be used to calculate frequency for these systems. -In that case, set frequency equal to the same equation, but inverted. -f=1/T= 1/2π √(k/m)

Real life examples of pendulums

Grandfather clock, swing

Adding, subtracting

Head to tail method https://www.youtube.com/watch?v=giGCuCN6OsY Determine the magnitude of the resultant: align vectors (with appropriate subtraction and addition) and use pythagorean theorem or Component Method Ay=Asinθ Ax=Acosθ Rx= Ax +Bx Ry= Ay+By R= √x^2 + y^2 angle and direction of resultan(inverse tangent): tanθ-1 (Ry/Rx)

Range

Horizontal distance traveled -A new value asked for in projectile motion. It is the product of velocity in the x-direction (Vx) and time (t). Range = Vx t

MCAT Question 2

Horizontal velocity is constant, no acceleration *normal force is perpendicular to the surface-when there is a surface these two will cancel out!*

Apply Bernoulli's Equation to the flow of an ideal fluid through a horizontal pipe

If fluid velocity increases, pressure decreases. -This can occur where the fluid encounters a section of pipe with a smaller cross-sectional area. In order to maintain flow rate, the velocity must increase in the narrower section of pipe. -In order for all three energy terms to continually sum to K, an increase in the (1/2)ρv^2 term must be accompanied by a decrease in one or both of the other terms. In this case, it is the P term. -If the increase in velocity were also associated with a decrease in height, both the P term and the ρgh term would decrease, and both would contribute to an increase in the (1/2)ρv2 term.

MCAT QUESTION

If no, velocity use x=1/2at^2 to find t

Pascal's Law:

If pressure increases at any point in a confined, incompressible fluid, it increases by that same amount at every other point within that fluid.

Density ratio of objects floating in water

If the liquid in which it is submerged is water, the fraction submerged is equal to the specific gravity.

Mass

Mass = a measure of an object's inertia

Do Ideal gases have intermolecular forces?

NO

Specific gravity of a liquid

SG of the liquid is DLiquid/DH2O For any object floating in any liquid, the ratio of SGobject to SGliquid will exactly equal the fraction of the object submerged in the liquid.

Scalars

Scalars have only magnitude. Ex: mass, temperature, speed, work, energy, charge, time, density

If velocity and acceleration have differed signs

Slowing Down (either direction)

If velocity and acceleration have the same sign

Speeding up (either direction

Flow Rate Biology Connection

This formula is often used to describe fluid flow in the cardiovascular system. -The variable A is the cross-sectional area of the blood or lymph vessel, and -V is velocity -Q is a function of cardiac output. *Cardiac output = stroke volume x heart rate*

Apparent Weight

This is an important point. The apparent weight of a submerged object is the actual weight minus the buoyant force Apparent Weight (AW) = Actual Weight (aW) - Buoyant Force (Fbuoyant) The difference between the actual weight and the apparent weight tells you: 1) the buoyant force 2) the weight of that volume of fluid.

Ideal, Non-Viscous Flow

This is how ideal, non-viscous fluids flow. -There is assumed to be no friction (drag) between the fluid and the walls of the pipe, or between fluid molecules themselves. -Fluid near the wall of the pipe flows with the same velocity as fluid at the center of the pipe. -*This is assumed on the MCAT if they do not specify otherwise*

Poiseuille Flow

This is how real, viscous fluids flow in pipes. -Real fluids exhibit laminar flow and have a leading edge that is parabolic in shape.

Acceleration DOWN THE INCLINE PLANE(derived from other equations)

a = gsinθ

Pressure is a constant for any one vertical depth within the same fluid (horizontally pressure is the same)

even if a convoluted pipe leads to that depth; even if the surface of the fluid is not directly above the point of measurement; and whether the container is 5 mm or 5 miles wide. In other words, if a pipe exits horizontally from an enormous 1,000-meter-tall water storage tank, the fluid pressure inside that pipe is the same as the fluid pressure inside the tank at that same vertical distance below the surface of the fluid.

Explain conceptually why an entire pin can float on water, but a small section cut from the end of the very same pin will not float. Doesn't the small section of the pin weigh less?

To break through the surface of the liquid, a certain number of these bonds must be broken. The greater the ratio of length to mass, the greater the chance that the mass of the object can be supported by a large number of bonds. This is the case when the pin is whole. -For the small section of the pin, however, it does not cover a very large area of the surface and therefore fewer intermolecular forces are involved. In this second case, the mass is greater than what can be supported by the small number of intermolecular bonds, and it is therefore able to break through the surface. -A water skeeter can stay afloat atop the water because of this principle. However, if the water skeeter tried to balance on one leg, he would quickly sink!

