Physics 2
Conservation of Relative Speed
for completely elastic collisions only! The relative speed before will equal the relative speed after.
"body in equilibrium" clue words
"constant speed" "constant velocity" "terminal velocity"
Formula for "gravity" AKA the "strength of the gravitational field" AKA "acceleration due to gravity"
g=Gm/r²
General formula of elasticity:
(ME)=stress/strain As such, a stiffer material will have a higher elastic modulus
Units for Torque
Nm
Thermal Expansion equation
(delta)L = (alpha) L(time zero) (delta)T
Formula for thermal expansion:
(delta)L=(alpha)L₀(delta)T
Strain
(delta)dimension/original dimension
A boy throws a 1 kg rubber ball toward a cement wall with a velocity of 10m/s. If the ball experiences a completely elastic collision, what will be the velocity of the ball after the collision and the impulse experienced by the ball, respectively?
-10 m/s -20 kgm/s
a 120 kg rocket is accelerating toward the ground at 8m/s². The engine creates a downward force of 200N. A small parachute is attached to the rocket and slows its descent. What is the force due to air resistance against the parachute? (assume all air resistance is due to the parachute)
...
See the following, THINK WORK
1) Change in velocity (change in KE = work) 2) Change in height (change in gravit.PE =work) 3)Change in position of masses/planets/etc in space (change in gravitation PE = work) 4) Change in position of charge (change in electrical PE = work) 5) Compression of a spring (change in elastic PE = work) 6) Friction (change in internal energy = work) 7) Air resistance (change in internal energy = work)
True or False. 1) A ball moving with twice the kinetic energy can compress a spring twice as far. 2) A ball moving with three times the velocity can compress a spring three times as far.
1) False. Whatever KE the ball has will be transferred completely into elastic PE so we can write KE = 1/2kx2. 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. 2) True. We can also write 1/2mv2 = 1/2kx2. This shows that velocity and the distance of compression, x, both have a square and are therefore directly and linearly related.
Power
1. P = (delta)E/t 2. P = W/t 3. P = Fdcos(theta)/t 4* P(i) = Fvcos(theta) **only for instantaneous power
Formula for Potential Energy Stored by a Capacitor
1/2CV² or 1/2VQ C=capacitance and Q=the ratio of charge on each conductor
Formula for Elastic Potential Energy
1/2kx²
Formula for Kinetic Energy?
1/2mv²
Formula for Elastic Collisions
1/2m₁v₁² + 1/2m₂v₂² = 1/2m₁v₁² + 1/2 m₂v₂² conservation of energy (INGNORE signes)
T (period) =
1/frequency
pi radians
180°
How many radians in a full circle?
2(pi) = roughly 6
A certain lever is known to impart a five-fold mechanical advantage to its user. If it requires 2,000 Joules of work to move a 20 kg box onto a table 10 meters high, how much work will it require when using the described lever?
2,000 J
2(pi) radians
360°
Vehicle A has a mass of 1000kg and is traveling at 50m/s east. Vehicle B has a mass of 1500kg and is traveling at 100m/s west. Vehicles A & B collide head-on, stick together, and continue moving in one direction. What is the velocity of the combined wreckage?
40 m/s west
What is the tension in a rope being pulled from opposite ends with identical forces of 50N?
50
Momentum
= inertia increased by velocity ALWAYS conserved (remains constant) in an isolated system.
Air bags increase the time during which the driver comes to rest during a collision. How or why would this prevent injury?
According to Impulse = F(avg)t, if time increases force must decrease. Less force experienced by the driver would result in less injury. Impulse must remain the same because this is simply the difference between the initial and final momentum—which will not be impacted by the presence or absence of the air bag.
A 500kg elevator is being accelerated upward by a cable with a tension of 6000N. 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 6,000N of force, then the elevator must be pulling on the rope with a force of 6,000N.
Objects Exchange Velocities
if the objects are equal masses and the collision is perfectly elastic, the two objects will exchange velocities before and after the collision?
Rotational Equilibrium
An object is in rotational equilibrium if 1) it is NOT rotating or 2) it is rotating with a constant angular velocity.
Systems NOT in equilibrium
Any problem where the object in question has a non-zero acceleration, or net force. SOLVE: in the same way as an equilibrium problem, but add "ma" to the losing side.
Simple Harmonic Motion
Anything that oscillates back and forth and can be represented by a sine wave graphically constitutes SHM.
A clock maker is designing a gandfather clock in which the hamonic motion of a pendulum regulates the movement of the hands of the clock. In calibrating his current prototype, the clock maker determines that the minute hand of the clock moves once every 52 seconds. To calibrate the clock to the correct time, the clock maker could: (A) Increase the mass of the bob. (B) Increase the length of the pendulum (C) Decrease the pendulum constant (D) decrease the length of the pendulum
B
In Practice, the acceleration due to gravity is not a constant 9.8m/s², but varies with distance form the center of the earth. Taking this into consideration, as a falling object approaches the earth it will: (A) exhibit uniform acceleration, but its velocity will increase (B) exhibit an increasing rate of velocity change (C)exhibit a decreasing rate of velocity change (D) exhibit uniform acceleration, but its velocity will decrease
B G is inversely related to r, as the object approaches the earth the rate of velocity change will increase
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.
As the angle of incline of a plane increases, what happens to the value of a? What happens to the value of cos and sin? What happens to the normal force and to the force down the plane? What are the min and max 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 increase 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 and the minimum would be a plane with no angle of incline, where acceleration down the plane would be zero.
Examples of reverse collisions
Bombs, ice skaters pushing off one another, or off the wall, *radioactive decay*, a skateboard slipping out from under the rider, a rocket firing its engines in space, etc.
Direction of centripetal acceleration
Both the centripetal force and centripetal acceleration vectors point radially toward the center of the circle circumscribed by the motion.
