Physics 2

Formula for "gravity" AKA the "strength of the gravitational field" AKA "acceleration due to gravity"

g=Gm/r²

What is the Equations for the Universal Law of Gravitation?

F=Gm₁m₂/r² (G is the gravitational constant and is always given)

What is a formula relating displacement from a spring's equilibrium point to force? (Hooke's Law)

F=k(delta)X

Formula for the force down an inclined plane, parallel to the surface

F=mgsin(theta)

True or False - A ball moving with twice the kinetic energy can compress a spring twice as far

False

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).

elasticity Bulk Modulus

bulk stress (simultaneous compression from all sides

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.

General formula of elasticity:

(ME)=stress/strain As such, a stiffer material will have a higher elastic modulus

Formula for thermal expansion:

(delta)L=(alpha)L₀(delta)T

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)

...

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

T (period) =

1/frequency

Formula for centripetal force:

F(c)=mv²/r

Work

Think of work in EXACTLY this way and in the EXACT order: W=(delta)Energy W=Fdcos(theta)

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)

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?

12kg

A 150 kg toy rocket is falling toward the earth with a constant velocity of 20m/s. A small amount of fuel still present in the cone creates a downward force of 30 N. What is the force due to air resistance?

1530

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

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

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

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.

Which way does the friction vector point for a car driving east down a straight road? For a gecko climbing a wall? For a car going around a bend in the road?

East Up Towards the center of the curve (note:Friction opposes SLIDING, not MOTION)

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)

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.

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)

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)

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

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 frictioni

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

Formula for the PE in a compressed spring

PE=1/2K(delta)X²

Formula for Electrical Potential Energy

K(e)q₁q₂/r or q(delta)V

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?

K=40 (delta)x=1/2

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!

Equation for Gravitational Potential Energy

PE=mgh (near earth) PE=pgh (For Liquids; Row, thhe greek ; =density) ) PE= -Gm₁m₂/r (in space, or near earth of NOT assuming g=10m/s²)

Units for Torque

Nm

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!

For a ball on a string in circular motion, does the centrifugal force act on the ball or the string?

On the string

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.

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.

elasticity Sheer Modulus

Shear stress (simultaneous pushing or pulling forces; two forces are NOT aligned)

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

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)

Formula for simple Harmonic Motion of a mass on a spring

T=2pi Square-root(m/k)

Formula for Torque

T=FL where L=the lever arm = rsin(theta)

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.

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

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.

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)

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.

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.

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.

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)

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

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

Formula for acceleration down an inclined plane

a=gsin(theta)

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

Momentum: what is it? what is the formula?

inertia increased by velocity, It is always conserved in an isolated system p=mv

Formula for Gravitational Potential Energy

mgh of -Gm₁m₂/r

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

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)