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MCAT Physics

Vectors

Scalar quantities, such as temperature, have magnitude only and are specified by a number with a unit, 67 degrees Celsius and obey the rules of ordinary algebra.  Vectors, such as displacement, have both magnitude and direction, six meters west and obey the special rules of vector algebra.

X and Y Coordinates

Two vectors A and B may be added geometrically by drawing them to a common scale and placing them head to tail.  The vector connecting the tail of A to the head of B is the sum vector.  To subtract B from A, reverse the direction of B and then add to A.  The component AX and AY of any vector A are the perpendicular projections of A on the coordinate axes.

Pythagoras’ theorem

Analytically, the components are given by AX=A (cos theta) and AY=A (sine theta).  Given the component, we can reconstruct the vector from:  A is given by the square root of the quantity, AX2 + AY2, which is a derivation of the Pythagoras’ theorem.

Kinematics

The motion of a body is described by giving its position or displacement, its velocity and its acceleration.  The average speed is defined as the distance traveled divided by the elapsed time.  The average velocity is the displacement vector divided by the elapsed time.

Displacement

Displacement is the vector representing the position of an object relative to its position at some chosen earlier time, or its point of origin.  Whereas speed is a scalar quantity, velocity is a vector.  The instantaneous velocity whose magnitude is the same as the instantaneous speed is the average velocity taken over an indefinitely short period of time.

Acceleration

Velocity as well other qualities describing motion are always measured with respect to some frame of reference.  Acceleration is the rate of change of velocity.  The change of velocity divided by the elapsed time, it is a vector.  If an object moves in a straight line with constant acceleration, the velocity, V, and the acceleration, A, are related to the initial velocity V0 and the displacement, D, and the time, T, by the equations-V=V0 + AT; D, the displacement, equals V0T + ½AT2; V2=V02 + 2AD.  The mean velocity equals V + V0/2.  Objects allow to fall freely without air resistance all fall with the same constant acceleration, G=9.8 meters/second2.

Motion and force.

Newton’s 3 Laws

Number 1

Newton’s three laws of motion are the basic laws explaining motion.  Newton’s first law states that if the net force on an object is zero, the object at rest remains at rest and an object in motion remains in motion in a straight line with constant velocity.  The tendency of a body to resist a change in motion is called inertia.  Mass is a measure of inertia.  Weight refers to the force of gravity on an object.

Number 2

Newton’s second law states that the acceleration of a body is directly proportional to the net force acting on it and inversely proportional to its mass.  F=ma, where F is the force, m is the mass, and A, the acceleration.  Force, which is a vector, is a push or a pull.  More precisely, Newton’s second law can be used as a definition of force as that action which is capable of accelerating an object.  Net force refers to the vector sum of all forces acting on a body. The force of gravity acting on a body is the product of its mass times the acceleration of gravity.

Number 3

Newton’s third law states that when every one body exerts a force on a second body, the second exerts an equal force on the first in the opposite direction.  A consistent set of units must always be used when making calculations.  SI unit s are the standard ones used for scientific work and these include the meter, kilogram and second.

Friction

When two bodies are in contact or slide over one another, the force of friction each exerts on the other can be written force of friction equals mu N, where N is the normal force, the force each body exerts on the other perpendicular to the surface in contact.  Mu is the coefficient of kinetic friction if the bodies are moving relative to each other.  If they are not moving, the above equation gives the maximum friction force where mu is the coefficient of static friction.

Projectile motion

That of an object moving through the air can be analyzed as two separate motions in the horizontal and vertical directions.  If air resistance can be ignored, the horizontal motion is at constant velocity where the vertical motion is uniformly accelerated and is the same as for the body falling vertically under the action of gravity.

Circular motion and gravitation

Angular quantities are defined in analogy with linear quantities.  Angles can be measured in degrees, revolutions or radians, where two pi radians is equal to one revolution, which is equal to 360 degrees.  A particle moving with constant speed, V, in a circle of radius, R, has a linear centripetal, which means towards the center acceleration.  That acceleration is given by V2/R.  Because the velocity vector is continually changing in direction, that’s why there is an acceleration.  A force acting towards the center is thus needed to keep a particle revolving in a circle.  If the particle is revolving in a circle with non-uniform speed, it will have both centripetal and tangential linear acceleration.