Centripetal Force vs Centrifugal Force
Centripetal force is actually caused by some other force that acts to pull things into circular motion Centrifugal force is the action-reaction pair with centripetal force. The string's force on the ball is centripetal and the ball's force on the string is centrifugal
When I see Centripetal Force, i think...
Centripetal forces are ALWAYS caused by some other responsible force (friction, tension, gravitational force,etc) -categorical name for all the forces that act to pull things into circular motion
Describe the energy of transformations that take place as a rocket ignites and combusts its rocket fuel, launches into the air, rises to a max height, then falls back to earth and strikes the ground.
Chemical energy stored in the bonds of the rocket fuel is transferred into kinetic and gravitational potential energy as the rocket rises. If air resistance is taken into account some energy will also be dissipated as heat due to drag. Gravitational PE reaches a max at the max height and is then transferred back into kinetic energy (and heat if considering air resistance) as the rocket falls back to earth. When the rocket strikes the ground its kinetic energy is transferred into heat energy.
Name the type of energy possessed or created by each of the following: an explosion, a chemical reaction, a collision, any moving object, any object with height, a spring, a battery, two positive charges, water in a tank.
Chemical energy turns into heat energy and the kinetic energy of any flying debris; In a chemical reaction chemical energy is transferred between reactants and products—in an exothermic reaction heat is released; in a collision the KE of the objects before the collision is transferred into heat energy used to create damage/deformity/etc.; any moving object has kinetic energy; any object with height has gravitational potential energy; a spring stores elastic potential energy; a battery stores chemical energy; two separated charges contain electrical potential energy; water in a tank has gravitational potential energy and kinetic energy of the molecules.
Closed systems
Closed systems exchange energy but not matter with an outside system. Though they are typically portions of larger systems, they are not in complete contact. The Earth is essentially a closed system; it obtains lots of energy from the Sun but the exchange of matter with the outside is almost zero.
What is energy
Energy = the capacity to do work (this is a good general definition for the MCAT, although it may not be the most technically perfect definition of energy)
1st Law of Thermodynamics
Energy change is not always completely due to work; some energy is often lost to heat. Work and heat are the ONLY two ways energy can be transferred in or out of a system. (delta)E = W + Q Work = energy transfer via a force Heat = energy transfer via energy flow from hot to cold
What is Heat Energy
Energy dissipated as heat. On the MCAT, this is usually heat dissipated from a collision, or from a current-carrying wire. The terms "heat energy" and "internal energy" are used almost interchangeably. The authors seem to favor the term "internal energy" when referring to an increase in temperature, and the term "heat energy" when referring to energy lost in collisions.
Law of Conservation of Energy
Energy in an ISOLATED system is always conserved. often transferred back and forth between forms, but never lost.
Circular Motion Formula
F(c) = mv²/r
Formula for centripetal force:
F(c)=mv²/r
Formulas for fiction
F(f)=U(s)F(n) or F(f)=U(s)mgcos(theta) F(f)=U(k)F(n) or F(f)=U(k)mgcos(theta)
Levers
F(m) = mg(L1/L2) L1 - lever arm for the mass L2 - lever arm for the applied force
Ramps
F(m) = mg(h/d) h-height of ramp d-distance along its hypotenuse F(m) - Force necessary to do work with the machine (less than doing it without the machine)
Hydraulic Lifts
F(m) = mg(h1/h2) h1 & h2 - distance traveled by the large and small plunger, respectively. F(m) = mg(A1/A2) A1 & A2 - cross sectional areas of the small plunger and larger plunger, respectively.
Pulley
F(m) = mg/(# of vertical ropes directly lifting the mass)
Equation for the Force necessary to use a lever
F(m)=mg(L₁L₂) L₁= lever arm for the mass L₂=lever arm of applied force
Equation for the Force necessary to go up a ramp
F(m)=mg(h/d) h=height of ramp d=distance along its hypotenuse
Equation for the Force necessary to use a hydraulic lift
F(m)=mg(h₁/h₂) or F=mg(A₁/A₂) h₁=the distance traveled by the large plunger h₂=the distance traveled by the small plunger
Equation for the Force necessary to use a pulley
F(m)=mg/(#of vertical ropes directly lifting the mass) Note: not every rope that is vertically oriented should be counted and entered into the above equation. To be counted, a vertical section of rope must life the mass directly, either by being attached to the mass, or by lifting a pulley that is attached to the mass.
Formula for the Normal Force on an inclined plane
F(n)=mgcos(theta)
What is the Equations for the Universal Law of Gravitation?
F=Gm₁m₂/r² (G is the gravitational constant and is always given) "inverse square law" because F varies inversely with the distance r.
What is a formula relating displacement from a spring's equilibrium point to force? (Hooke's Law)
F=k(delta)X (delta)X is the displacement of the spring from its equilibrium point, NOT the length of the spring)
Formula for the force down an inclined plane, parallel to the surface
F=mgsin(theta)
Ice cubes in a full glass of water... will it overflow when melted?
First we ignore thermal expansion of the liquid. The melted water with no ice cubes will remain at the exact same level as the water with ice cubes floating in it. The key to understanding this is to understand the principle that the fraction of an object that is submerged in a liquid is equal to the ratio of the densities between the object and the liquid. So, if an object is 1⁄2 as dense as the liquid in which it is submerged, it will float with one half of its volume submerged and one half above the liquid. Ice is 9/10 as dense as water so it will float with 9/10 of its volume submerged. However, for the same mass, ice occupies 10/9 the volume that water does. Multiplying these together (9/10*10/9 = 1) demonstrates that the volume occupied by 9/10 of the ice is the same volume that will be occupied by the entire ice cube once it is melted. Now let's include thermal expansion of the liquid because it has cooled down as the ice melted. The volume of the liquid then becomes ever so slightly smaller due to that cooling. So in principle, the water level would go down.
A 100kg satellite is orbiting 4,000km above the earth. If the mass of the earth is 6.0 x 10^26kg and the radius of the earth is 6.4 x 10^3km, what is the instantaneous velocity of the satellite? (assume that G = 6.7 x 10-11).
