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College Physics (AP101)

College Physics (AP101). Dr.Haitao Huang ( 黄海涛 ) Department of Applied Physics, Hong Kong PolyU Tel: 27665694; Office: CD613 Lecture Notes can be downloaded from http://ap.polyu.edu.hk/apahthua Textbooks: Physics for Scientist & Engineers , D.C.Giancoli, Prentice Hall

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College Physics (AP101)

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  1. College Physics (AP101) Dr.Haitao Huang (黄海涛) Department of Applied Physics, Hong Kong PolyU Tel: 27665694; Office: CD613 Lecture Notes can be downloaded from http://ap.polyu.edu.hk/apahthua Textbooks: Physics for Scientist & Engineers, D.C.Giancoli, Prentice Hall Fundamentals of Physics, D.Halliday et al., John Willey & Sons College Physics, A.Giambattista et al., McGraw Hill University Physics, H.D.Young et al., Pearson Physics for Scientist & Engineers, R.D.Knight, Pearson Resources: Multimedia Lab: CD706 College Physics----by Dr.H.Huang, Department of Applied Physics

  2. Highly Recommended References: For beginners: Berkeley Physics Course (V.1-5) 力学，赵景员，王淑贤 热力学，王竹溪； 电磁学，赵凯华，陈熙谋； 光学，赵凯华，钟锡华 Intermediate: The Feynman Lectures on Physics, R.P.Feynman, et al. Classical Mechanics, H.Goldstein et al. Classical Electrodynamics, J.D.Jackson Quantum Theory of the Solid State, J.Callaway Principles of Optics, M.Born et al. 分析力学， 电动力学， 热力学与统计物理，王竹溪 量子力学，曾谨言 Advanced: Course of Theoretical Physics, (V.1-10), L.D.Landauet al. The Principles of Quantum Mechanics, P.A.M.Dirac Assessment Weighting: Continuous Assessment: Lab (and report): 16% Quiz (and midterm exam): 24% Final Exam: 60% (Minimum requirement: D) Minimum requirement for overall grade: D College Physics----by Dr.H.Huang, Department of Applied Physics

  3. Classical Mechanics Electrodynamics Physics Optics Thermodynamics Quantum Mechanics The fundamentals of physics need to be understood by anyone who hopes to make a career in the sciences or technology: physicists, engineers, chemists, astronomers, mathematicians, geologist, or biologists. The study of physics uses skills that are useful in other fields as well. College Physics----by Dr.H.Huang, Department of Applied Physics

  4. Essence of Physics Classical Mechanics: Newton’s law Electrodynamics: Maxwell’s equations Optics: Fermat’s principle Thermodynamics: First, second and third laws Quantum Mechanics; Schrödinger’s equation College Physics----by Dr.H.Huang, Department of Applied Physics

  5. Measurement Measurement in Physics: Physics is based on measurement of physical quantities. Certain physical quantities have been chosen as base quantities, such as length (长度), time (时间), and mass (质量), each has been defined in terms of a standard (标准) and given a unit (单位) measure, such as meter, second, and kilogram. Other physical quantities are defined in terms of the base quantities and their standards and units. The standard of one meter is the length of the path traveled by light in vacuum (真空) during a time interval of 1/299,792,458 of a second (1983). The standard of one second is the time taken by 9,192,631,770 vibrations of the light (of a specified wavelength (波长)) emitted by a cesium-133 atom (1967). The standard of one kilogram is the mass of a platinum-iridium cylinder kept at the International Bureau of Weights and Measures near Paris (1889). A second mass standard is the carbon-12 atom, which has been assigned a mass of 12 atomic mass units (u) SI Unit: In 1971, the 14th General conference on Weights and Measures picked seven quantities as base quantities, thereby forming the basis of the International System of Units, abbreviated SI from its French name and popularly known as the metric system. Many SI derived units are defined in terms of these base units. College Physics----by Dr.H.Huang, Department of Applied Physics