Units of Pressure

UNITS = Pascals, mmHg, atm or Torr

MCAT QUESTION

Using a barometer, atmospheric pressure can be calculated by simply measuring the height to which a fluid rises in an evacuated cylinder that has been allowed to equilibrate at atmospheric pressure—and then calculating the fluid pressure at the base of that column using P = pgh. Plug the info given into pgh, remembering to change centimeters into meters. Recall that the density of water is 1,000 kg/m3 and thus the density of mercury is 14 times that number. This gives: (14,000 kg/m3)(10 m/s2)(0.5 m) = 70,000 Pascals.

Gauge pressure

measured with respect to atmospheric pressure, where atmospheric pressure is defined as zero gauge pressure. -gauge pressure is the amount of pressure in excess of the ambient atmospheric pressure

The time it takes to complete one pendulum cycle

the time to do this complete, non-repetitive motion is defined as one period.

Equation for displacement

x=1/2at^2 *Memorize for the purposes of manipulating equations and seeing relationships between variables only*

Friction

*Friction opposes sliding, NOT motion*

The Bernoulli Effect

*If fluid velocity increases, pressure decreases* -In an airplane, the pressure difference arises between upper and lower surfaces of the wing because the air is traveling at different speeds above and below the wing. -The effect of fluid speed on pressure is called the Bernoulli effect

Convert between Pascals, mmHg, atm and Torr

100,000 Pascals = 1 atm = 760 mmHg = 760 Torr

Density of water

1000 kg/m3 or 1.0 g/cm3

Flow Rate

Q=AV -A = total cross sectional area *if a large pipe splits into to smaller pipes you must add the cross-sections of both new pipes to get the new area* -V = velocity

-Air density, like air pressure, ____________ with increasing altitude. It also changes with variation in temperature and humidity.

Air density and air pressure decrease with increasing altitude. -It also changes with variation in temperature and humidity.

Archimedes' Principle

Any object displaces an amount of fluid exactly equal to its own volume (if fully submerged), or to the volume of whatever fraction of the object is submerged (if floating). -The weight of the displaced fluid is exactly equal to the buoyant force pushing up on the object.

Density

D=m/v

Bernoulli's Equation

K=P+ ρgh+(1⁄2) ρv2 -Bernoulli's Equation demonstrates the Law of Conservation of Energy: -the random vibrational energy of the fluid molecules is given by P (a.k.a. "Pressure Energy") -the gravitational potential energy per volume of the fluid is given by ρgh, where h is *height* -the kinetic energy per volume of moving fluid molecules is given by (1⁄2)ρv2 -The sum of these three forms of energy in an ideal fluid is always equal to a constant (K). Energy is transferred from one form to the other, but the sum of the components will never change.

Fluid Pressure Formula

P= ρgh **ρ IN kg/m^3 Fluid pressure can be thought of as the average force of molecular collisions per unit area,or as the weight of the column of fluid above the point of measurement. Note that in this case h is depth, NOT height. *ANSWER IS IN PASCALS*

General Pressure Formula:

P=F/A -equation can be used for all units

Absolute Pressure

The actual total pressure of the atmosphere, plus gauge pressure. An example most students are familiar with would be tire pressure. If your tire had a pressure of 1 atm at sea level, it would appear to you to have no pressure whatsoever. However, it clearly has some pressure—it is just the same pressure as the air around you. When you inflate the tire to 35 psi, you are really inflating it to 35 psi ABOVE atmospheric pressure. Atmospheric pressure is about 15 psi at sea level, so the actual total pressure is 50 psi. -It is the gauge pressure that is 35 psi.

What value must be low for a pendulum to exhibit Simple Harmonic Motion?

The angle of displacement must be small for SHM

Atmospheric pressure

The fluid pressure due to the earth's atmosphere at that location. -You could conceptualize it as the weight of the column of air above that point—explaining why atmospheric pressure decreases with increased elevation.

Fluid pressure

The force per unit area at some point within a fluid. -atmospheric pressure is really just a type of fluid pressure wherein air is the fluid. -If we examine common usage, however, when people refer to "fluid pressure" they are usually referring to the pressure at some depth within a liquid.

Surface Tension (intermolecular force of *liquid*)

The intensity of intermolecular forces, per unit length, at the surface of a liquid - The greater the ratio of length to mass, the greater the chance that the mass of the object can be supported by a large number of bonds.

Why does the displacement of the pendulum gradually decrease over time?

The pendulum will not continue oscillating to the same height because of the influence of non-conservative forces such as air resistance. Energy lost to these sources is not available to the bob, so it cannot travel to its original height (or attain its original PE).