First, we would recommend changing all of the km measurements into meters by multiplying them by 10^3. Then, find the formula by setting the gravitational force equation equal to mv^2/r and simplifying to get v = squarerootGM/r. Don't forget that the actual radius between the satellite and the center of the earth is earth's radius PLUS the altitude of the satellite. If you add up the distances given in the stem you get approximately 1 x 10^4 kilometers. This is equivalent to 1 x 10^7 meters. Your formula with values plugged-in should be v = squareroot(6.7x10-11)(6x1025)/(1x107), which gives: 2.0X10^4 m/s
Which way does the friction vector point for a car driving east down a straight road? For a car skidding with locked brakes? For a gecko climbing a wall? For a car going around a bend in the road?
For a car driving east the friction vector between the road and the tires also points east. The tendency of the tires is to slide past the road toward the back of the car and friction opposes that sliding. When a car locks its wheels, however, friction points opposite the direction of the car's tendency to slide forward. Friction would point up the wall for a gecko because the geckos hands would tend to slide down the wall. For a car going around a bend momentum makes it want to continue tangent to the circle but friction points toward the center of that circle keeping it in circular motion (an example of a centripetal force).
Why do objects exchange velocities in perfectly elastic collisions?
For equal masses there are two cases -- one ball moving and collides with a ball at rest, and both balls moving and collide. The first case where the balls exchange velocities clearly conserves both momentum and kinetic energy. The initial KE and momentum of the moving ball are simply transferred to the second ball. There is no doubt then that both momentum and KE are conserved. The second case can be understood from the first case. Imagine a reference frame that is moving along with the second (moving) ball. In that reference frame we are back to the first case again where the second ball is at rest. As we've seen, the first case conserves both momentum and energy, so it does also in the second case.
Why must the collision by elastic to conserve relative speed?
For the relative speed to be conserved, there must be no loss of energy. The kinetic energy associated with the two masses traveling at their initial velocities before the collision must be available after the collision. This is in addition to the conservation of momentum. Combined these two conservations results in the curious result that the relative speed is conserved. (The relative velocities differ in sign.) In an inelastic collision energy is not conserved. Some energy goes to creating breakage, deformity, etc. and therefore maintaining the relative speed is no longer possible.
Stress
Force/Area
Friction
Friction is a force that is created whenever two surfaces move or try to move across each other. Friction always opposes the motion or attempted motion of one surface across another surface. Friction is dependant on the texture of both surfaces. Friction is also dependant on the amount of contact force pushing the two surfaces together (normal force).
Impulse equations
I = (delta)p I = m(delta)v I = F(avg)t
Elastic Potential Energy
If a mass of velocity v strikes a spring compressing it, all of its kinetic energy will turn into elastic potential energy. Setting the initial kinetic energy equal to the final PE allows you to predict how far the spring will compress.
Work-Energy Theorem
If a net force does work on a rigid object, the work done on that object is equal to the change in the kinetic energy of the object. W = KE(final) - KE(initial)
The Work-Energy Theorem
If all motion is in the horizontal plane, and no other forms of energy (elastic, electrical, etc.) are involved, the work done is simply the change in kinetic energy of the object W= KE(f) - KE(i)
Kinetic Friction vs. Static Friction:
If there is sliding, its Kinetic friction; If there is no sliding, its static friction
A 10kg block falls from a height of 10m onto a table that is 5m high. Was the momentum conserved during the fall? Was momentum conserved during its collision with the table?
If you define the ball as your system, then gravity is an external force so momentum will not be conserved. However, if you defined your system to include the earth and the ball then momentum would be conserved. At rest the ball had no momentum and whatever momentum it gains as it increases in velocity will be matched by an increase in momentum of the earth. Because they will be vectors in opposite directions they will cancel to zero and momentum will have been conserved: zero before, zero after. In the collision with the table, momentum will be conserved for the ball-table-earth system, but not the ball-table system. All of this, of course, assumes no external forces.
Block B weighs 2/5 less than Block A. If Block A is placed a distance x from a fulcrum, where should Block B be placed to balance the fulcrum?
If you go too quickly on this one, you'll probably miss it . . . and hopefully learn a lesson. Block B weighs 2/5 LESS than A, which does not mean it weighs 2/5 AS MUCH AS Block A; subtract 2/5 from one and you get 3/5. Since B weighs 3/5 as much as A, it will need to be 5/3 as far away from the fulcrum
Elastic vs Inelastic Collisions
In an elastic collision momentum and energy are both conserved. In an inelastic collision momentum is conserved but energy is NOT
If an object is deformed during a collision it must have undergone which type of collision?
Inelastic.
Isolated systems
Isolated systems can exchange neither energy nor matter with an outside system. While they may be portions of larger systems, they do not communicate with the outside in any way. The physical universe is an isolated system; a closed thermos bottle is essentially an isolated system (though its insulation is not perfect).
Units of Work
Joules N*m or kg*m²/s²
Formula for Electrical Potential Energy
K(e)q₁q₂/r or q(delta)V
Kinetic Energy
KE = 1/2mv²
500N is applied to an object and doesn't move. 501N 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.
Kinetic friction is always less than static friction for the same two surfaces. 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.
Mechanical Energy
ME = KE + PE (aka - Total Energy) In the absence of non-conservative forces (friction, drag, air resistance) ME is always conserved.
Moduli of Elasticity
ME = stress/strain
Formula for Mechanical Energy
ME=KE + PE In the absence of non-conservative forces such as friction, drag, air resistance, et., mechanical energy is always conserved (AKA: total energy)
Machines
Machines reduces the force necessary to perform a given amount of work. For example a 5x machine allows a person of a maximum force, F, to create a force of 5F Machines NEVER reduce or change the amount of WORK!