  6. Measurement SI Unit: In 1971, the 14th General conference on Weights and Measures picked seven quantities as base quantities, thereby forming the basis of the International System of Units, abbreviated SI from its French name and popularly known as the metric system. Many SI derived units are defined in terms of these base units. Significant Figures (有效数字) The number of reliably known digits in a number is call the number of significant figures. In general, no final result should have more significant figures than the original data from which it was derived. If multiple steps of calculation are involved, you should retain more significant figures than the original data have. However, when you come to the final result, you should round off according to the original data with the least significant figures. To express the very large and very small quantities, we use the so called scientific notation. We write numbers in “powers of ten”. One advantage of scientific notation is that it allows the number of significant figures to be clearly expressed. Example: A friend asks to borrow your precious diamond for a day to show her family. You are a bit worried, so you carefully have your diamond weighed on a scale which reads 8.17 grams. The scale’s accuracy is claimed to be±0.05 grams. The next day you weigh the returned diamond again, getting 8.09 grams. Is this your diamond? College Physics----by Dr.H.Huang, Department of Applied Physics

  7. Measurement h A d B r r  Example: How to estimate the radius of the Earth? Suppose that you watch the Sunset over a calm ocean while lying on the beach, starting a stopwatch just as the top of the Sun disappears. You then stand, elevating your eyes by a height h=1.70m, and stop the watch when top of the Sun again disappears. If the elapsed time on the watch is t=11.1s, what is the radius r of the Earth? The angle between the tangent points A and B is . College Physics----by Dr.H.Huang, Department of Applied Physics

  8. Measurement • Homework: • Horses are to race over a distance of 4.0 furlongs. What is the race distance in (a) rods and (b) chains? (1 furlong = 201.168 m, 1 rod = 5.0292 m, and 1 chain = 20.117 m.) • The micrometer (1 m) is often called the micron. (a) How many microns make up 1.0 km? (b) What fraction of a centimeter equals 1.0 m? • The earth is approximately a sphere of radius 6.37106 m. (a) What is its circumference in kilometers? (b) What is its surface area in square kilometers? (c) What is its volume in cubic kilometers? • The standard kilogram is in the shape of a circular cylinder with its height equal to its diameter. Show that, for a circular cylinder of fixed volume, this equality gives the smallest surface area, thus minimizing the effects of surface contamination and wear. • (a) Assuming that the density (mass/volume) of water is exactly 1g/cm3, express the density of water in kilograms per cubic meter (kg/m3). (b) Suppose that it takes 10 h to drain a container of 5700m3 of water. What is the “mass flow rate”, in kilograms per second, of water from the container? • The density of an iron atom is 7.87g/cm3, and the mass of an iron atom is 9.2710-26kg. If the atoms are spherical and tightly packed, (a) what is the volume of an iron atom and (b) what is the distance between the centers of adjacent atoms? College Physics----by Dr.H.Huang, Department of Applied Physics

  9. Motion Along a Straight Line The description of the motion of moving bodies is called kinematics (运动学). Particle (质点) is a physical model that we neglect the size and shape of a body. It is treated as a geometric point. Position(位置): The position x of a particle on an axis locate the particle with respect to the origin, or zero point, of the axis. The position is either positive or negative, according to which side of the origin the particle is on, or zero if the particle is at the origin. Displacement (位移): The displacement x of a particle is the change in its position: Displacement is a vector (矢量) quantity. It is positive if the particle has moved in the positive direction of the axis, and negative if it has moved in the negative direction. Average Velocity (平均速度): The algebraic sign of the average velocity indicates the direction of motion. The average velocity does not depend on the actual distance a particle covers, but instead depends on its original and final positions. College Physics----by Dr.H.Huang, Department of Applied Physics

  10. Motion Along a Straight Line Average Speed (平均速率): Instantaneous Velocity (瞬时速度): The instantaneous velocity (at a particular time) may be found as the slope (at that particular time) of the graph of x versus t. Speed is the magnitude of instantaneous velocity. Average Acceleration (平均加速度): Instantaneous Acceleration (瞬时加速度): Constant Acceleration: Basic equations: Free-Fall (自由落体) Acceleration: near the Earth’s surface: College Physics----by Dr.H.Huang, Department of Applied Physics

  11. Motion Along a Straight Line Example: The right figures are x(t), v(t), and a(t) plots for an elevator cab that is initially stationary, then moves upward (which we take to be the positive direction), and then stops. For interval bc, the slope in x(t) plot is constant so that the velocity is constant and the acceleration is zero. For interval ab, the velocity increases linearly with time so that the acceleration is constant and positive. For interval cd, the velocity decreases linearly with time so that the acceleration is constant but negative. College Physics----by Dr.H.Huang, Department of Applied Physics