Another common application of this principle is to the velocity of water exiting a spigot. Derive a formula for the velocity of water exiting a Spigot

The potential energy of the water in the tank near the top of the tank, the ρgh term, is converted to the kinetic energy of the flowing water at the spigot, the 1/2ρv2 term. To derive this equation, set 1/2ρv2 = ρgh and solve for v. This gives: v = √(2gh).

500 N is applied to an object and it does not move. 501 N is applied and it just begins to slide. Describe the amount of force that must be applied to the object continuously to move it at a constant velocity across the surface.

*Between the same two surfaces, kinetic friction is always less than static friction* -To maintain constant velocity, the applied force must exactly counterbalance the kinetic friction. Therefore, we would expect that some force less than 501N will be required. If the same force of 501N remains on the object, we can predict that it will accelerate -It typically takes more force to budge an object into motion than it does to maintain the motion once it has been started.

As the angle of incline of a plane increases,what happens to the value of a? What happens to the value of sinθ and cosθ? What happens to the normal force and to the force down the plane? What are the minimum and maximum values for acceleration down an inclined plane?

-Because the acceleration down a plane is directly related to the sine of the angle, the greater the angle, the closer the sine of the angle will be to one. -Therefore, the larger the angle, the closer the acceleration will be to 9.8m/s2. -The normal force is related to the cosine of the angle, so as the angle increases, this value gets closer to zero. Therefore, as the angle increases, the normal force decreases. -The force down an inclined plane is also related to the sine of the angle, so it too will increase as the angle of incline increases. -The theoretical maximum incline is 90 degrees, where acceleration would be 9.8 m/s2. The theoretical minimum would be a plane with no angle of incline, where acceleration down the plane would be zero.

If a projectile has an initial vertical velocity of 30 m/s, how many seconds will it take to reach its max height?

. 3 seconds

COMMON MISCONCEPTION #1

1) A constant force will NOT cause an object to accelerate faster and faster; it will cause a constant (non-changing) acceleration. -*Only a changing force can cause a changing acceleration*. 2) That being said, a constant force can cause an object to move faster and faster in which case: -the displacement is changing non-linearly -the velocity is changing linearly, -the acceleration is not changing at all (i.e., it is constant).

As an example consider the situation depicted in the diagram at the right. The free-body diagram shows the forces acting upon a 100-kg crate that is sliding down an inclined plane. The plane is inclined at an angle of 30 degrees. The coefficient of friction between the crate and the incline is 0.3. Determine the net force and acceleration of the crate.

Begin the above problem by finding the force of gravity acting upon the crate and the components of this force parallel and perpendicular to the incline. The force of gravity is 980 N and the components of this force are Fparallel = 490 N (980 N • sin 30 degrees) and Fperpendicular = 849 N (980 N • cos30 degrees). Now the normal force can be determined to be 849 N (it must balance the perpendicular component of the weight vector). The force of friction can be determined from the value of the normal force and the coefficient of friction; Ffrict is 255 N (Ffrict = "mu"*Fnorm= 0.3 • 849 N). The net force is the vector sum of all the forces. The forces directed perpendicular to the incline balance; the forces directed parallel to the incline do not balance. The net force is 235 N (490 N - 255 N). The acceleration is 2.35 m/s/s (Fnet/m = 235 N/100 kg).

Gravity

Gravity is a field that exists between any two objects with mass. *Field = an invisible influence capable of exerting a force on a mass or charge*

Calculate the distance (or height) traveled using Distance = rate*time.

Distance = Avg velocity(rate) x time

Newton's Universal Law of Gravitation

F = Gm1m2/r^2 -Often referred to as the "inverse square law" because force varies inversely with the square of the distance between their center of masses r.. -*G, the gravitational constant, is 6.67 x 10-11* Nm2/kg2, but this value is usually given. -The formula F = G m1m2/r2 gives the force *due* to gravity NOT gravity itself. -The Universal Law of Gravitation is true everywhere. Near earth, however, we make an assumption that gravity is a constant 10 m/s2, despite the fact that this law shows that gravity actually varies ever so slightly with height - Based on the near-earth assumption, we can simplify the equation to: F = mg (m2/r^2 is estimated to be 10m/s^2)

Hooke's Law

F = k∆x *where ∆x is the displacement of the spring from its equilibrium point, NOT the overall length of the spring* -compress spring x is negative -stretch spring x is positive -k is the proportionality constant, often referred to as the spring constant. -The spring constant is a positive constant whose value is dependent upon the spring which is being studied. A stiff spring would have a high spring constant. This is to say that it would take a relatively large amount of force to cause a little displacement. Springs, and many other items such as resilient solids, rubber, and even bonds between atoms, follow Hooke's Law. -spring accelerates to equilibrium pt then decelerates

Inclined Plane Formulas

F = mgsinθ ; Force down an inclined plane, parallel to the surface F(normal) = mgcosθ ; Normal Force on an inclined plane Vf = √(2gh) ; Velocity of a particle at the base of an inclined plane

True or False? 1) A ball moving with twice the kinetic energy can compress a spring twice as far.

