Force and Newton s Laws Chapter 3

Transcription

1 Force and Newton s Laws Chapter Classical Mechanics Galileo ( ) and Isaac Newton ( ) developed the current approach we use to understand the motion of objects. The minimal number of laws required to describe the motion of all systems of particles is three, and these are emobdied in Newton s three laws of motion as stated in his Principia Mathematica. In this chapter, we will discover how to calculate the acceleration of an object under the influence of external forces. We will also introduce the concept of mass in order to relate the net external forces to the acceleration of the object. 3-2 Newton s First Law Newton s first law simply stated is: Consider a body on which no net force acts. If the body is at rest, it will remain at rest. If the body is moving with constant velocity, it will continue to do so. No external force is required to keep a body in motion. The reason that everyday objects come to rest is due to some net external force such as air resistance, friction, etc. Inertial Reference Frames Each observer such as (1) a person standing on the ground, (2) a person in a car, (3) a person in a plane, etc. each define a reference frame. Each reference frame requires a coordinate system and a set of clocks, which enable everyone in the coordinate system to agree on the time and place for an event. By measuring the space-time coordinates (x, y, z, and t) of multiple events, everyone in a particular reference frame can agree on the location (x), time (t), average velocity (x 2 x 1 )/(t 2 t 1 ), and average acceleration (v 2 v 1 )/(t 2 t 1 ) of an object, that is, the motion of an object. There is a special reference frame called the inertial reference frame. In general, the acceleration of a body depends on the reference frame in which it is measured. However, the laws of classical mechanics are valid only in a certain set of reference 1

2 frames namely, those in which all observers would measure the same acceleration for a moving body. Definition of an inertial frame If the net force acting on a body is zero, then it is possible to find a set of reference frames (in fact, an infinite number of them) in which that body has no acceleration. Here is a subtle but very important point. While observers in all inertial frames will not necessarily agree on the velocity of an object, they will agree on the acceleration of an object. If you really understand this, then you re beginning to understand the concept of an inertial frame. The tendency of a body to remain at rest or in uniform linear motion is called inertia and Newton s first law is often called the law of inertia. 3-3 Force According to Newton s first law, the absence of force leads to the absence of acceleration. There are different kinds of forces. There are push-pull forces due to physical contact with an object. Other contact forces include air resistance and friction. There is also tension, where a force is communicated through a massless rope or string (idealized situtation). There are action-at-a-distance forces such gravity and electromagnetism, where there doesn t appear to be physical contact between the objects. The study of the nature of forces and their influence on systems of particles, is the cornerstone of physics. Today, we know of four forces in nautre: 1. gravity 2. electromagnetism 3. weak radioactive decays 4. strong the forces between quarks which make up protons and neutrons 2

3 Any force tension, air resistance, friction, gravitation, etc., can be described in terms of these four fundamental forces. 3-4 Mass From experiments, we apply a fixed force to three objects (m, 2m, and 3m), and notice that the accelerations of these objects differ according to their respective masses. a 1 = a o a 2 = 1 2 a o a 3 = 1 3 a o We can quickly see that the acceleration is inversely proportional to the mass. a 1 = F m = a o a 2 = F 2m = 1 2 a o a 3 = F 3m = 1 3 a o 3-5 Newton s Second Law From the previous section, we observed that for a constant force, the acceleration was inversely proportional to the mass, namely, a 1/m. In general, we experimentally observe that the vector sum of the external forces is equal to mass times the acceleration vector. F ext = m a Newton s 2 nd law (1) Newton s first law is embodied in Newton s second law. If the sum of the external forces is zero, then the acceleration vector is 0. Equation 1 is a compact way of writing three scalar equations: 3

4 Fx = ma x (2) Fy = ma y (3) Fz = ma z (4) and these are the equations we normally use to solve problems. These equations can be applied to the same system simultaneously to find a x, a y, and a z. Once we know the accelerations, for example a x, we can use the equations of constant acceleration to solve for other kinematical quantities: x = 1 2 (v ox + v x )t (5) v x = v ox + a x t (6) x = v ox t a xt 2 (7) v 2 x = v 2 ox + 2a x x (8) There are some simple rules to follow when applying Newton s laws to a problem. 1. Draw a picture. 2. Identify the system of interest. 3. Draw all the external forces acting on the system. 4. Pick a coordinate system, preferably one where one of the axes (x or y) lines up with the expected acceleration if there is one). 5. Resolve all the forces in the x and y directions. 6. Apply Newton s 1 st to find an unknown quantity in the problem, or apply Newton s 2 nd law to find a x or a y. Exercise 3: An electron travels in a straight line from the cathode of a vacuum tube to its anode, which is 1.5 cm away. It starts with zero speed and reaches the anode with a speed of m/s. Assume constant acceleration and compute the force on the electron. This force is electrical in origin. The electron s mass is kg. 4

5 Exercise 10: (a) Neglecting gravitational forces, what force would be required to accelerate a 1200-metric-ton spaceship from rest to one-tenth the speed of light in 3 days? In 2 months? (One metric ton = 1,000 kg.) (b) Assuming that the engines are shut down when this speed is reached, what would be the time required to complete a 5-light-month journey for each of these two cases? (Use 1 month = 30 days.) 3-6 Newton s Third Law Newton s 3 rd law simply states: When one body exerts a force on another, the second exerts a force on the first. These two forces are always equal in magnitude and opposite in direction. F AB = F BA (9) The force on A due to B is equal and opposite to the force on B due to A. Examples: 1. Book on a table 2. Rocket thrust 3. Moon orbiting the earth N.B. It takes two to have a force. 3-7 Weight and Mass Near the surface of the earth, we assume that the acceleration due to gravity g is close to 9.80 m/s 2. The weight of an object near the surface of the earth is due to the mutual attraction (i.e., gravitational force) between the object and the earth (Newton s 3 rd law). The mass of an object (or its inertia) is a measure of its resistance to acceleration. We measure the mass in kilograms. 5

6 The weight of an object near the surface of the earth is simply written as: W = mg Weight near the surface of the earth (10) Question: Question: Does a satellite orbiting the earth have weight? An astronaut is apparently weightless in the International Space Station. Does the earth still exert an attractive force on the astronaut? Explain. Exercise 14: A space traveler whose mass is 75.0 kg leaves Earth. Compute his weight (a) on Earth, (b) on Mars, where g = 3.72 m/s 2, and (c) in interplanetary space. (d) What is his mass at each of these locations? 3-8 Applications of Newton s Laws in One Dimension Exercise 23: A man of mass 83 kg (weight 180 lb) jumps down to a concrete patio from a window ledge only 0.48 m (=1.6 ft) above the ground. He neglects to bend his knees on landing, so that his motion is arrested in a distance of about 2.2 cm (=0.87 in). (a) What is the average acceleration of the man from the time his feet first touch the patio to the time he is brought fully to rest? (b) With what average force does this jump jar his bone structure? Problem 7: A child s toy consists of three cars that are pulled in tandem on small frictionless rollers (Fig. 3-33). The cars have masses m 1 = 3.1 kg, m 2 = 2.4 kg, and m 3 = 1.2 kg. If they are pulled to the right with a horizontal force P = 6.5 N, find (a) the acceleration of the system, (b) the force exerted by the second card on the third car, and (c) the force exerted by the first car on the second car. 6