  12. Motion Along a Straight Line Example: The position of a particle moving on the x axis is given by What is the velocity at t=3.5s (x in meter)? Is the velocity constant, or is it continuously changing? Example: A particle’s position is given by where x is in meter. (a) Find v(t) and a(t). (b) Is there ever a time when v=0? (c) Describe the particle’s motion for t0. Example: Spotting a police car, you brake a Porsche from 75 km/h to 45 km/h over a distance of 88 m. (a) What is the acceleration, assumed to be constant? (b) What is the elapsed time? (c) If you continue to slow down with the acceleration calculated in (a), how much time will elapse in bringing the car to rest for 75 km/h? What distance will be covered? (d) Suppose that later, using the acceleration calculated in (a) but a different initial velocity, you bring your car to rest after traveling 200m. What is the total braking time? College Physics----by Dr.H.Huang, Department of Applied Physics

  13. Motion Along a Straight Line Example: A worker drops a wrench down the elevator shaft of a tall building. (a) Where is the wrench 1.5 s later? (b) How fast is the wrench falling just then? Example: A pitcher tosses a baseball straight up, with an initial speed of 12m/s. (a) How long does the ball take to reach its highest point? (b) How high does the ball rise above its release point? (c) How long will it take for the ball to reach a point 5.0m above its release point? Concepts: Can an object have zero velocity and still be accelerating? Can an object have constant velocity and still have a varying speed? Can the velocity of an object reverse direction when the object’s acceleration is constant? Can an object be increasing in speed as its acceleration decreases? College Physics----by Dr.H.Huang, Department of Applied Physics

  14. Motion Along a Straight Line • Homework: • You drive on Interstate 10 from San Antonio to Houston, half the time at 56km/h and the other half at 89km/h. On the way back you travel half the distance at 56km/h and the other half at 89km/h. What is your average speed (a) from San Antonio to Houston, (b) from Houston to back to San Antonio, and (c) for the entire trip? (d) What is your average velocity for the entire trip? (e) Graph x versus t for (a), assuming the motion is all in the positive x direction. Indicate how the average velocity can be found on the graph. • Two trains, each having a speed of 30km/h. are headed at each other on the same straight track. A bird that can fly 60km/h flies off the front of one train when they are 60km apart and heads directly for the other train. On reaching the other train it flies directly back to the first train, and so forth. (a) How many trips can the bird make from one train to the other before they crash? (b) What is the total distance the bird travels? • A man stands still from t=0 to t=5.0min; from t=5.00min to t=10.0min he walks briskly in a straight line at a constant speed of 2.20m/s. What are his average velocity and average acceleration during the time intervals (a) 2.00min to 8.00min and (b) 3.00min to 9.00min? (c) Sketch x versus t and v versus t, and indicate how the answers to (a) and (b) can be obtained from the graphs. • An electron with initial velocity v0=1.50×105m/s enters a region 1.0cm long where it is electrically accelerated. It emerges with velocity v=5.70×106m/s. What was its acceleration, assumed constant? • An object falls from a bridge that is 45m above the water. It falls directly into a model boat, moving with constant velocity, that was 12m from the point of impact when the object was released. What was the speed of the boat? College Physics----by Dr.H.Huang, Department of Applied Physics

  15. Vectors s b a Scalars (标量)and Vectors (矢量) Scalars, such as temperature, have magnitude only. They are specified by a number with a unit and obey the rules of arithmetic and ordinary algebra. Vectors, such as displacement, have both magnitude and direction and obey the special rules of vector algebra. The Sum (Resultant) of Two Vectors 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 the first to the head of the second is the sum vector s. Components (分量)of a Vector The (scalar) components ax and ay of any two dimensional vector a are given by, where  is the angle formed from the positive direction of x axis to the direction of a. y a  x College Physics----by Dr.H.Huang, Department of Applied Physics

  16. Vectors Unit-Vector (单位矢量；基矢)Notation i, j, and k are unit vectors whose magnitudes are unity and whose directions are those of the x, y, and z axes, respectively, in a right-handed coordinate system. A vector a can be written as: Adding Vectors in Component Form Suppose a and b are the vectors to be added, the scalar components of the vector sum r are: Commutative law: Associative law: Subtraction: Vectors and Physical Laws The laws of physics are independent of the choice of coordinates. College Physics----by Dr.H.Huang, Department of Applied Physics