Gravity

Newton’s law of universal gravitation states that every body in the universe attracts every other body with a force proportional to the product of their two masses and inversely proportional to the square of the distance between them.  It is this force of gravity that keeps the moon in its orbit around the Earth and the planets in their orbits around the sun.  The dynamics of rotation is analogous to the dynamic of linear motion.  Force can be replaced by torque which is defined as the product of force times the perpendicular distance from the pivot point.

Inertia

Mass is replaced by the moment of inertia which depends not only on the mass of the body, but also on how the mass is distributed about the axis of rotation.

Angular Acceleration

And linear acceleration is replaced by angular acceleration.  So, instead of seeing F=ma as we previously saw, now we have the rotational equivalent of Newton’s second law, which is tau, the torque force, equals I, the moment inertia times alpha, the angular acceleration.

Center of Gravity

The center of gravity of a body is that point at which the force of gravity can be considered to act for purposes of determining the motion of the body as a whole.  The complete motion of a body can be described as the translational motion of its center of gravity, plus the rotation about its center of gravity.

Equilibrium

We will now discuss bodies in equilibrium.  A body at rest or one in uniform motion at constant velocity is said to be in equilibrium.  The determination of the forces within a structure at rest is the field called statics.  The two necessary conditions for a body to be in equilibrium are:  one, the vector sum of all the forces on it must be zero and, two, the sum of all the torques calculated about any arbitrary point as axis must also be zero.  It’s important when doing statics problems to apply the equilibrium conditions to only one body at a time.

A body in static equilibrium is said to be in stable, unstable or neutral equilibrium, depending on whether a slight displacement leads to a return to the original position, that would be stable equilibrium, or further movement, which would be unstable, or a rest in the new position, which would be neutral equilibrium.  An object in stable equilibrium is also said to be in balance.

Elasticity

Hooke’s law applies to many elastic solids, and states that the change in length of an object is proportional to the applied force.  If the force is too great, the object will exceed its elastic limit, which means it will no longer return to its original shape when the distorting force is removed.  If the force is even greater, the ultimate strength of the material can be exceeded and the object fractures.

Stress

The force per unit area acting on a body is called the stress.  And the resulting fractional change in length is called the strain.  The stress on a body is present within the body and can be of three types-compression, tension and shear.  The ratio of stress to strain is called the elastic modulus of the material.  Young modulus applies for compression and tension and the shear modulus for shear.

Both moduli apply to an object whose volume changes as a result of pressure on all sides.  All three moduli are constants for a given material when distorted within its elastic region.

The subject of statics is especially useful for calculating forces within muscles and bones and in structures such as buildings and bridges.

Momentum and energy

The momentum, P, of a body is defined as its mass times its velocity where P=MV.  In terms of momentum, Newton’s second law can be written, F, the force equals dP over dT, which is the rate of change of a momentum equals the net applied force.

Momentum is a conserved quantity.  The law of conservation of a momentum states that the total momentum of an isolated system of objects remains constant.  An isolated system is one on which the net external force is zero.

Work

Work is done on an object by a force when the force moves the object through a distance, D.  If the direction of the force makes an angle, theta, with the direction of motion, the work done by this force is given by W=FD (cos theta).

Energy

Energy is defined as the ability to do work.  Both work and energy are measured in Joules, where one Joule equals one Newton meter.  Kinetic energy is energy of motion.  A body of mass, m, and speed, V, has translational kinetic energy equal to ½ MV2.

An object can have potential energy by virtue of its position or shape.  Examples are gravitational potential energy, which is equal to mgh, where h is the height of the object of mass, m, above an arbitrary reference point.  An object can also have elastic potential energy, such as a compressed spring.  An object can also have chemical, electrical, or nuclear energy.

The change of potential energy of an object when it changes position is defined as the work needed to take it from one position to the other.  The work energy theorem states that the net work done on a body by the net force equals the change in kinetic energy of that body.

