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Physics Day 1 Translational Motion-concept of distance, speed, velocity, and acceleration to describe the location and motion of an object at a point in time. 1. Units and dimensions Basic SI units (System International) Length in meters (m) Time in seconds (s) Mass in kilograms (kg) Temperature in Kelvin (K) SI prefixes Pico (10-12) Nano (10-9) Micro (10-6) Milli (10-3) Centi (10-2) Kilo (103) Mega (106) Giga (109) Problem 1.1 2. Vectors, components Vectors use arrows to signify direction. The length of the arrow represents the magnitude of velocity. A scalar is a physical quantity with a magnitude but not direction (i.e. speed) 3. Vector addition Put arrows head to tail, the order does not matter. In the case of right angles, use trigonometry (i.e. sine, cosine, tangent of the angle) Pythagorean theorem A2 + B2 = C2 Velocity and Force tend to be used most often in vector form Note: This is an essential concept that will be used extensively to examine the x and y components of velocity, force, etc. Problem 2.1, 2.2, and 2.4 4. Speed, Velocity (average and instantaneous) Speed is the rate at which an object covers a distance -Average speed (v) is a distance traveled in a time interval (v = x/t) where x is the distance traveled and t is the time interval 1 -Instantaneous speed (v = x/t) according to the tangent line (draw graph). If the instantaneous speed does not change, then the object is moving at a constant speed (draw graph). The MCAT will always assume speed is constant unless stated otherwise. Problem 1.3 and 1.4 Velocity is a term that accounts for both speed and direction of a moving object. -Average velocity is the change in position over a period of time (v = x/t) -Instantaneous velocity is equal in magnitude to the instantaneous speed…remember to add in the direction the object is traveling. Think about it: Suppose an airplane leaves NYC and flies a distance of 640 km in an hour and then lands. What can you say about its average speed and average velocity? What can you say about its instantaneous speed and instantaneous velocity? #1. An ultrasonic wave enters a body with a speed of 1,500 m/s, and a reflection is noted at the same position on the surface 4.0 x 10-5 sec later. What is the distance between the surface and the reflecting surface? A. 1.3 x 10-8 m B. 2.7 x 10-8 m C. 3.0 x 10-2 m D. 6.0 x 10-2 m MCAT Note: This practice question illustrates a classic feature of the MCAT where answering the question requires complete understanding of the system. In this case, it requires knowing that the wave is traveling to the reflecting surface AND back. Simply plugging in numbers will usually end up in an incorrect answer. 5. Acceleration-an increase or decrease in the speed or change in direction of a moving object (a = v/t) thus a change in velocity over time. All accelerations are assumed to be constant unless otherwise noted. Problem 1.5 *Equations to memorize* Remember that x and y components are interchangeable vf = vo + at where vf is the final velocity, vo is the original velocity, a is acceleration, and t is time. Problem 1.6 x = vot + ½(at2) Problem 1.7 vf2 = vo2 + 2ax Problem 1.8 x = {(vo +vf)/2}(t) 2 Problem 1.10 Generally speaking, the sign for acceleration is negative when decelerating and positive when accelerating. More importantly, understanding the system in question will keep signs clear. 6. Freely falling bodies- here the acceleration is constant (gravity) and speed varies. Objects reach a terminal speed where the drag force due to air resistance does not allow the object to go any faster. An object’s terminal speed depends on its mass, size, and shape. With no air resistance (i.e. a vacuum), all objects fall at the same speed. In the absence of air resistance, the speed that a dropped object has when it reaches the ground is the same as the speed with which it must be thrown upward from the ground to rise to the same height. MCAT Note: You must assume that air resistance is absent unless you are told otherwise. Passages and problems are very clear on whether you take air resistance into account. Passage #13 Problem 1.11 and 1.12 Projectiles are a type of free falling bodies. The initial velocity is in two vector components: x and y where vox = vo cos and voy = vo sin . The velocity at some later time point also has two components: vx = vox (assumes no air resistance) and vy = voy – at. Lastly, the position of the project can be determined, again in two components: x = xo + voxt and y = yo + voyt – 1/2gt2. The last two equations must be *memorized*! #2. If a projectile leaves a gun near the surface of Earth with a speed of 2.0 km/s horizontally, how far will it fall in 0.01 s? A. 4.9 x 10-4 m B. 4.9 x 10-2 m C. 9.8 x 10-2 m D. 2.0 x 101 m Problem 2.9 Passages #11 and #10 3 Force and Motion, Gravitation-describes movements of objects under the influence of some force. 