  17. Vectors c a b Multiplying a Vector by a Scalar The product of a scalar s and a vector v is a new vector whose magnitude is sv and whose direction is the same as that of v is s is positive, and opposite that of v if s is negative. To divide v by s, multiply v by (1/s). The Scalar (or Dot) Product (标量积) The Vector (or Cross) Product (矢量积) The vector product of a and b is a vector c whose magnitude is in which  is the smaller of the angle between the directions of a and b. The direction of c is perpendicular to the plane defined by a and b and is given by a right-hand rule. a  b College Physics----by Dr.H.Huang, Department of Applied Physics

  18. Vectors Example: Find the vector r which is the sum of the following three vectors: Example: What is the angle  between ? Example: If , where and , what is c? Example: Describe two vectors a and b such that (a) and ; (b) ; (c) and . Concepts: (a) Can a vector have zero magnitude if one of its components is not zero? (b) Can two vectors having different magnitudes be combined to give a vector sum of zero? Can three vectors give a vector sum of zero if they (c) are in the same plane, and (d) are not in the same plane? (e) Can the magnitude of the difference between two vectors ever be greater than the magnitude of one of the vectors or their sum? College Physics----by Dr.H.Huang, Department of Applied Physics

  19. Motions in 2- and 3-Dimensions Position Vector: Displacement: Average Velocity: Instantaneous Velocity: The instantaneous velocity v of a particle is always tangent to the path of the particle. Average Acceleration: Instantaneous Acceleration: College Physics----by Dr.H.Huang, Department of Applied Physics

  20. Motions in 2- and 3-Dimensions Projectile Motion: Projectile motion is the motion of a particle that is launched with a initial velocity v0 and then has the free-fall acceleration g. If v0 is expressed as a magnitude (v0) and an angle 0, the equations of motion along the horizontal x axis and vertical y axis are The horizontal motion and the vertical motion are independent of each other. The path of (trajectory) a particle in projectile motion is parabolic: The particle’s range (射程) R is the horizontal distance from the launch point to the point at which the particle returns to the launch height: The range is maximum for a launch angle of 45°. College Physics----by Dr.H.Huang, Department of Applied Physics

  21. Motions in 2- and 3-Dimensions Uniform Circular Motion: (匀速圆周运动) If a particle travels along a circle or circular arc with radius r at constant speed v, it is in uniform circular motion. The direction of a is toward the center of the circle or circular arc, and a is said to be centripetal (向心). The time for the particle to complete a circle is called the period of revolution, or simply the period, of the motion. Example: A rescue plane is flying at a constant elevation of 500m with a speed of 198km/h toward a point directly over a person struggling in the water. At what angle of sight  should the pilot release a rescue capsule if it is to strike very close to the person in the water? As the capsule reaches the water, what is its velocity? College Physics----by Dr.H.Huang, Department of Applied Physics

  22. Motions in 2- and 3-Dimensions Example: (The Earth’s radius is 6370km) A satellite is in circular Earth orbit, at an altitude h=200km above the Earth’s surface. There the free-fall acceleration g is 9.20m/s2. What is the orbital speed v of the satellite? Example: The compass in a plane indicates that the plane is headed (pointed) due east; its airspeed indicator reads 215km/h. (Airspeed is speed relative to the air.) A steady wind of 65.0km/h is blowing due north. (a) What is the velocity of the plane with respect to the ground? (b) If the pilot wishes to fly due east, what must be the heading? Homework: 1. The position of an electron is given by r=3.0ti–4.0t2j+2.0k (where t is in seconds and the coefficients have the proper units for r to be in meters). (a) What is v(t) for the electron? (b) In unit-vector notation, what is v at t=2.0s? (c) What are the magnitude and direction of v just then? 2. What is the maximum height to which a baseball player can throw a ball if his maximum throwing range is 60m? 3. An airplane, diving at an angle of 53.0 with the vertical, releases a projectile at an altitude of 730m. The projectile hits the ground 5.00s after being released. (a) What is the speed of the aircraft? (b) How far did the projectile travel horizontally during its flight? (c) What were the horizontal and vertical components of its velocity just before striking the ground? College Physics----by Dr.H.Huang, Department of Applied Physics

