Chapter 5. Force and Motion

Transcription

1 Chapter 5 Force and Motion

2 What do we learn in this chapter? Newtonian Mechanics Newton s three laws Applications Normal force Gravitational force Friction Tension Bouyancy Some basic concept Force (Interaction) Mass Their relation with motion

3 5.2 Newtonian Mechanics Newtonian Mechanics: Knowledge about the relation between force and motion of an object with mass

4 5.2 Newtonian Mechanics: Historic remarks Aristotle: All objects have a natural place in the universe. Ø Heavy objects wanted to be at rest on the Earth. Ø Light objects wanted to be at rest in the sky. Ø Stars wanted to remain in the heavens. Ø A body was in its natural state when it was at rest. Ø Moving body (including in a straight line at a constant speed) needs to be propelled by force. Galileo Galilei: Motion and inertia ü A force is necessary to change the velocity of a body. ü No force is needed to maintain its velocity. ü This insight was refined by Newton, who made it into his first law, the "law of inertia. Aristotle (384 BC-322 BC). Greek philosopher and polymath. Galileo Galilei: Italian physicist, mathematician, astronomer, and philosopher

5 5.2 Newtonian Mechanics: Historic remarks The three laws of motion were first compiled by Sir Isaac Newton in his book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy") in Newton used these laws to explain and Newton's own physical first investigate the motion of many edition of his objects and systems, suchcopy as planetary Principia. motion. Quite successful! Kepler s three laws of planetary motion. Isaac Newton ( )

6 English poet Alexander Pope s famous epitaph: Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light. Newton himself: If I have seen further it is by standing on the shoulders of giants. Newton another memoir quite famous all over the world: I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

7 Importance and range of validity Newton's laws were extensively verified by experiment and observation, with excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena from what going on the Earth to what going on in the Heaven. These three laws hold to a good approximation for macroscopic objects under almost everyday conditions. However...

8 Importance and range of validity Newton's laws (with universal gravitation and classical electrodynamics) are inappropriate at very small scales very high speeds very strong gravitational fields. Another big difference: In quantum mechanics concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state. It is a completely different picture.

9 Mechanics of All

10 5.3 Newton s First Law Newton s First Law: If no net force acts on a body, the body s velocity cannot change; that is, the body cannot accelerate Scenario-1: If the body is at rest, it stays at rest. Scenario-2: If it is moving, it continues to move with the same velocity (same magnitude and same direction).

11 5.4 Force: What is it? Ø A force is measured by the acceleration it produces (one definition) Ø Forces have both magnitudes and directions. Ø When two or more forces act on a body, we can find their net, or resultant force, by adding the individual forces vectorially. Ø The net force acting on a body is represented with the vector symbol F! net

12 5.4 Force: Simplest operation Which of the following figures correctly show the vector!! addition of forces F 1 and F2 to yield! the third vector, which is meant to represent their net force Fnet?

13 5.4 Force The force that is exerted on a standard mass of (1 kg) to produce an acceleration of (1 m/s 2 )has a magnitude of (1 Newton) (abbreviated as N) Newton: SI, derived units.

14 5.3 Newton s First Law: Simple but with critical rich information. Newton placed the first law of motion to establish frames of reference for which the other laws are applicable. It postulates the existence of at least one frame of reference (to be called a Newtonian or inertial reference frame), relative to which the motion of a particle not subject to forces is a straight line at a constant speed. The first law can also be restated as: In every material universe, the motion of a particle in a preferential reference frame is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in the frame. Newton's laws are valid only in an inertial reference frame. Any reference frame that is in uniform motion with respect to an inertial frame is also an inertial frame.

15 Inertial Reference Frames: Where are they? An inertial reference frame is one in which Newton s laws hold. (a) The path of a puck sliding from the north pole as seen from a stationary point in space. Earth rotates to the east. (b) The path of the puck as seen from the ground. If the puck is sent sliding along a long ice strip extending from the north pole, and if it is viewed from a point on the Earth s surface, the puck s path is not a simple straight line. The apparent deflection is not caused by a force, but by the fact that we see the puck from a rotating frame. In this situation, the ground is a noninertial frame.

16 Foucault pendulum A simple device conceived as an experiment to demonstrate the rotation of the Earth. There is one in Fermilab:

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18 5.5: Mass: One definition Mass is an intrinsic characteristic of a body The mass of a body is the characteristic that relates a force on the body to the resulting acceleration. Empirical observations: The ratio of the masses of two bodies is equal to the inverse of the ratio of their accelerations when the same force is applied to both.