The law of conservation of energy states that energy can be transformed from one type to another, but the total energy remains constant.  It is valid even when friction is present since the heat generated by friction can be considered a form of energy.  Momentum is conserved in any collision between objects.  Energy is conserved, too, but kinetic energy only in so-called elastic collisions in which other forms of energy do not change.

Power

Power is defined as the rate at which work is done or the rate in which energy is transformed.  The SI unit of power is the watt, where 1 watt equals one Joule per second.

Fluids

Now, let us look at fluids.  The three common phases of matter are solid, liquid and gas.  Liquids and gases are collectively called fluids.  Meaning they have the ability to flow.  The density of a material is defined as its mass per unit volume.  Specific gravity is the ratio of the density of the material to the density of water.  Pressure is defined as force per unit area.  The pressure at a depth, h, in a liquid is given by rho gh, where rho is the density of the liquid and g is the acceleration due to gravity.

In addition, if an external pressure is applied to a confined fluid, this pressure is transmitted throughout the fluid.  This is known as Pascal’s principle.  Pressure is measured using manometer or other types of gauge.  A barometer is used to measure atmospheric pressure.  Standard atmospheric pressure, which is the average at sea level, is 1.01 x 105 Newton per meter squared.

Archimedes Principle

Archimedes principle states that an object submerged wholly or partially in a fluid is bouoyed up by a force equal to the weight of fluid it displaces.  This principle is used in a method to determine specific gravity and explains why objects whose density is less than that of liquid will float in that liquid.

Fluid flow rate is the mass or volume of fluid that passes a given point per unit time.  The equation of continuity states that for an incompressible fluid flowing in an enclosed tube, the product of the velocity of flow and the cross-sectional area of the tube remains constant.  AV is a constant.

Bernoulli’s Equation

Bernoulli’s equation tells us that where the velocity of a fluid is high, the pressure in it is low.  And where the velocity is low, the pressure is high.  Bernoulli’s principle explains many common phenomena.  Fluid flow can be characterized either as streamline, sometimes called laminar in which the layers of fluid move smoothly and regularly along paths called streamlines, or it can be characterized as turbulent in which case the flow is not smooth and regular, but it’s characterized by irregularly shaped whirlpools.

Viscocity

Viscosity refers to friction within a fluid that prevents the fluid from flowing freely and is essentially a frictional force between different layers of fluid as they move pass one another.

Temperature and the kinetic theory.  The atomic theory of matter postulates that all matter is made up of tiny entities called atoms.  Some substances are made up of only one type atom, and these are called elements.  Atoms can combine to form molecules and substances made up of a single type of molecule are called compounds.  A substance made up of more than one type of molecule is called a mixture.

Atomic and molecular masses are specified on a scale that’s compared to Carbon 12.  The distinction between solid, liquid and gases can be attributed to the strength of the attractive forces between the atoms and molecules and depends on their average speed.

Temperature is a measure of how hot or cold a body is.  Thermometers are used to measure temperature on the Celsius, Fahrenhiet and Kelvin scales.  Two standard points on each scale are the freezing point of water, which is zero degrees Celsius, 32 degrees Fahrenhiet, and 273 Kelvin, and the boiling point of water which is 100 degrees Celsius, 212 degrees Fahrenhiet, and 373 degrees Kelvin.  A change in temperature of one Kelvin equals a change of one Celsius degrees or 9/5 Fahrenhiet degrees.

The change of length, L, of a solid when its temperature changes by an amount, T, is directly proportional to the temperature change and to its original length, L0. That is L is equal to alpha L0 T, where alpha is the coefficient of linear expansion.

The change in volume of most solids, liquids and gases is proportional to the temperature change and to the original volume, V0, where V is equal to beta V0 T.  The coefficient of volume expansion, beta, is approximately equal to 3 times alpha for solids.  Water is unusual because unlike most materials whose volume increases with temperature, its volume actually decreases as the temperature increases from 0 degrees Celsius to 4 degrees Celsius.