  23. Force and Motion I Newtonian Mechanics (牛顿力学): Newtonian mechanics relates accelerations and forces. Newton’s First Law: If there is no net force on a body, the body must remain at rest if it is initially at rest, or move in a straight line at constant speed if it is in motion. For such a body, there are reference frames, called inertial frames, from which the body’s acceleration a will be measured as being zero. Newton’s Second Law: F is the net force (净力) or resultant force (合力). A free-body diagram is helpful in solving problems with the second law: only one body is considered in the diagram. That body is represented by a dot. The external forces on the body are drawn as vectors, and a coordinate system is superimposed, oriented so as to simplify the solution. Newton’s Third Law: Action force (作用力) and reaction force (反作用力) are equal in magnitude and opposite in direction. They act on different bodies. College Physics----by Dr.H.Huang, Department of Applied Physics

  24. Force and Motion I Some Particular Forces: A body’s weight W (重力) is the force on the body from a nearby astronomical body (usually the Earth) : where g is the free-fall acceleration. A normal force N (压力) is the force exerted on a body by a surface against which the body is pressed. The normal force is always perpendicular to the surface. A frictional force f (摩擦力) is the force on a body when the body slides or attempts to slide along a surface. The force is parallel to the surface and directed so as to oppose the motion of the body. A frictionless surface (无摩擦表面，光滑表面) is one where the frictional force is negligible. A tension T (张力，拉力) is the force on a body from a taut cord at its point of attachment. The force points along the cord, away from the body. For massless cords (their mass is negligible), the pulls at both ends of the cord have the same magnitude T, even if the cord runs around a massless, frictionless pulley (滑轮) (the pulley’s mass is negligible and the pulley has negligible friction on its axle opposing its rotation). College Physics----by Dr.H.Huang, Department of Applied Physics

  25. Force and Motion I Example: A student (with cleated boots) pushes a loaded sled whose mass m is 240 kg for a distance d of 2.3 mover the frictionless surface of a frozen lake. He exerts a constant horizontal force F, with magnitude F=130 N, as he does so. (a) If the sled starts from rest, what is its final velocity? (b) The student now wants to reverse the sled’s velocity in 4.5 s. With what constant force must he push on the sled to do so? Example: A block of mass M=3.3 kg is free to move along a horizontal frictionless surface such as an air table. The sliding block is connected by a cord that extends around a massless, frictionless pulley to a second block (the hanging block), whose mass m is 2.1 kg. The hanging block falls and the sliding block accelerates to the right. Find (a) the acceleration of the sliding block, (b) the acceleration of the hanging block, and (c) the tension in the cord. College Physics----by Dr.H.Huang, Department of Applied Physics

  26. Force and Motion I Example: A block whose mass M is 33 kg is pushed across a frictionless surface by a stick whose mass m is 3.2 kg. The block is moved (from rest) a distance d=77 cm in 1.7 s at constant acceleration. (a) What force must the hand apply to the stick? (b) With what force does the stick push on the block? (c) What is the net force on the block? Example: A block of mass m=15 kg is hanging from three cords. What are the tensions in the cords? Example: Two blocks connected by a cord that passes over a massless, frictionless pulley. Let m=1.3 kg and M=2.8 kg. Find the tension in the cord and the (common) magnitude of the acceleration of the two blocks. Example: A block of mass m=15 kg is held by a cord on a frictionless inclined plane. What is the tension in the cord if =27? What force does the plane exert on the block? College Physics----by Dr.H.Huang, Department of Applied Physics

  27. Force and Motion I • Homework: • An elevator and its load have a combined mass of 1600 kg. Find the tension in the supporting cable when the elevator, originally moving downward at 12 m/s, is brought to rest with a constant acceleration in a distance of 42 m. • A 1.0 kg mass on a 37 incline is connected to a 3.0 kg mass on a horizontal surface. The surfaces and the pulley are frictionless. If F=12 N, what is the tension in the connecting cord? • A 100 kg crate is pushed at constant speed up the frictionless 30.0 ramp. (a) What horizontal force F is required? (b) What force is exerted by the ramp on the crate? • The figure shows two blocks on a frictionless plane, with inclination angle , and a third block. Blocks m1 and m2 are connected by a cord with tension T1; blocks m2 and m3 are connected by a cord (around a frictionless, massless pulley) with tension T2. If =20, m1=2.00 kg, m2=1.00 kg, and m3=3.00 kg, find T1, T2, and the acceleration of the blocks. College Physics----by Dr.H.Huang, Department of Applied Physics