1. Center of mass- point at which we can consider all of an objects mass to be concentrated. This is done to allow for treatment of objects as particles. 2. Newton’s first law, inertia Inertia is the reluctance of an object to change its state of rest or of uniform motion. It is a property of all matter that is determined by the objects mass. Mass is the measure of inertia at rest…the more mass an object has, the harder it is to get the object to accelerate, decelerate, or change direction. Simply stated, an object at rest will remain at rest and an object in motion will continue in motion at constant velocity in the absence of any interaction with something else. 3. Newton’s second law, F = ma (*memorize*), where F is the force, m is the object mass and a is the acceleration. Example: one Newton of force will accelerate a 1 kg object by 1 m/s2. One Newton equals 0.225 lbs. (about a quarter pound). #3. Starting from a resting position at one end of frictionless table, a constant force of 3.0 N is applied to an object with a mass of 60 g. How long will it take for the object to move 1.0 m? A. 0.40 sec B. 6.0 sec C. 0.20 sec D. 40.0 sec Problems 3.1, 3.2, 3.3 4. Newton’s third law, forces equal and opposite- for ever action there is an equal and opposite reaction. Problem 3.8 5. Concept of a field- a region of space with a continuous distribution of a certain property 6. Law of gravitation- every object in the universe attracts every other object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance separating them. Fgrav = G {(mamb)/r2)} (*memorize*). Problem 6.7 7. Uniform circular motion (UCM)- an object moving in a circle at constant speed undergoes UCM: v = (2r)/T where T is the period (*memorize*). Passage #9 8. Centripetal acceleration- can produce a change in direction without a change in speed of an object (ac = v2/r) 4 Problem 6.1 9. Weight- is the force that the earth attracts on an object (W = mg). 10. Friction, static and kinetic Frictional forces (Ff)are forces that act to impede relative motion between two surfaces in contact. Useful in terms of walking, nails and screws in wood, etc. Undesirable in terms of efficiency (i.e. car engines waste half of their power overcoming friction). Static friction- force keeping two stationary surfaces in contact from moving Starting friction- is the max static friction. Once the force applied exceeds the starting friction, one surface breaks away and begins to move relative to the other surface. Kinetic friction- is less than the starting friction, therefore starting is always harder than keeping something going. Rolling friction- occurs between wheels and some surface. Coefficient of friction ()- is a constant for a given pair of surfaces. Ff = FN = mg where FN is the normal force (*memorize*). #4. An object is sliding at a constant speed of 10 m/s across a level surface. If the coefficient of kinetic friction between the object and the surface is 0.7, which of the following statements correctly describes the forces acting on the object? A. Friction is the only horizontal force acting on the object. B. There is no net horizontal force acting on the object. C. There is no work done by the frictional force. D. There are no vertical forces acting on the object. Problem 3.9 11. Motion on an inclined plane W/F = L/h = Fout/Fin = MA where W is the work, L is the length of the plane, h is the height of the plane, Fin is the input force, Fout is the output force, and MA is the mechanical advantage. Problem 8.13 12. Analysis of pulley systems nFin = Fout where n is the number of times the pulley rope overlaps. Much less force must be applied to move a given object with a particular