Static vs Dynamic Equilibrium
Objects at rest are in static equilibrium; objects moving at a constant velocity are in dynamic equilibrium. In both cases the net force experienced by the object must be zero
A massless 12cm board is perfectly balanced on a fulcrum placed 5cm from its left end. A mass of 20kg is hanging from a point 2cm from its left end and a mass of 10kg is hanging exactly from its right end. What is the net torque on the board?
On purpose, this problem doesn't work out exactly right. If you sum all the torques given in the problem you come up with a net torque. However, the question tells you that the board is "perfectly balanced." Whenever you see this on the MCAT, you know the net torque must be zero. We made the numbers not work so that if you tried to do it the long way you would get it wrong and read this. Remember, for torque/fulcrum/pivot point situations, if there is no rotation, there is NO NET TORQUE!
What causes friction on a microscopic level? On the molecular level?
On the microscopic level even the very smoothest surface is actually extremely rough. The peaks, valleys, protrusions , etc. of the two surfaces literally collide with one another. 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 will just call "contact forces" for the MCAT are often classified as an example of electromagnetic forces. This interaction is actually due to repulsion between electrons.
For a ball on a string in circular motion, does the centrifugal force act on the ball or the string?
On the string
What portion of the movement of a pendulum represents one cycle?
One cycle for a pendulum would be movement of the bob from one side to the other and then back to the first side. Many students make an error in thinking that one swing to the other side would be one cycle. The clue is that the action must cycle or repeat. When the bob is swinging back it is doing something it has not done before. Once it gets back to the starting point it is then repeating the same motion, and the time to do this is one period.
Conceptual definitions of open, closed, and isolated systems.
Open System = both mass and energy can be exchanged with the surroundings; Closed System = energy, but not mass, can be exchanged. Isolated System = neither mass nor energy can be exchanged.
Open systems
Open systems can exchange both matter and energy with an outside system. They are portions of larger systems and in intimate contact with the larger system. Your body is an open system.
Potential Energy Stored by a Capacitor
PE(capacitor) = 1/2QV or 1/2CV2 or 1/2Q2/C
Elastic Potential Energy
PE(elastic) = 1/2kx²
Electrical Potential Energy
PE(electrical) = Kqq/r or qEd or qV
Gravitational Potential Energy
PE(grav) = -Gmm/r or mgh
Formula for the PE in a compressed spring
PE=1/2K(delta)X²
Equation for Gravitational Potential Energy
PE=mgh (near earth) PE=pgh (For Liquids; Row (p)=density) PE= -Gm₁m₂/r (in space, or near earth of NOT assuming g=10m/s²)
Ball A (mass x) and Ball B (mass 2x) undergo a completely elastic collision on a frictionless surface. The initial velocity of Ball A is 4 m/s and the initial velocity of Ball B is -5 m/s. Which of the following gives the velocities of Ball A and Ball B, respectively, following the collision? A) -8 m/s and 2.5 m/s B) -8 m/s and 2 m/s C) -6.5 m/s and 3.5 m/s D) -6.5 m/s and 2.5 m/s
Recall that for two masses in a completely elastic collision, relative velocity is always conserved. The relative velocity before the collision was 9 m/s. The only answer choice with a relative velocity of 9 m/s is choice D
Which object has the higher ME, a marshmallow or a rock?
Rock
The generator-turbine system must be in rotational equilibrium when: I. It has zero angular momentum II. All external forces sum to zero III. All external torques sum to zero
Rotational equilibrium is closely related to translational equilibrium. For translational equilibrium all external forces must sum to zero. For rotational equilibrium, all external torques must sum to zero. This can be either because the object is not rotating, or because it is rotating with a constant angular velocity. This makes statement III true. Statement I is also true because zero angular momentum is only possible when something has zero angular velocity (i.e., it is not rotating). Finally, statement II is false because the external forces need not sum to zero; if they have varying lever arms, their torques can still sum to zero.
Angular Frequency: scalar or vector?
Scalar. It is the magnitude of the angular velocity vector. Describes rate of rotation in rad/s.
elasticity Sheer Modulus
Shear stress (simultaneous pushing or pulling forces; two forces are NOT aligned)
Derive a formula for the minimum coefficient of static friction between a car and the road to keep a car going around a circular track.
Similar to above, set the formula for static friction equal to the formula for centripetal force to get: u(s)=v2/(rg). This assumes the road is not banked.
Stress vs strain
Stress=Force/Area Strain=change in dimension/original dimension
Formula for simple Harmonic Motion of a pendulum
T=2pi Square-root(L/g) (T is period. Frequency is the inverse of period)
Formula for simple Harmonic Motion of a mass on a spring
T=2pi Square-root(m/k) (T is period. Frequency is the inverse of period)
Formula for Torque
T=Fl T=mgl (l=lever arm) T=Frsin(theta) (r=distance btwn the force and the point of rotation)
elasticity Young's modulus
Tensile strength (simultaneous pushing or pulling forces on both sides of an object; the forces must exactly align in both the vertical and horizontal planes.
Examples of Equilibrium
Terminal Velocity Constant Velocity Objects at rest Balanced fulcrums or boards hanging from strings Objects floating in liquid
What value must be low for a pendulum to exhibit Simple Harmonic Motion?
The angle of displacement must be small for SHM.
In what direction does the angular velocity vector point?
The angular velocity vector points along the axis of rotation. Use the right hand rule to decide if it is up or down. Curl the fingers of your right hand around the axis of rotation such that your fingers point in the direction of rotation and your thumb will be pointing in the direction of the vector.
Impulse: What is it? what are the formulas?
The change in an object's momentum Impulse=(delta)p Impulse =m(delta)v Impulse=F(average)xtime If there is no change in velocity, there can be no impulse The greater the change in velocity, the greater the impulse
A ball with a radius of 11cm is spinning at a rate of 24rad/s. How far does a particle on the equator of the ball travel each second?
