Chapter 4 Dynamics: Newton s Laws of Motion

Transcription

1 Chapter 4 Dynamics: Newton s Laws of Motion

2 Force Newton s First Law of Motion Mass Newton s Second Law of Motion Newton s Third Law of Motion Weight the Force of Gravity; and the Normal Force Applications Involving Friction, Inclines

3 Recalling Last Lectures

4 Relative Velocity Consider a person walking on the boat with velocity relative to the boat. His velocity relative to the shore will be given by: using: Then (3.19)

5 Relative Velocity Notes: In general, if is the velocity of A relative to B, then the velocity of B relative to A, will be in the same line as but in opposite direction: (3.20) If is the velocity of C relative to A, then, the velocity of C relative to B, will be given by: (3.21)

6 Relative Velocity Problem 3.37 (textbook) : Huck Finn walks at a speed of 0.60 m/s across his raft (that is, he walks perpendicular to the raft s motion relative to the shore). The raft is travelling down the Mississippi River at a speed of 1.70 m/s relative to the river bank (Fig. 3 38). What is Huck s velocity (speed and direction) relative to the river bank? Solution developed on the blackboard

7 H uck H uck raft rel. rel. bank rel. raft bank ( ) Relative Velocity 37. Call the direction of the flow of the river the x direction, and the direction of Huck walking relative to the raft the y direction. r r r v = v + v = 0, 0.60 m s , 0 m s = 1.70, 0.60 m s ( ) ( ) v r Huck rel. raft θ v r Huck rel. bank v r r a f t r e l. b a n k ( c u r r e n t) 2 2 Magnitude: v = = 1.80 m s Huck rel. bank o Direction: θ = tan = 19 relative to river 1.70

8 Relative Velocity Problem 5.50 (textbook) : An unmarked police car, traveling a constant 95 Km/h is passed by a speeder traveling 145 Km/h. Precisely 1.00 s after the speeder passes, the policeman steps on the accelerator. If the police car s acceleration is 2.00 m/s 2. How much time elapses after the police car is passed until it overtakes the speeder (assumed moving at constant speed)? Solution developed on the blackboard

9 Relative Velocity 52. Take the origin to be the location at which the speeder passes the police car, in the reference frame of the unaccelerated police car. The speeder is traveling at 1 m s 145 km h = m s 3.6 km h relative to the ground, and the policeman is traveling at 1m s 95 km h m s 3.6 km h = relative to the ground. Relative to the unaccelerated police car, the speeder is traveling at m s=vs and the police car is not moving.

10 Relative Velocity Do all of the calculations in the frame of reference of the unaccelerated police car. The position of the speeder in the chosen reference frame is given by x = vt s s The position of the policeman in the chosen reference frame is given by ( ) 2 1 x = a t 1, t > 1 p 2 p (Note that he policeman does not move relative to the reference frame until he starts accelerating his car). The police car overtakes the speeder when these two distances are the same.; i.e.: x = x s p

11 Relative Velocity Then,. ( ) ( ) ( )( ) s p s 2 p 2 x = x vt= a t m s t= 2m s t 2t+ 1 = t 2t+ 1 2 t t t ± = 0 = = s, s 2 Since the police car doesn t accelerate until t = 1.00 s, the correct answer is t = 15.8 s

12 Force We have introduced the concept of motion and obtained a set of equations that describe the motion of an object given some initial conditions. We have also introduced the concept of vector and have shown that velocity and acceleration are vectors (have magnitude and direction). But.. we have not discussed what set an object in motion, or change its state of motion. This is what we will be learning in this chapter.

13 Force The area that studies the connection between motion and its cause is called dynamics. The cause of the motion is force. But what force is?? Force is the interaction between two objects. For example: When you push a grocery cart which is initially at rest, you are exerting a force on it. The force you apply on the cart changes its state of motion

14 Force Force is the interaction between two objects. Side note: You may ask me: but how do two objects interact? For example: You have seen that magnets attract some materials such as a piece of iron. But the magnet does not need to touch the iron to attract it. So, how do these two objects interact? The answer to this question requires advanced physics. However, let s say that there are some special particles that operates as messengers between the two objects. These messengers are the carries of the force between these objects and therefore responsible for their interaction. There are four fundamental kinds of forces: Gravitational Responsible for having the Moon orbiting the Earth. Electromagnetic Responsible for the attraction between the magnet and iron. Weak Main responsible for the processes that make stars to shine Strong Responsible for the nuclear interaction

15 Force Note that the direction of the grocery cart depends on the direction of the force you apply on it: You can push it on a straight line, or to the left, or to the right, etc.. The strength you push the cart will determine how fast the cart goes or how fast you will change its direction. So, it is clear that you should assign not only a direction, but also a magnitude to the force you apply on the cart. In other words, Force is a vector. And we will represent it using the standard vector representation: with the arrow representing the direction a force is applied on an object.