False. Whatever KE the ball has will be transferred completely into elastic PE so we can write KE = (1/2)kx^2. We see that KE is related to the square of x, so it will require four times the kinetic energy to compress the spring twice as far

Static Friction Equations

Ff=μs x F(normal force exerted on the object) or Ff= μs x mg x cosθ θ is the angle between the force, mg, and aline perpendicular to the sliding surface (i.e., the angle between the mg vector and a line normal to the surface)

Maximum Static Friction

In cases of static friction, the friction created before an object begins to slide will always remain equal to the net applied force which the friction is opposing. -For example, if you push on a boulder with 20 N of force, there will be 20 N of static frictional force opposing you. If you increase the force you apply to 100 N, the static friction will also increase to 100N. This continues up to the "maximum static friction." -Once the value of maximum static friction is exceeded, the object will begin to slide and we then have a case of kinetic friction, NOT static.

What speed to objects in a vacuum fall at?

Objects fall at the same speed because there is no air or other objects present (EMP video of man dropping feather and book on moon) -most of the universe is made up of the vacuum of space -Objects in a vacuum do not attain a terminal velocity because there is no air resistance. They continually accelerate.

What causes friction on a microscopic level?

On the microscopic level, even the smoothest surface is actually quite rough. The peaks, valleys, protrusions, etc., of the two surfaces literally collide with one another

When is an object in free fall?

Only if gravity is pulling it down and no other forces are acting on it(only where there is no air)

Elastic Potential Energy

PE = (1/2)k∆x^2 The potential energy stored in a compressed spring (or in any other object that follows Hooke's Law) -Although the above formula can obviously be used to solve for k, ∆x or PE, it is more likely to be used in connection with the conservation of energy. If a ball with velocity v strikes a spring, compressing it, all of the ball's kinetic energy will turn into elastic potential energy. Setting the initial kinetic energy equal to the final potential energy allows you to predict how far the spring will compress (∆x).

Why does V=√(2gh) work for either falling bodies or a mass on an inclined plane?

The formula V = √(2gh) is derived from the law of conservation of energy by equating mgh to (1/2)mv2 and solving for v. As long as friction and air resistance are ignored, energy will be conserved, whether the object falls directly to the ground, or rolls down a plane.

MCAT QUESTION

This question has two important parts. First, you must decide if the equations presented are actually a valid way to arrive at g. Second, you must decide which one involves the most "easily measurable" experimental data. Actually, all of the equations are accurate. Answer A is a derivation of F = mg, answer B is a derivation of P = pgh, answer C is a derivation of F = Gmm/r2, and answer D is a derivation of v = √(2gh). Answer A is false because it would require the measurement of the force due to gravity at some instant during free fall— almost impossible to measure practically. Answer B is false because it requires the measurement of fluid pressure, which is most accurately described as P = (the sum of all of the forces on a submerged object due to the collisions it experiences with molecules of the liquid)/surface area (any ideas on how to get that one in the lab?). Answer C has a constant, G, which could not be measured directly, and the radius of the earth. This leaves D, which is the correct answer. It would be reasonably simple in the lab to measure the final velocity of an object using radar or via a photo trigger and a timer. The height, of course, could be measured using a meter stick, tape measure or some other advanced device =).

Calculating the Spring Constant from Hanging Weights:

To calculate the spring constant, solve for k using Hooke's Law. -For ∆x, enter the displacement from the equilibrium point for one trial, or the difference in displacement between two trials. -For F, use the force applied in one trial, or the difference in the force applied between two trials. *CAUTION: It is a common mistake to plug in the mass of the block hanging on a spring for the force. You need to convert that mass into a force using F = mg* -Be careful here; always note on this type of problem that you can only calculate the spring constant if it is the same, or "identical" spring used in both trials.

Vectors

Vectors have both *magnitude* and *direction*. Ex: Velocity, displacement, acceleration, force, weight, electric field, magnetic field, momentum, torque, impulse

Track the velocity of a ball thrown by a pitcher—from the moment the pitcher begins his pitching motion until the ball hits the ground (no catcher, ignoring air resistance).