The distance is vt which is rt. For t=1s, this results in a distance of (11)(24)(1)cm=264cm, or about 2.6m. An approximate solution is that 24 rad/s is the equivalent of approximately 4 cycles or revolutions each second. The circumference of the ball is 2(pi)r or (2)(3.14)(11) = 69 cm. The distance traveled by a particle on the equator of the ball each second is therefore 4 revolutions x 69 cm/rev. = 276cm or about 2.8meters.
What force does the earth experience due to a falling rock?
The earth experiences the same force that the rock experiences. The difference in mass is just so unfathomably large that the earth would take eons of time to move an observable distance under that tiny force.
What is Chemical Energy
The energy contained within chemical bonds, or the energy stored/released due to the separation and/ or flow of electrons (i.e., a battery)
What is Internal Energy
The energy of the internal vibrations and random motions of molecules and/or atoms within a system. Non-conservative forces such as friction or drag acting on a moving object result in the transfer of kinetic energy into internal energy
When do you need to use T=Frsin(theta)
The force applied is not perpendicular to r. In most fulcrum and board-string problems the forces do act at 90 degrees, hence sin(90degrees)=1 and T=Fr.
An elevator weighing 1,000kg is being accelerated upward at 2m/s2. What is the tension in the cable holding the elevator?
The force down on the elevator is mg, or 10,000 N, if the tension in the cable were just 10,000 N there would be no acceleration, but there is an acceleration of 2 m/s2. In order to accelerate a 1,000kg elevator at 2 m/s2 you'll need an additional force of 2,000 N (according to F=ma). 10K + 2K = 12K N total tension in the cable.
A 120kg rocket is accelerating towards the ground at 8m/s². The engine creates a downward force of 200N. A small parachute is attached to the rocket and slows its decent. What is the force due to air resistance against the parachute?
The forces acting down on the rocket are: mg (1200N) + 200N = 1400N. The forces acting up on the rocket are: Fair + ma (120kg*8m/s2) = Fair + 960N. Therefore: Fair + 960 = 1400 and Fair = 440N.
A 15kg toy rocket is falling towards earth with a constant velocity of 20m/s. A small amount of fuel still present in the cone creates a downward force of 30N. What is the force due to air resistance?
The forces pushing the rocket down are: mg (150N) + 30N = 180N. The force due to air resistance is the only upward force, so it must equal 180N.
Why does V=squareroot(2gh) work for either falling bodies or a mass on an inclined plane?
The formula V = √2gh is derived from conservation of energy by equating mgh to 1/2mv2 and solving for v. As long as friction, air resistance, etc. are ignored, energy will be conserved in an identical way whether the object falls directly to the ground or rolls down a plane.
A student attaches a 4kg mass to a spring and it stretches 1m. He removes the first mass and replaces it with a 2kg mass. what is the spring constant, k, and how far will the spring stretch the second time?
The gravitational force downward equals the spring force upward, mg= k ∆x; solving for k, k=mg/∆x=4*10/1 N/m gives 40N/m. The gravitational force is halved so the balancing spring force must also be halved. Since the spring force is proportional to displacement it too is halved. Thus the displacement is 1/2m.
Why must we include a negative in PE=-Gm¹m²/r?
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 cannot be. A rock gets more PE as it gets farther from the center of the earth, not less. The negative makes it so that as r increases we get a smaller negative number, which is actually a larger value.
An atom of 56-Fe undergoes alpha decay. The velocity of the exiting alpha particle is 2.0x10³ m/s. What is the identity and ejection velocity of the decay product?
The new particle formed has 52amu (Cr-52). This can be determined by subtracting the A and Z number for Helium from Fe-56. The momentum of the ejected particle must equal the momentum of the decay product, therefore: (2 x 103 m/s)(4 amu) = (v)(52 amu). This simplifies to8x103/5.2x101 =154m/s.
Why does the displacement of a the pendulum gradually decrease over time?
The reason the pendulum will not continue oscillating and reaching the same height is because of nonconservative forces such as air resistance. Energy lost to these sources is not available to the bob so it cannot travel to its original height.
How much work is done when a 1000kg car traveling at 40m/s applies its brakes and comes to a complete stop?
The work done is equal to the change in energy from the KE of motion at 40m/s to zero KE at rest. KE = 1/2mv2 = (.5)(1,000)(40)2 = 8 x 105J.
What is the work done on a spring compressed by .5m?
The work done is equal to the change in potential energy of the spring. PE = 1/2kx2 = (.5)(10)(.5)2 = 1.25J.
Examples of Simple harmonic motion
There are many possible examples of SHM. 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. Essentially any movement that oscillates about an equilibrium position and shows the characteristic sinusoidal pattern.
When is work +? When is work -?
There are two conventions for the sign of work. In physics, we say that if the force and the displacement are in the same direction, work is positive (e.g., pushing a barbell up). If force and displacement are in opposite directions we say work is negative (e.g., lowering a barbell). In both physics and chemistry a common convention is used that says: If work is done on the system it is positive; if it is done by the system it is negative.
Power
Think of power in EXACTLY this way and in this EXACT order: 1) P=(delta)E/t 2) P=W/t 3)P=Fdcos(theta)/t 4)P(i)=Fvcox(theata) (instantaneous power) Units= watts (J/s)
Work
Think of work in EXACTLY this way and in the EXACT order: W=(delta)Energy W=Fdcos(theta)
Show via dimensional analysis how m(delta)v and F(avg)t are equivalent.
This can be done by simply looking at the units. The units for force are: kg*m/s2. When you multiply this by time in seconds, as in the third equation, one of the seconds cancels and you get kg*m/s—which are the units for the second equation.
19. Two gymnasts jump simultaneously from a platform and land very close to each other in the center of a trampoline. Gymnast A weighs 50kg and Gymnast B weighs 100kg. The trampoline obeys Hooke's Law and projects both gymnasts into the air. Ignoring air resistance, which of the following statements is true? A) Gymnast A will reach twice the height of Gymnast B. B) Gymnast B will reach twice the height of Gymnast A. C) Both Gymnasts will reach the same height. D) Gymnast A will reach four times the height of Gymnast B.