  28. Force and Motion II Friction: A frictional force (摩擦力) is exerted on the body by the surface. It is parallel to the surface and directed so as to oppose the sliding. If the body does not slide, the frictional force is a static frictional forcefs (静摩擦力). If there is sliding , the frictional force is a kinetic frictional force fk (动摩擦力). Properties of Friction: If the body does not move, then the static frictional force fs and the component of F that is parallel to the surface are equal in magnitude and opposite in direction. The magnitude of fs has a maximum value fs,max=sN, where s is the coefficient of static friction (静摩擦系数) and N is the magnitude of the normal force. If the body begins to slide along the surface, the magnitude of the frictional force rapidly decreases to a constant value fk= kN, where k is the coefficient of kinetic friction (动摩擦系数). College Physics----by Dr.H.Huang, Department of Applied Physics

  29. Force and Motion II Uniform Circular Motion (匀速圆周运动): The centripetal acceleration (向心加速度) and the centripetal force (向心力) are: Example: The inset shows a coin resting on a book that has been tilted at an angle  with the horizontal. When  is increased to 13, the coin begins to slide down the book. What is the coefficient of static friction s between the coin and the book? Example: A woman pulls a loaded sled of mass m=75 kg along a horizontal surface at constant velocity. The coefficient of kinetic friction k between the runner and the snow is 0.10, and the angle  is 42 . (a) What is the tension T in the rope? (b) What is the normal force with which the snow pushes vertically upward on the sled? College Physics----by Dr.H.Huang, Department of Applied Physics

  30. Force and Motion II Example: Two crates with the same mass m1=m2=14 kg are connected through a taut, massless cord that runs around a frictionless, massless pulley. The hanging crate descends with constant velocity. (a) What are the magnitude and direction of the frictional force exerted on m1 by the plane? (b) What is k? Example: A raindrop with radius R=1.5 mm falls from a cloud that is at height h=1200 m above the ground. The drag coefficient C for the drop is 0.60. Assume that the drop is spherical throughout its fall. The density of water w is 1000 kg/m3, and the density of air a is 1.2 kg/m3. (a) What is the terminal speed of the drop? (b) What would have been the speed just before impact if there had been no drag force? Example: The bob of a conical pendulum has a mass of 1.5 kg, whirls around in a horizontal circle at constant speed v at the end of a cord whose length L, measured to the center of the bob , is 1.7 m. The cord makes an angle  of 37 with the vertical. As the bob swings around in a circle, the cord sweeps out the surface of a cone. Find the period  of the pendulum bob. College Physics----by Dr.H.Huang, Department of Applied Physics

  31. Force and Motion II  F Example: A car of mass m=1600 kg traveling at a constant speed v=20 m/s along a flat, circular road of radius R=190 m. What is the minimum value of k between the tires of the car and the road that will prevent the car from slipping? Example: Suppose that the above car now moves around a banked circle. What bank angle  makes reliance on friction unnecessary? Concepts: In the right figure, if the angle  of force F on the stationary box is increased, do the following quantities increase, decrease or remain the same: (a) Fx; (b) fs; (c) N; (d) fs,max? If the force F angled upward, how would the above quantities change? College Physics----by Dr.H.Huang, Department of Applied Physics

  32. Force and Motion II 30 B A 4.0 kg 5.0 kg F r m M • Homework: • Block B in the figure weighs 711 N. The coefficient of static friction between block and horizontal surface is 0.25. Find the maximum weight of block A for which the system will be stationary. • A 4.0 kg block is put on top of a 5.0 kg block. To cause the top block to slip on the bottom one, while the bottom one is held fixed, a horizontal force of at least 12 N must be applied to the top block. The assembly of blocks is now placed on a horizontal, frictionless table. Find (a) the magnitude of the maximum horizontal force F that can be applied to the lower block so that the blocks will move together and (b) the resulting acceleration of the blocks. • A mass m on a frictionless table is attached to a hanging mass M by a cord through a hole in a horizontal table. The mass m is in a circular motion with a radius of r. Find the speed with which m must move in order for M to stay at rest. College Physics----by Dr.H.Huang, Department of Applied Physics

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