(Note: This problem assumes that the pitcher throws the ball perfectly horizontally, with no vertical component to the velocity.) -Ignoring air resistance, the ball starts at rest in the pitcher's hand. -The ball experiences a force created by the pitcher that accelerates it forward. -This acceleration only exists during the time that the pitcher's hand (the source of the force) is in actual contact with the ball. -Once the ball is released, it becomes a projectile in free fall. It will maintain the exact horizontal velocity it had at the moment of release throughout its flight. - Gravity will accelerate the ball toward the earth at 9.8m/s2 in the vertical direction only. There will be no acceleration in the horizontal direction, and therefore horizontal velocity will be constant at all times. -When the ball strikes the ground, it will have the vertical component of final velocity created by its fall from the height of release to the ground, plus the same constant horizontal velocity it had throughout the flight. *NOTE that it will be in the air for the same amount of time as it would have been had the pitcher dropped it from the same height and let it fall directly to his feet.- once released gravity is the only force acting on it*

When can speed be treated the same as velocity?

*On the MCAT, you can treat "speed" the same as "velocity" if and only if the question makes it clear that the distance traveled is along a straight line* When this is the case, the magnitude of the distance equals the magnitude of the displacement and the magnitude of the speed = the magnitude of the velocity. The MCAT authors seem to have a small penchant for using terms like "constant speed" instead of "constant velocity," so don't let that surprise or confuse you.

Resolving Vectors into Components

*PAY ATTENTION TO THE ANGLE* -If measured with respect to (w.r.t.) the x-axis, Vcosθ gives the x-component and Vsin gives the y-component (where V is the magnitude of the original vector); if measured w.r.t. the y-axis, however, Vsinθ gives the x-component and Vcos gives the y-component.

Distance vs Displacement

-Distance is a measurement of travel along any path, curved, linear or otherwise. -Displacement is a vector indicating the change in position from start to end. -The magnitude of the displacement is the shortest distance between the starting and ending points. *Distance is dependent on path* *Displacement is path-independent* Distance= (Rate)(Time)

Kinetic Friction vs. Static Friction:

-If there's sliding, it's kinetic friction; -if there's no sliding, it's static friction.

What causes friction on the molecular level?

-On the molecular level, the electron clouds of atoms do occupy real space. -When we try to get them to occupy the same space, or bring them very close together, they will repel one another. -This is why what we call "contact forces" are often classified as an example of electromagnetic forces. -This interaction is actually due to repulsion between electrons. -Everyday objects do not actually touch; rather, contact forces are result of the interactions of the electrons at or near the surfaces of the objects (exchange force). This includes friction.

Suppose that an elephant and a feather are dropped off a very tall building from the same height at the same time. Why does the elephant fall faster?

-The elephant and the feather are each being pulled downward due to the force of gravity. When initially dropped, this force of gravity is an unbalanced force. Thus, both elephant and feather begin to accelerate (i.e., gain speed). As the elephant and the feather begin to gain speed, they encounter the upward force of air resistance. Air resistance is the result of an object plowing through a layer of air and colliding with air molecules. The more air molecules which an object collides with, the greater the air resistance force. Subsequently, the amount of air resistance is dependent upon the speed of the falling object and the surface area of the falling object. Based on surface area alone, it is safe to assume that (for the same speed) the elephant would encounter more air resistance than the feather. -Answering these questions demands an understanding of Newton's first and second law and the concept of terminal velocity. According to Newton's laws, an object will accelerate if the forces acting upon it are unbalanced; and further, the amount of acceleration is directly proportional to the amount of net force (unbalanced force) acting upon it. Falling objects initially accelerate (gain speed) because there is no force big enough to balance the downward force of gravity. Yet as an object gains speed, it encounters an increasing amount of upward air resistance force. In fact, objects will continue to accelerate (gain speed) until the air resistance force increases to a large enough value to balance the downward force of gravity. Since the elephant has more mass, it weighs more and experiences a greater downward force of gravity. The elephant will have to accelerate (gain speed) for a longer period of time before there is sufficient upward air resistance to balance the large downward force of gravity.

Tension

-The force in a rope, string, cable, etc. -The tension force is directed along the length of the wire and pulls equally on the objects on the opposite ends of the wire. - In most cases, you can ignore tension by replacing it with a force vector on the object to which the rope, string, or cable is attached.

Velocity vs. Speed

-Velocity is the rate of change of displacement -Speed is the rate of change of distance. -They will NOT be equal in most situations. Ex: The racecar driver may have very high speed, but has an average velocity of zero every time he passes the starting line.