16 Force Forces being vectors can be added using the methods discussed in chapter 3. Example: In the figure you and your mother are pushing the same grocery cart with forces represented by and, respectively. The total force applied on the cart,, will be the vector addition of both forces: You can then use, for example, the method of components for vector addition after having selected an appropriate reference frame and coordinate system.

17 Force In general we can state that if a system of two or more forces are applied on a object, the net force on this object will be given by: (4.1) Where the subscript i in denotes a particular force acting on the object where i can be any number between 1 and N, where N is the total number of forces being applied on the object. The Greek symbol Σ denotes a summation over all possible forces from 1 to N. Example: Determine the net force acting on an object subject five different forces: ; ; ; ; and y x

18 Relationship between force and motion Newton s Laws of Motion We have seen that given a reference frame, an object can be found in any of the following states of motion: at rest moving with constant velocity accelerated But an applied force on an object can change its state of motion: The object can starting moving from rest Can change velocity from a state of constant velocity Can change direction (and therefore the vector velocity), and may be the magnitude of its velocity. It is clear that if NO forces are applied on an object, it will tend to remain on its original state of motion.

19 Newton s First Law of Motion Newton s Laws of Motion We can generalize the previous statement to say that: If the NET FORCE of all forces applied on a object is ZERO, then the object will tend to remain in its state of rest, or constant velocity. This follows from the fact that a non-zero net force will change the state of motion of any object, which implies in changing its velocity either in direction or magnitude or both. But, by definition, change in velocity implies in acceleration. So, if no acceleration is present, the object will remain in its original state of motion. We can then state the Newton s first law of motion as: Every Object continues in its state of rest, or uniform velocity in a straight line, as long as no net force acts on it

20 Newton s Laws of Motion Example: An object initially at rest is suddenly subjected to two forces, and, of same magnitude and opposite direction. What is the net force acting on this object? Will it move? Answer: Let be the net force, then: But, then Since the net force is zero, the object will NOT move.

21 Newton s Laws of Motion Note 1: You can always represent the forces on the previous diagram as instead of This can be generalized to any system of forces.

22 Newton s Laws of Motion Note 2: Newton s first law ONLY holds in inertial frames. Inertial frames are frames FIXED on a system which is either at rest or moving with constant velocity (no changes in direction and magnitude). Or, in other words, inertial frames are those where the Newton s first law is valid. Frames other than the inertial frames are called non-inertial frames (for obvious reasons). An example of non-inertial frame is that of a frame fixed on an accelerated system (example: a car). Example: if you are in a car and fix the reference frame on it. You then decide to accelerate it to increase its speed from a state of constant velocity. You will then notice that anything free on the passenger sit will tend to move, though no net force has been applied on them. If you crash your car on a wall, you will tend to move forward even if no force has been applied on you.

23 Newton s Laws of Motion Example: Assume you are in a truck and have fixed the reference frame on it. You then decide to accelerate the truck to increase its speed from a state of constant velocity. You will then notice that anything free on the passenger sit on anywhere on the truck will tend to move, though no net force has been applied on them. If you crash your truck on a wall, you will tend to move forward even if no force has been applied on you. Now, for a person at rest relative to the ground, the box will see to continue moving with constant velocity while the car is being accelerate (ignore friction effects).

24 Newton s Laws of Motion The Concept of Mass: Mass measures that quantity of matter in a body (or object). It is given in units of Kg. But then, you may ask me: What is matter???? The answer is even more complicate than that to understand how objects interact. We will not discuss about it here. A alternative definition can be given as: Mass is a measure of the inertia of an object. Inertia is the tendency an object has to continue in its state of motion. You may have experienced the fact that it gets harder to move a box of some material as you add more of the same material to the box. So you are adding mass. This means that adding mass in the box makes it more difficult to change its state of inertia. So, mass is related to inertia: The larger is the mass of an object, the bigger is its inertia.

25 Newton s Laws of Motion

26 Newton s Second Law of Motion: Newton s Laws of Motion We have seen that force and acceleration are related In fact, acceleration is the product of an applied force. We have also noticed that force and mass are also related more mass implies that a stronger force is needed to change the state of motion of a body). But how are these quantities related??? From the equation of motion we know that: If you triple (or double, or etc) your acceleration a, you expect to triple the velocity added to your motion, v - v 0. It is also observed that if you triple the applied force on an object, its change in velocity will be triple. It is then clear that force and acceleration are directly proportional. We say: The symbol means proportional to

27 Newton s Second Law of Motion: Newton s Laws of Motion It also has been observed that for the same applied force, the change in velocity will depend on the mass of the object: the greater the mass, the smaller is the change in velocity smaller is the acceleration. This, together with let Newton to state his second law of motion The acceleration of an object is directly proportional to the net force acting on it, and is inversely proportional to its mass. The direction of the acceleration is in the direction of the net force acting on the object

28 Newton s Second Law of Motion: Newton s Laws of Motion Newton s second law can be summarized with the following equation: where I have omitted the subscript i and the limits of summation: 1 and N (they are implicitly assumed to be present). This equation can be re-written to yield the well known relationship between force, mass and acceleration: (4.2) Note that you can also write:, where