This is a good test to see if you can think thru a difficult question on the MCAT. You will get some like these that just aren't that straight forward. The key here is to think about what will happen when the trampoline is fully compressed. It will exert a force upward on the two gymnasts. Were the gymnasts alone, it could accelerate the lighter gymnast at a faster rate, but since they are both on the trampoline, the lighter gymnast cannot be accelerated any faster than the heavier gymnast. Essentially, the heavier gymnast is holding the trampoline down on its way up and keeping it from pushing the lighter gymnast up any faster. As a result, they will stay together on the trampoline until they both leave the trampoline at the same time and with the same initial velocity. This means, ignoring air resistance, they'll both reach the same height
27. A hydraulic apparatus is being used to launch projectiles into the air. The original design of the hydraulic chambers features two cylinders of equal cross-sectional area. The one cylinder receives a downward force. A hydraulic line connects the fluid in this chamber to a second chamber, causing that cylinder to simultaneously rise. If the original design could propel a rock to a height x, what will be the new height if the apparatus is reconfigured such that the cylinder receiving the applied force has one-half the cross-sectional area of the one launching the projectile? A) 2x, because initial velocity leaving the apparatus has doubled. B) 2x, because initial velocity leaving the apparatus has increased 1.4 times. C) 4x, because both the force created by the apparatus and the acceleration have doubled. D) 4x, because both the force created by the apparatus and the initial velocity leaving the apparatus have doubled.
This is another excellent question to test how carefully you can think thru a situation described to you on the MCAT. Don't get intimidated if it takes two or three steps, just make sure you have the right relationship established at each step. First, the change in the hydraulic lift should result in a doubling of the force. Use F=ma to see that the acceleration during the time for which the rock is still in contact with the apparatus will also double. Now use V = √(2ax) to find that if the acceleration during this period doubles the final velocity right when the rock leaves the apparatus will go up by 1.4 (the square root of 2). This final velocity will also be the initial velocity for the rock as a projectile. Finally, use V = √(2gh) to see that if the initial velocity increases by a factor of 1.4, we must put something in for h that, once square rooted, will also equal 1.4, that number would be 2. Answer A gives this factor correctly, but for the wrong reason because we know velocity was 1.4 times greater leaving the apparatus, not two times greater. B is thus the best answer.
A man who must lift a very heavy 200kg box creates a make-shift pulley system by looping a rope over an upper pipe, under and around a lower pipe and back up and over the upper pipe again. He then ties one end to the box. Assuming that the pipes are frictionless and the rope is massless, what force will be required to lift the box?
This man has essentially created a worthless pulley system. Remember that for a pulley to reduce the force necessary to lift something, the number of vertical sections of rope that directly pull up on the box must exceed the number being pulled. In this case, one piece of rope is being pulled down and one pulls up on the box, for a net benefit of zero. It will take mg to lift the box, or 2,000N.
A ramp at a large factory is used to gradually slow down large boxes dropping from an upper floor conveyor belt. Two surfaces are tested as a covering for the ramp. The coefficients of kinetic friction of surfaces A and B are 0.4 and 0.8, respectively. Which surface has the greatest power? A) Surface B, because it causes a smaller change in velocity per unit time. B) Surface B, because it transfers more kinetic energy into internal energy per unit time. C) Surface A, because it causes a greater change in velocity per unit time. D) Surface A, because it transfers more kinetic energy into internal energy per unit time
This question illustrates an MCAT favorite—using a very unique application of a principle that most examinees will never have considered. "Ramps have power?" you may ask. Well, power is defined as change in energy per unit time. Any ramp with friction can certainly be used to slow down a moving object, which would in very deed change its kinetic energy over time (change in energy/time = power). This question also illustrates a common way the MCAT presents answer choices—to have two of the four choices give the same exact answer, but tied to a different explanation. Either A or B must be the correct answer because surface B has the greater coefficient of friction and therefore the ability to cause a greater change in KE per time to a moving object. The explanation with B is exactly what friction does—change KE into internal energy.
Equilibrium on an Inclined Plane
Use an altered coordinate system. Use "T" - call all forces acting down the plane "down forces" and all forces acting up the plane "up forces" The force down the plane due to gravity is always F=mgsin(theta) The force of friction is always parallel to the plane opposite the direction of the motion. Never any acceleration perpendicular to the plane so you can ignore these forces.
Elastic Collisions
Use conservation of energy and IGNORE signs. 1/2m1v1² + 1/2m2v2² = 1/2m1v1² + 1/2m2v2² (KE of object 1 before + KE of object 2 before = KE of object 1 after + KE of object 2 after)
A new planet is discovered in a neighboring solar system that has twice the mass, but half the radius of earth. How does the strength of the gravitational field on earth (gE) compare to the strength of the gravitational field on the newly discovered planet (gN)? A) gE = gN B) gE = 8gN C) 2gE = gN D) 8gE = gN
Using the formula, g = Gm/r2, if we increase the mass of the planet by a factor of two, this increases the gravitational field by a factor of 2. If the radius is also cut in half, due to the square on the radius, the force will also be increased by a factor of 4. Multiply these together to get the total effect on F: 2*4 = 8. Thus the field on the new planet will be 8 times stronger than on earth. Answer D is the only relationship that reflects this fact.
Formula for the Velocity at the base of an inclined plane
V(f)=Squareroot(2gh)
2nd Work Equation
W = Fdcos(theta) Any time a Force is applied across a displacement work has been done. block pushed across table, ball falling from height h
Power units?
Watts (J/s)
Why solve systems in equilibrium by adding "ma" to the losing side? What are you accounting for?