Calculate Average Velocity

-take the initial velocity and add it to the final velocity, then divide by two. -You will only consider the upward half of the motion, or downward half of the motion, and then one of them will always be zero. Vavg = (V1 + V2)/2

Principles of Projectile Motion

1) Horizontal velocity never changes (as long as you are ignoring air resistance) 2) Horizontal acceleration ALWAYS= 0 (same as 1) 3) Vertical acceleration always = 10 m/s2 downward 4) Vertical behavior is exactly symmetrical (i.e., if ignoring air resistance, a projectile's upward trip is identical to its downward trip) 5) Time in the air depends on the vertical component of velocity only 6) Range depends on both the vertical and horizontal components of velocity (range =Vxt -Vx its horizontal velocity but time is found from vertical velocity) 7) Time is always the same for both the x and y components of the motion.

Displacement vs. Time Graphs

1) On a displacement vs. time graph the slope represents velocity *It is the actual numerical value of the slope of the line that represents velocity or acceleration* in both these cases NOT "how the line looks." A straight line with a positive slope "looks" like it is "increasing" at first glance. However, if the line is straight, the slope is NOT increasing, the slope is constant* *2) A positive slope on a displacement vs. time graph tells us that velocity is positive. (Therefore, by convention, the motion is to the right) -a negative slope on a displacement vs. time graph tells us that velocity is negative. (Therefore, by convention, the motion is to the left)* 3) linear line = constant velocity(object in equilibrium) 4) If we see a flat line with zero slope, this means that displacement is constant. *5) On a displacement vs. time graph, when the line is above the x-axis, displacement is positive and the object is to the right of the origin. -When the line is below the x-axis, displacement is negative and the object must be to the left of the origin. -Crossing the x-axis means that we passed the origin -A corner tells us that the sign of velocity (which is the slope of a d vs. t graph) abruptly changed— in other words, the object turned around* A flat line tells us that displacement is constant, which means the object is standing still. It also tells us that velocity is zero because the slope of the line is zero—another proof that the object is not in motion at that instant.-

Velocity vs. Time Graphs

1) On a velocity vs. Time graph the slope represents acceleration *It is the actual numerical value of the slope of the line that represents velocity or acceleration in both these cases,* NOT "how the line looks." A straight line with a positive slope "looks" like it is "increasing" at first glance. *However, if the line is straight, the slope is NOT increasing, the slope is constant* 2) The sign of the slope on a velocity vs. time graph tells us the sign of acceleration: *This tells us which direction the acceleration vector points, and hence the direction of the net force*. 3) linear line = constant acceleration -acceleration is zero, then the velocity-time graph is a horizontal line 4) If we see a flat line with zero slope, this means that velocity is constant. 5)On a velocity vs. time graph, the location of the line above or below the x-axis tells you the sign of the velocity, *regardless of whether the object is moving to the right or to the left*. -When the line crosses the x-axis, the velocity changes from positive to negative. However, at the "exact point" where it crosses that axis, as the question asks, the velocity is zero, meaning it does come to an instantaneous stop (much like a ball at its peak height) 6) For example, for a ball being thrown up into the air, Velocity should have a large magnitude (i.e., high on the y- axis) at the beginning, and then decrease with time. Crossing the x-axis means velocity went from positive to negative; in other words, the particle turned around A corner (where slope abruptly changes from positive to negative or vice versa) tells us that the direction of the acceleration vector abruptly reversed LARGER SLOPE= LARGER VELOCITY

Two Requirements for Action-Reaction Pairs

1) Same type of force (gravity, electrostatic, electromagnetic) 2) The force is acting on two different objects

Steps for Interpreting Linear Motion Graphs

1) What does the slope represent? 2) Is this slope (+) or (-)? What does the sign of the slope tell you? 3) Is the slope constant (straight line) or non-constant (curved line)? What does this observation tell you? 4) What value is on the y-axis? Students tend to forget this important information. It is the quantity on the y-axis that is changing. If we see a flat line with zero slope, this means that whatever is on the y-axis is constant. 5) Is the y value (+) or (-) (i.e., is the line above or below the x-axis?). What does this observation tell you? 6) Do you expect the value on the y-axis to be large or small at the beginning?

Four Questions to Test Conceptual Understanding:

1. Can I visualize it? 2. Can I draw a picture, graph or diagram of it? 3. Can I explain it to someone else in layman's terms? 4. Can I think of and describe real-life examples?

If a man on a cliff first fires a gun straight down toward the ground below; and then fires a second round straight up into the air, which bullet will hit the ground with the greater velocity? (Assume both bullets hit the ground at the base of the cliff; neglect air resistance.)

23. They will both have the same final velocity. The muzzle velocity of the gun will be the initial velocity for the case where the gun is pointed straight up. Because projectile motion is symmetrical, the bullet will return to that exact same velocity as it passes the top of the cliff from which it was fired. This will be equivalent, therefore, to turning the gun downward and firing it straight at the ground from the top of the cliff.