mass. 13. Force Centripetal force is the force needed to make an object follow a curved path. It is perpendicular to the velocity and is directed toward the center of the curvature. Fc = (mv2)/r (*memorize*) 5 Problem 6.2 Equilibrium and Momentum A. Equilibrium- an object is acted upon by a set of forces whose resultant vector is zero. 1. Concept of force- any influence that can change the velocity of an object. Units are Newton’s. 2. Translational equilibrium- forces acting on an object have a vector sum of zero, with no net force there is no acceleration. F = 0 Fx = TBx + TAx = 0 Fy = TBy + TAy – mg = 0 (*memorize*) Problem 8.1 Passage #10 3. Rotational equilibrium- sum of torques acting on an object is zero. = +/- FL (*memorize*) where the sign corresponds to the direction of rotation (clockwise rotation means the sign is negative and vice versa for counterclockwise) and L is the distance between the pivot point and the point the force is acting upon. #5. When a downward force is applied at a point 0.60 m to the left of a fulcrum, equilibrium is obtained by placing a mass of 10-7 kg at a point 0.40 m to the right of the fulcrum. What is the magnitude of the downward force? A. 1.5 * 10-7 N B. 6.5 * 10-7 N C. 9.8 * 10-7 N D. 1.5 * 10-6 N #6. A 0.5 kg uniform meter stick is suspended by a single string at the 30 cm mark. A 0.2 kg mass hangs at the 80 cm mark. What mass hung at the 10 cm mark will produce equilibrium? A. 0.3 kg B. 0.5 kg C. 0.7 kg D. 1.0 kg Problem 8.5 4. Analysis of forces acting on an object 6 5. Newton’s first law-already covered above 6. Torques, lever arms- there are three classes of lever arms (I, II, and III). The first two classes allow for a mechanical advantage (i.e. less input force to move some object). In general, FinLin = FoutLout (Diagrams are drawn in class). Problem 8.11 (remember that the whole crate is not lifted) 7. Weightlessness- actual vs. apparent weight B. Momentum- the tendency of a moving body to pursue a straight path. 1. Momentum () = mv (*memorize*) Problem 5.1 2. Impulse (I) = Ft is a vector quantity that causes a change in momentum, thus mvFt = I Problem 5.2 3. Conservation of linear momentum memorize*) Problem 5.3 4. Elastic collisions- kinetic energy is conserved. m1v1 + m2v2 = m1v1’ + m2v2’ (*memorize*) The left side of the equation are the conditions before the collision and the right side are the conditions after the collision. Think about 3 scenarios: 1) when m1 = m2, v2 = 0, v1’ = 0 and v2’ = v1: The colliding object stops, and the target object moves off with the same speed. An example would be a pendulum with identical steel balls suspended by strings. 2) when m1 < m2, v1’ is opposite to v1: The lighter object bounces off the heavier one. An example would be bouncing a golf ball off of a brick wall. 3) when m1 > m2, the colliding object continues in the same direction after the impact but with reduced speed while the target object moves ahead of it at a faster pace. An example would be hitting a tennis ball with a racquet on a serve. #7. A fission neutron loses its kinetic energy through collisions. When a neutron in motion collides with a particle, what is the maximum percentage of its kinetic energy that can be transferred? A. B. C. D. 100% 50% 33% 25% 7 5. Inelastic collisions- kinetic energy is not conserved; it is lost as heat, sound, etc. For the MCAT, these collisions are most often completely inelastic (i.e. the objects stick together resulting in maximal kinetic energy loss). m1v1 + m2v2 = MV (*memorize*) where M is the total mass and V is the total velocity. #8. A 0.5 kg ball accelerates from rest at 10 m/s2 for 2 sec. It then collides with and sticks to a 1.0 kg ball that is initially at rest. After the collision, approximately how fast are the balls going? A. 3.3 m/s B. 6.7 m/s C. 10.0 m/s D. 15.0 m/s Passage #5 Problem 5.5 Work and Energy A. Work- a measure of the amount of change to which a force gives rise. It has scalar characteristics. 1. Derived units, sign conventions W = Fx (*memorize*) when the force applied and the distance traveled are parallel. When the two are not parallel but at some angle, W =Fxx = (F cos)x (*memorize*) Problem 4.1 2. Amount of work done in gravitational field is path-independent: discussed below. 3. Mechanical advantage- already discussed above. 