We are pretending that the net forces acting from either side are equal. In reality, because the object is accelerating we know there is a net force and they are not equal. That net force is what causes the object to accelerate. By adding ma to the weaker side we are making them equal again. We use ma instead of using a variable like Fnet simply because we usually know the mass of the object and the acceleration of the system, but don't necessarily know the net force. Of course, you could solve by summing all forces to the net force, then substitute ma for that net force. Most classes teach this method, but then you must carefully track signs.
Adding springs on springs
When multiple springs are attached parallel to one another and to the same bumper plate (such that each spring is compressed equally under any displacement, the effective spring constant is the sum of the individual spring constants. If the springs are attached end-to-end the springs add much like resistors in parallel, according to: 1/keffective = 1/k1 + 1/k2 + 1/k3 . . .
Describe the work done when a force, applied at an angle of 45° to the horizontal, is used to push a box across the floor. Describe the work done when a box is moved at constant velocity across a frictionless table. Describe the work done by the string when a ball on a string is swung in a circular motion.
When the force is not in the same direction as the displacement, only that component of the force that is aligned with the displacement does work. Thus the work done is Fdcos(45degrees). If the angle between force and displacement vectors is 90 degrees then no work can be done by that force. If a box is moved at constant velocity horizontally no work is done because there was no net force and the energy of the box was the same before and after the motion (ignoring some net force necessary to get the box moving). For a ball on a string no work is done by the centripetal force created by the string because this force is exactly 90 degrees to the tangential displacement vector at any instant.
A 100kg diver jumps directly off a cliff and into the water below. Immediately before hitting the water, his speed is 50m/s. He continues to a depth of 10m before being slowed to a stop by the water. Calculate the average force exerted on the diver by the water (note: The combined average force is due to both the drag [a friction-like force in fluids] and the buoyant force).
When you see "average force" on the MCAT, it is a dead give away that it will be dealing with either Impulse (Impulse = Favg*t) or with work (W = Favgdcos0). In this case we can calculate the work done on the diver in slowing him down from 50m/s to 0m/s. This will come from the difference in kinetic energy at the two points. Because we were given the distance over which this occurred and F and d are in the same direction, we can just divide work by distance and get the average force (F = W/d). Plug in the numbers and you should get 12,500 N
A worker must move a 50kg block onto a table that is 2m off the floor. How much work will be required without the use of a machine? If the worker constructs a ramp that is 2m high and 8m long, how much force will be required? How much work will be required with the use of the machine?
Without the machine, the man can lift the block onto the table at constant velocity simply by overcoming mg. So, the necessary force = 500N. The ramp has a mechanical advantage of 4, reducing the necessary force to only 125N. The work done will be identical with or without the machine.
What are the only two ways energy can be transferred into or out of a system?
Work and Heat (delta)E=W+q Work = energy transfer via force Heat = energy transfer via friction or energy flow form hot to cold
When is work positive or negative
Work done by a force is positive if the applied force has a component in the direction of the displacement. When a body is falling down, the force of gravitation is acting in the downward direction. The displacement is also in the downward direction. Thus the work done by the gravitational force on the body is positive. Consider the same body being lifted in the upward direction. In this case, the force of gravity is acting in the downward direction. But, the displacement of the body is in the upward direction. Since the angle between the force and displacement is 180o, the work done by the gravitational force on the body is negative.
Heat Energy
energy that is dissipated as heat. normally from a collision or current-carrying wire almost used interchangeable with internal energy
If a 20kg hangs exactly 3 meters from the fulcrum, what weight should hang on the other end, 5 meters from the fulcrum, to balance the board?
You could solve this problem by setting torques clockwise equal to torques counterclockwise, but it would be better if you could look at this situation conceptually and use your knowledge of manipulating equations. T = Fl shows us that force and lever arm are inversely related. Therefore, if a mass is a factor of 5/3 farther from the fulcrum than is the first mass, then that mass will need to be 3/5 as large as the first mass. 20kg*3/5 = 12kg.
Centripetal Acceleration?
a(c) = v2/r. This can be derived by substituting the formula for centripetal force into F = ma and solving for a.
Truck engine A has twice the power as Truck engine B. T or F? a) Truck A can accelerate the same trailer from 0 to 50m/s in half the time. b)Truck A can accelerate a trailer to a velocity v, in a time t, that is twice as massive as the trailer Truck B can accelerate to the same velocity in the same time period. c) Truck A can accelerate the same trailer to twice the velocity in the same amount of time.
a) Accelerating the same trailer from 0 to 50m/s would represent the same amount of work. Doing so in half the time would represent twice the power. Truck A does have twice the power of Truck B, so this is true. b) According to KE = 1/2mv2, twice as much energy would be required to accelerate at trailer twice as massive to the same speed. If this is accomplished in the same period of time, twice the power will be required. Truck A does have twice the power, so this statement is also true. c) Accelerating a trailer to twice the velocity will actually require four times the energy due to the square on velocity in the KE equation. Therefore, Truck A cannot accomplish this task and this answer is false.
What force equation should be set equal to the centripetal force in each situation? a) a car turning in a circle, b) satellite orbitting earth, c) ball on string, d) charged particle in magnetic field?
a) U(s)mg; b) Gmm/r²; c) tension (T); d) qvBsin(theta)
How will decreasing the following aspects of a pendulum change the frequency of oscillation? a) mass of the pendulum bob b) length of the pendulum, c) gravity Use T=2pi squareroot(L/g)
a) decreasing the mass of the bob will have no effect; b) decreasing the length of the pendulum will increase the frequency; c) decreasing gravity would decrease the frequency.
How will increasing the following aspects of a mass-spring system change the frequency of oscillation? a) mass on the spring b) length of the spring, c) mass of the spring, d) gravity, e) the spring constant. Use T=2pi squareroot(m/k)
a) increasing the mass will decrease the frequency; b) the length of the spring has no effect as long as it is designed not to change k; c) the mass of the spring itself has no effect assuming it is far less massive than the massive object attached to it [although this would probably alter k and it would have an effect]; d) increasing gravity would have no effect; e) increasing the spring constant would increase the frequency. Encourage students to simple substitute 1/f for T in the formula for these types of problems. This will make the correct relationships easy to see. Notice that the MCAT likes to tempt you to mix up the variables that uniquely impact mass-spring systems but not pendulums.