Kinetic Theory of Gases

According to the kinetic theory of gases, which is based on the idea that a gas is made up molecules that are moving rapidly and at random, the average kinetic energy of the molecule is proportional to the Kelvin temperature.  At any moment, there exists a wide distribution of molecular speeds within a substance.

Heat

Thermal energy or internal energy refers to the total energy of all the molecules in a body.  Heat refers to the transfer of energy from one body to another because of a difference of temperature.  Heat is thus measured in energy units such as Joules.  Heat and thermal energy are also sometimes specified in calories or kilocalories where one calorie is equal to 4.18 Joules, and one calorie is the amount of heat needed to raise the temperature of one gram of water by 1 degree Celsius.

Heat Capacity

The specific heat capacity, C, of a substance is defined as the energy or heat required to change the temperature of unit mass of substance by 1 degree.  In the equation, Q=mc T, where Q is the heat absorbed or given off, T the temperature rise or decline, and m, the mass of the substance, that is Q=mc T.  When heat flows within an isolated system, the heat gained by one part of the system is equal to the heat lost by the other part of the system.

Calorimetry

This is the basis for calorimetry, which is the quantitative measurement of heat exchange.  An exchange of energy occurs without a change in temperature whenever substance changes phase.  This happens because the potential energy of the molecules changes as a result of the changes in the relative positions of the molecules.

Heat of Fusion

The heat of fusion is the heat required to melt one kilogram of a solid into the liquid phase.  It is also equal to the heat given off when the substance changes from liquid to solid.  The heat of vaporization is the energy required to change one kilogram of a substance from the liquid to the vapor phase.  It is also the energy given off when the substance changes from vapor to liquid.

Heat Transfer

Heat is transferred from one place or body to another in three different ways.  In conduction, energy is transferred from higher kinetic energy molecules to a lower kinetic energy neighboring molecules when they collide.  Convection is a transfer of energy by the mass movement of molecules over considerable distances.  Radiation, which does not require the presence of matter, is energy transfer by electromagnetic way, such as from the sun.

All bodies radiate energy in an amount that is proportional to their surface area and to the fourth power of their Kelvin temperature.  The energy radiated or absorbed also depends on the nature of the surface, dark and absorbing versus brightly reflecting, which is characterized by the emissivity.

The first and second laws of thermodynamics

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system, Q minus the work, W, done by the system.  This is simply a restatement of the conservation of energy and it’s found to hold for all types of processes.

The second law of thermodynamics can be stated in several equivalent ways.  One, heat flows spontaneously from a hot object to a cold one but not the reverse.  Two, there can be no 100% efficient heat energy.  That is, one that can change a given amount of heat completely into work.  And, three, natural processes tend to move toward a state of greater disorder or greater entropy.  Entropy is a quantitative measure of the disorder of the system.  From statistical point of view, the most probable state of a system is that with the most entropy or disorder.

Vibrations and waves

A vibrating object undergoes simple harmonic motion if the restoring force is proportional to the displacement.  In other words, it obeys Hooke’s law.  The force constant, K, is the ratio of restoring force to the displacement.  The maximum displacement is called the amplitude.  The period, T, is the time required for one complete cycle back and forth and the frequency, F, is the number of cycles per second.  They are related by F=1/T, the period.  The period of vibration for a mass, m, on the end of a spring is given by the following relationship.  T=2 pi square root quantity m/K.

Harmonic Motion

Simple harmonic motion is sinusoidal, which means that the displacement as a function of time follows a sine or a cosine curve.  A simple pendulum of length, L, approximates simple harmonic motion if the amplitude is not too great.  Its period is given by, T=2 pi the square root of l/g, where g is the acceleration due to gravity.

Resonance

During a vibration, the energy continually alternates between kinetic and potential.  When friction is present, the motion is said to damped.  The displacement decreases in time and the energy is eventually all transformed to heat.  When an oscillating force is applied to a system capable of vibrating, the amplitude of vibration is very large if the frequency of the applied force equals or nearly equals the natural frequency of vibration of the object.  This is called resonance.