4. Work-kinetic energy theorem- the work done on an object may be converted into heat, sound, etc. thus KE nor PE are conserved but the overall energy is conserved. W = Wf + KE + PE (*memorize*) Problem 4.12 5. Power- rate at which work is being done. P = W/t = Fv (only if F and v are parallel, if not then Fv cos(*memorize*). Efficiency = (Poutput/Pinput) x 100 Problems 4.3, 4.4, and 4.5 (4.5 requires a calculator so don’t freak out, the MCAT will not have questions like this…it is just to give you the concepts) 8 B. Energy 1. Kinetic energy- energy due to motion KE = ½ mv2 = Fx (*memorize*) Problem 4.6 2. Potential energy- energy due to position PE = mgh = Fx (*memorize*) 3. Conservation of Energy- no violation has ever been found. The total amount of energy in a system isolated from the rest of the universe always remains constant, although energy transformations from one form to another may occur within the system. This concept is important in relating PE and KE. For example, the total energy of a falling ball remains constant as its potential energy is transformed into kinetic energy. Energy conversion is something that appears at least once on every MCAT: #9. When ultrasonic waves are detected by a transducer that feeds a signal into a display monitor, which of the following is the best description of the energy transformation that occurs at the transducer? A. Mechanical to electrical B. Kinetic to mechanical C. Potential to kinetic D. Potential to electrical #10. As the Hubble telescope was transported into orbit aboard Discovery, which of the following best describes the main energy conversion(s) taking place? A. Kinetic to gravitational potential B. Gravitational potential to kinetic C. Gravitational potential to kinetic to chemical D. Chemical to kinetic to gravitational potential Problem 4.10 4. Conservative Forces (such as gravity, magnetic forces, and elastic forces) - give rise to PE and are a state function. The work done by (or against) a force of this kind depends only on the endpoints of the motion of an object it acts on. The path taken between the endpoints does not matter. If the path is reversed, the work done can reappear as KE. KE and PE are conserved here. Non-conservative force (such as friction)- depends on the path taken and does not give rise to PE. Reversing the path cannot return the work done by the force. Here the work is not lost but merely is changed into a form of energy (heat, light, sound, etc.) that is not recoverable. KE and PE are not conserved here but the total energy is conserved. 9 5. Power, units- watts (joules/sec) #11. What physical quantity can be determined by multiplying kW by an amount of time? A. Current B. Charge C. Power D. Energy Waves and Periodic Motion A. Wave Characteristics 1. Transverse waves- are characteristic in stretched string because of individual segments of string vibrate perpendicular to the direction the wave is traveling. T = mv2/L where L is the length of the string, T is the tension, m is the mass of the string, and v is the velocity of the wave. Problem 12.1 Longitudinal waves- occur when particles (individual) of a medium vibrate back and forth parallel to the direction that the waves travel. They are density fluctuations, i.e. sound. Waves may be transverse, longitudinal, or a combination of both. Waves on the surface of a body of water (or other liquid) have characteristics of both. #12. A longitudinal wave could NOT travel through which of the following? A. Water B. A vacuum C. Nitrogen gas D. Steel #13. Longitudinal waves travel by causing air molecules to: A. Flow for long distances. B. Vibrate perpendicular to the direction of motion. C. Rotate around their centers of mass. D. Strike nearby particles and then return to their original positions. 