Formula for acceleration down an inclined plane
a=gsin(theta)
Centrifugal Force
action-reaction pair to centripetal force Ex: string pulling ball in circular motion, the ball also is pulling on string. Centrifugal force is equal and opposite (away from the center of the circle)
For inelastic collisions...
always use conservation of momentum. YOU MUST USE SIGNS! any velocity vector to the left or down MUST be given a negative sign. m1v1 + m2v2 = m1v1 + m2v2
In physics, what is a "Field?" Is Gravity a Field?
an invisible influence that can exert a force on a mass or charge ( Gravity is a field that exists between any two objects with mass
Pendulum KE
at a maximum at the bottom of the arc and at a minimum at the max height of the bob
Pendulum PE
at a maximum at the max height of the bob and at a minimum at the bottom of the arc
elasticity Bulk Modulus
bulk stress (simultaneous compression from all sides
Bulk Modulus
bulk stress/ strain modulus (simultaneous compression from all sides)
Energy
capacity to do work
Work
change in Energy =(delta)Energy look for any change to the total energy of an object of system... if E changes, think WORK.
Impulse
change in a objects momentum if there is no change in velocity there can be NO impulse. the greater the change in velocity the greater the impulse. (assuming constant mass)
Elastic vs. Inelastic Collisions
elastic collision = momentum and KE are both conserved. Inelastic collision = momentum is conserved but not KE
Chemical Energy
energy contained within chemical bonds, or the energy stored/released due to the separation and/or flow of electrons (ex: battery)
Internal Energy
energy of internal vibrations and random motions of molecules and/or atoms within a system. Nonconservative forces such a friction or drag acting on a moving object result in the transfer of KE into internal energy.
Max 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. If you push a boulder at 20N there will be 20N of static force opposing you. If you increase the force to 100N the static friction will also increase to 100N. Until the value exceeds to "maximum static friction" then it will begin to slide and then have kinetic friction, not static.
Momentum: what is it? what is the formula?
inertia increased by velocity, It is always conserved in an isolated system p=mv
Tension
is the force in a rope, string, cable, etc.
The Universal Law of Gravitation
is true everywhere. However, we make an assumption that gravity is a constant 10m/s² even though it should vary every so slightly with height. Based on the assumption, we can simplify to F=mg
Units of momentum
kg∙m/s
Formula for Gravitational Potential Energy
mgh of -Gm₁m₂/r
Formula for perfectly inelastic
m₁v₁ + m₂v₂ = (m₁ + m₂)v₃ "perfectly inelastic" indicates that the two objects collided and then stuck together. Use this formula for reverse collisions
Formula for inelastic collisions
m₁v₁ + m₂v₂ = m₁v₁ + m₂v₂ conservation of momentum (YOU MUST USE SIGNS. any velocity vector to the left of down mus be given a negative sign.
Solving angular motion problems
often can do it linearly by taking the circumference of the circle as the translational distance: C = 2(pi)r = (pi)d
Momentum Equation
p=mv
Machines
reduce the amount of force necessary to performa given amount of work NEVER reduce or change the amount of work! Find force needed without a machine then apply correct ratio..
Solving Equilibrium Problems
set the forces or torques equal to each other: F(right) = F(left) ; F(up) = F(down) ; T(counterclockwise) = T(clockwise) Make a "T" on paper, in the right column write down all forces that push the object to the right and in the left column all the push it to the left.
How to solve centripetal motion problems
set the general centripetal force equation to the equation for whatever force is actually causing the force in that situation
Shear Modulus
shear stress/strain modulus simultaneous pushing or pulling forces - the two forces are NOT aligned)
Velocity of a mass on a spring
sinusoidal... at maximum points away from equilibrium velocity equals 0. At equilibrium point velocity is at its maximum.
Friction opposes...
sliding, NOT motion.
Calculating the Spring Constant by hanging weights
solve for k using Hooke's Law. For (delta)x enter the displacement from the equilibrium point for one trial OR the difference in displacement between two trials. For F, use the FORCE (not kg) applied in one trial or the difference in force between two trials.
"Zero net torque" clue words
stationary or exactly balanced
Young's Modulus
tensile or compressive stress/strain modulus (simultaneous pushing or pulling forces on both sides of an object; the two forces must be exactly aligned in both vertical or horizontal planes)
Which object has the higher ME (moduli of elasticity), a marshmallow or rock?
the rock.
Inclined Planes
there are always at least two forces acting upon any object that is positioned on an inclined plane - the force of gravity and the normal force. The force of gravity (also known as weight) acts in a downward direction; yet the normal force acts in a direction perpendicular to the surface
Reverse Collisions
two objects are together and come apart
reverse collisions
two objects are together and come apart. The MCAT often uses radioactive decay to test students on reverse collisions Use the formula for perfectly inelastic collisions
Perfectly inelastic
two objects collide and then stick together, then the equation becomes... m1v1 + m2v2 = (m1+m2)v3
Pendulum Gravitational PE
usually assumed to be 0 for a bob at the lowest point of its arc. (we assume h to = 0)
Derive a formula for velocity of a satellite orbiting the moon.
v = √(Gm/r); This is derived by setting mv2/r equal to the universal law of gravitation and solving for velocity.
Formulas for angular motion:
w=v/r w=2(pi)f where "w" (omega) is angular frequency (in Rad/s), v is tangential velocity, r is radius and f is the frequency (Hz)
Water is weird
when almost frozen it starts expanding due to its highly-ordered lattice structure of ice (alternating H bonds) thus it floats in its liquid.
Thermal Expansion
when solids are heated, they expand. when solids are cooled, the shrink.