MCAT QUESTION

A; Thinking of equations to use to tackle this problem, two options might be X = 1/2at2 or t = 2v/g, since they both involve time. The former equation describes how distance varies with time. It may be tempting to use, but since the height will be different on the Earth compared to on the Moon (for the same initial velocity), it's not easy to apply it to this situation. The second equation describes the time for a round-trip of a projectile going up and coming back down, for a given initial velocity. That's more similar to this situation, since we are told the projectile has the same initial velocity in both places. From that equation, you can see that if g is 1/6 as much, t must be 6 times larger. That yields answer A.

Newton's 3rd Law

Action Reaction -Whenever one object exerts a force (action) on a second object, the second object always exerts an equal and opposite force (reaction) on the first object. ' SIZE DIFFERENCE DOES NOT MATTER Examples of action-reaction pairs: -any contact force between two objects; -gravitational force between any two objects (regardless of mass) -electrostatic force between two charged particles (regardless of charge) *The mass of the earth is so large that the acceleration is negligible so you can't notice the force that the earth is exerting back*

Air resistance

Air resistance is the force exerted on projectiles or falling bodies due to actual physical collisions with air molecules. *Always assume air resistance is being ignored, unless it specifically states otherwise in the question stem or passage*

MCAT QUESTION

An important requirement for this question is to pick the right sign convention. On the MCAT, it is always assumed that to the right and upward are positive, and to the left and downward are negative, unless otherwise specified. A "straight line with a negative slope on a displacement vs. time graph" means a constant negative velocity. Thus, the answer should involve two left-pointing (negative) velocity vectors, which do not change in size. Answer C is the only possible choice. The other answers involve velocity vectors which change in magnitude and/or direction, which on a displacement vs. time graph would indicate a change in slope.

MCAT QUESTION

Be careful here; always note on this type of problem that you can only calculate the spring constant if it is the same, or "identical" spring used in both trials. It is in this case, so we just take F = kx and rearrange for k, giving k = F/x. In this question, the spring has both an un-stretched length and a stretched length, so F and x must be interpreted correctly. F is the additional force created by the second weight (10 N), NOT the total force, nor the difference in kg between the two weights. The distance x, is the additional distance traveled after adding the second weight compared to the first weight, or 4 m. Plugging these numbers into k = F/x gives 2.5 N/m.

Imagine two planets of masses A and B, where A = 2B. What is the ratio of the forces between them? What is the ratio of the acceleration between them?

Because A is 2B, A is the larger planet by a factor of 2. The ratio of the forces between them is still 1:1 according to Newton's Third Law. However, under the same force, the smaller planet will have twice the acceleration. This means the ratio of their accelerations would be 1:2.

If there is no acceleration is, or could there be, a force?

If there is no acceleration, there could be a force acting on that object. However, if there is a force, we would know that it must be exactly canceled out by the sum of the other force vectors on that same object

If a projectile has an initial vertical velocity of 30 m/s, how long will it be in the air, how high will it go, and what will be its average velocity during the entire trip?

It will take 3 seconds to reach max height, and will be in the air for six seconds total. It will travel an average velocity of 15m/s during the upward trip for 3 seconds, which means max height will be 45m. Its average velocity for the return trip will be equal in magnitude to its average velocity for the upward trip, but opposite in direction, so the average velocity during the entire trip will be 0m/s. Alternately, you could reason that since average velocity is displacement/time, and the displacement of the complete trip is zero, the average velocity is also zero.

Gravitational potential energy in space or near earth NOT assuming g=10 m/s^2

PE = -Gm1m2/r (in space, or near the earth if one is NOT assuming g = 10m/s2 -The negative sign is necessary because, without it, the formula would predict that as r increases, PE decreases. If you have an even basic concept of PE, you will see that this cannot be. A rock gets more PE as it gets farther from the center of the earth, not less. The negative sign makes it so that as r increases, we get a smaller negative number, which is actually a larger value.

Gravitational Potential Energy

PE = mgh (near earth) -Anything with mass can have gravitational potential energy -For example, fluids have mass, so they can also have potential energy. However, because they don't always move as a single unit, it is more useful to replace the mass term in the formula with density ρ (mass/volume) to give PE per unit volume of fluid: PE = ρgh -energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the Earth for the object. -There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object. The higher that an object is elevated, the greater the gravitational potential energy.

One projectile is launched in a vacuum and a second identical projectile is launched in the presence of air resistance. Both projectiles are given identical initial velocities and launch angles. What will be the differences in range, max height and final velocity for the two projectiles?