2. Wavelength, frequency, velocity v = f(*memorize*) where f is the frequency, is the wavelength and v is the velocity of the wave. T = 1/f (*memorize*) where T is the period of the wave. 10 #14. A tsunami in the open ocean has a wavelength of 5 * 105 m and travels at 140 m/s. What is the frequency of this wave? A. B. C. D. 2.8 * 10-4 Hz 6 * 102 Hz 3.6 * 103 Hz 7 * 107 Hz Problems 12.2 and 12.3 3. Amplitude and intensity are interchangeable; they are the amount of displacement from the resting position of the medium and both are independent of the speed, frequency, and wavelength of a wave. 4. Supposition of waves, interference, addition When two pulses travel past a point in a string at the same time, the displacement of the string at that point is the sum of the displacements each pulse would produce there by itself. This holds for all types of waves included waves in a stretched string, sound waves, water waves, and light waves. Out of phase vs. In phase Constructive vs. Destructive interference #15. Two mechanical waves of the same frequency pass through the same medium. The amplitude of Wave A is 3 units, and the amplitude of Wave B is 5 units. Which of the following describes the range of amplitudes possible when the 2 waves pass through the medium simultaneously? A. Always 4 units B. Between 2 and 8 units C. Between 3 and 5 units D. Between 5 and 8 units 5. Resonance (different from chemistry)- All structures have a natural frequency of oscillation. A periodic force at the right frequency can induce vibrations in a constructive fashion. The rattling of a car at certain speeds is a familiar example of resonance. 6. Standing waves are reflected back and forth between fixed points; nodes are where there is no displacement of the string from the resting position. Pipe closed at one end and open at the other: = 2L/n where n is an integer Pipe open at both ends: = 4L/n where n is an integer 11 #16. Consecutive resonances occur at wavelengths of 8 m and 4.8 m in an organ pipe closed at one end. What is the length of the organ pipe? (Note: Resonances occur at L = n/4, where L is the pipe length, is the wavelength, and n = 1, 3, 5, …) A. 3.2 m B. 4.8 m C. 6.0 m D. 8.0 m Harmonics have the general equation of fn = nf1 (*memorize*) where n is the number of the harmonic (second, third, etc.) and f1 is the fundamental frequency (the lowest possible frequency or the longest wavelength). f2, f3 and so on, are called overtones. 7. Beat frequencies- are regular loudness pulsations. There are two ways to think of a beat frequency: First is fbeat = (f2 – f1)/2, second is fbeat = (f2+f1)/2. Unfortunately, the MCAT gives no indication of when to use one of the two equations. However, the answers will never come from both equations so ultimately it is pretty straightforward to determine which equation they want you to use. #17. When a trumpet is tuned by comparison with a 512-Hz note from a piano, a beat frequency of 4 Hz is produced. The trumpet could have produced which of the following pairs of frequencies? A. 504 Hz and 508 Hz B. 508 Hz and 512 Hz C. 508 Hz and 516 Hz D. 512 Hz and 516 Hz 8. Refraction- bending of light when it goes from one medium to another. Diffraction- bending of light around the edge of an obstacle in its path. B. Periodic Motion 1. Amplitude, period, frequency 2. Phase 3. Hooke’s law- F = kx (*memorize*) where k is a constant for the spring and x is the distance of the spring from its resting position. The elongation of a spring is proportional to the stress (force) applied if the elastic limit is not exceeded. When the stress is removed, the spring returns to its original length. 12 #18. A vertically oriented spring is stretched by 0.15 m when a 100 g mass is suspended from it. What is the approximate spring constant of the spring? A. 0.015 N/m B. 0.15 N/m C. 1.5 N/m D. 6.5 N/m Elastic limit is the maximum stress that can be applied to an object without the object being permanently deformed as a result (think of a slinky). Springs can be joined in series or in parallel (think of capacitors in electrical circuits). If they are joined in series, then k = (k1k2)/(k1+k2). If they are in parallel, then k = k1 + k2… Problem 11.1 A spring can also hold potential energy: PE = ½ kx2 (*memorize*) Problem 11.2 4. Simple harmonic motion- without friction, the spring-object system will keep oscillating back and forth indefinitely. There are two equations that deal with this concept, I will put them up in class but they are not to be memorized Problem 11.3 To determine the maximum speed of a stretched spring that is released, relate KE and PE. 5. Motion of a pendulum- two types: simple and physical. I will write both equations in class…they are not to be memorized. Problem 11.7 6. General periodic motion: velocity, amplitude Passages #2 and #7 13