Range: The projectile experiencing air resistance will have a shorter range for two reasons. First, air resistance will create a force that impedes the horizontal component of the projectile's velocity. The projectile's horizontal velocity will be less, and therefore its range will be less. Second, air resistance will also impede the vertical component of the projectile's velocity. This will decrease max height and time in the air. Less time in the air allows for less horizontal travel before hitting the ground. Final velocity will also be less in the presence of air resistance because (as already stated) 1) horizontal velocity will be less, 2) vertical velocity will be less, and, therefore, 3) the total final velocity will be less.

Describetwoskydiversjumpingfromaplane,oneofmass100kg,andoneofmass150kg. Describe all vectors affecting them at all times, when and how each jumper reaches terminal velocity, how their terminal velocities compare, who hits the ground first, and why. Assume both skydivers have the same cross-sectional area.

Takes longer for heavier person to reach terminal velocity-accelerate longer because air resistance must match a greater force of gravity -Absent air resistance, they would both accelerate at exactly g. However, when air resistance is taken into account, the force due to air resistance will act to slow this acceleration. Because both masses are the same shape and size, this force will be the same for both balls. However, according to F = ma, the larger mass will be slowed down to a lesser degree by the same force of air resistance.

Track the velocity of a ball as it is shot from a cannon and then returns to the ground (include air resistance).

Taking air resistance into account, the cannonball experiences a force due to the gases from the explosion inside the cannon that pushes it down the barrel of the cannon. The cannonball simultaneously experiences a force due to friction that opposes its motion. There is obviously a large net force oriented toward the end of the barrel when these are summed together. Once the cannonball leaves the barrel, it becomes a projectile. Unlike the previous example, air resistance will create an opposing force on the cannonball throughout its flight, and therefore a negative acceleration. This will occur in both the vertical and horizontal directions. The cannonball will not go as far, as high, or be in the air as long, as it would have been without air resistance. When it strikes the ground, it will have a horizontal velocity less than it did when leaving the barrel. *The vertical velocity will also be less than the original vertical velocity because projectile motion is only symmetrical without air resistance. With air resistance, projectiles fall more steeply than their launch angles, decreasing total velocity at impact compared to initial total velocity.* (We'll assume the end of the barrel and the ground are at the same height, though they logically would not be. If the ground is lower than the end of the barrel, the final velocity will be slightly greater.) -force of gravity cannot be as strong as the force of the cannon so it won't hit the ground with the same initial velocity -in bottom of cannon barrel there is a chemical(gas) -when chemical lights on fire, run causes temperature to increase and gas to increase in volume, forcing it out of the barrel

Newton's 1st Law:

The Law of Inertia: -An object in motion tends to stay in motion (in the same direction and at the same speed), and an object at rest tends to stay at rest, unless acted upon by some net external force. -*Objects with a constant velocity maintain a constant velocity unless acted upon by a net outside force* -*This applies to both the "at rest" and the "in motion" cases, as well as any question about changing direction--because a change in direction is a change in velocity*.

Inertia

The ability of an object to resist a change to its velocity

A rock is thrown into the air at an angle of 30° *from vertical* at a velocity of 40 m/s. Resolve this vector into its vertical and horizontal components.

The mistake most students will make on this easy problem is to use the wrong formula for each component. Note that if measured with respect to the x-axis, the x-component is given by Vcos; but if the angle is measured with respect to the y-axis, then the x-component is given by Vsin. We suggest that it is easiest (and the least error-prone) to always convert angles so that they are measured with respect to the x-axis, then always use Vcos for the x-component and Vsin for the y-component. For this question, the x-component is: Vcos = (40m/s)(cos60°) = (40)(.5) = 20m/s. The y-component is: Vsin = (40m/s)(sin60°) = (40)(.9) = 36m/s.

MCAT QUESTION

This is a "constant velocity" problem. Hopefully, you picked up in the question stem that it is moving at a "constant speed." At constant velocity, all of the forces must cancel to zero. Thus, if the rope on the left is exerting a 100 N force, the x-component of the upper rope must also pull with a force of 100 N. Given the angle of 45, and knowing that one of the sides is 100 N, there are multiple ways to solve for the hypotenuse. Here's one: Fcosθ = 100N, so F(√2/2) = 100, so F = 100(√2) = 141. You can also use the Pythagorean Theorem.

True or False? 2) A ball moving with three times the velocity can compress a spring three times as far.

True. We can also write ( 1/2)mv^2 = (1/2)kx^2. This shows that velocity and the distance of compression, x, both have a square; and are therefore directly and linearly related.

Equation for gravity itself "gravity" "strength of the gravitational field" "acceleration due to gravity"

represented by a lowercase g and is described by the formula: g = Gm/r2.

Used this equation *only* to calculate "round trip" times, or in other words, the total time in the air.

tair= 2V/g The variable V must be the vertical component of initial velocity *Memorize for the purposes of manipulating equations and seeing relationships between variables only*