19 5.5: Toward Newton s 2 nd Law m x a x = m 0 a 0 m x a x = m 0 a 0 = m 1 a 1 = m 2 a 2 = m x a x = m 0 a 0 = m 1 a 1 = m 2 a 2 = =Constant What is this constant? Cannot be the shape of the object. Cannot be the material of the object. Cannot be the smell of the object. It has to be the cause of the acceleration, the force!

20 5.6: Newton s Second Law: In component notation, Newton s Second Law: The net force on a body is equal to the product of the body s mass and its acceleration. F net = m a The acceleration component along a given axis is caused only by the sum of the force components along that same axis, and not by force components along any other axis.

21 5.6: Newton s second law Again, the SI unit of force is Newton (N): 1 N =(1 kg)(1 m/s 2 ) = 1 kg m/s 2.

22 5.6: Newton s second law: Free body diagram ü A coordinate system is usually included, and the acceleration of the body is sometimes shown with a vector arrow (labeled as an acceleration). One of the most important steps in Newtonian Mechanics ü In a free-body diagram, the only body shown is the one for which we are summing forces. ü Each force on the body is drawn as a vector arrow with its tail on the body. The figure here shows two horizontal forces acting on a block on a frictionless floor.

23 Example

24 Example

25 Example

26 Example

27 Example, 2-D forces:

28 Example, 2-D forces:

29 Example, 2-D forces:

30 5.7: Some particular forces The weight, W, of a body is equal Gravitational Force: to the magnitude Fg of the gravitational force on the A gravitational force on a body is body. a certain type of pull that is directed toward a second body. W = mg (=F) Suppose a body of mass m is in free fall with the free-fall acceleration of magnitude g. The force that the body feels as a result is: Fg = mg.

31 5.7: Some particular forces Normal Force: When a body presses against a surface, the surface (even a seemingly rigid one) deforms and pushes on the body with a normal force, F N. Direction: F N is always perpendicular to the surface. In the figure, forces F g and F N and are the only two forces on the block and they are both vertical. Thus, for the block we can write Newton s second law for a positiveupward y axis, (F net, y = ma y ), as: The free-body diagram for the block.

32 5.7: Some particular forces Friction If we either slide or attempt to slide a body over a surface, the motion is resisted by a bonding between the body and the surface. The resistance is considered to be single force called the frictional force, f. Direction: This force is directed along the surface, opposite the direction of the intended motion.

33 5.7: Some particular forces Friction 1. Static friction: between non-moving surfaces. 2. Kinetic friction: between moving surfaces. Force The (static) friction increases as the applied force increases until the block moves. After the block moves, it experiences kinetic friction, which is less than the maximum static friction.

34 5.7: Some particular forces Tension When a cord is attached to a body and pulled taut, the cord pulls on the body with a force T. Direction: T directed away from the body and along the cord. (a) The cord, pulled taut, is under tension. If its mass is negligible, the cord pulls on the body and the hand with force T, even if the cord runs around a massless. (b), (c) Frictionless pulley.

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36 5.7: Some particular forces Buoyancy - The magnitude of buoyancy is proportional to the difference in the pressure between the top and the bottom of the column. - It is equivalent to the weight of the FLUID that would otherwise occupy the column, i.e. the displaced fluid. (The Archimedes' principle) Let s looks at an example that requires some thoughts on the system and force analysis T 0 N 0 W 1 W 2

37 5.7: Some particular forces Buoyancy Two systems & The forces exerting on them - Option-1 Normal force Tension T B N Buoyancy Weight of water W 1 Weight of the object W 2 T+B-W 1 =0 N-B-W 2 =0

38 5.7: Some particular forces Buoyancy Two systems & The forces exerting on them - Option-2 T+N-W 1 -W 2 =0 T = W 1 -B Tension Weight of water Weight of the object T Normal force N T W W 1 2

39 5.7: Some particular forces Buoyancy T+B-W 1 =0 à T=W 1 -B N-B-W 2 =0 à N=W 2 +B T 0 =W 1, N 0 =W 2 T 0 -T=W 1 -(W 1 -B) = B >0 N 0 N=W 2 - (W 2 +B) = -B <0 T+N-W 1 -W 2 = 0 à T = W 1 +W 2 -N T = W 1 -B T 0 =W 1, N 0 =W 2 W 1 -B = W 1 +W 2 -N N = B+W 2 T 0 -T=W 1 -(W 1 +W 2 -N ) = N-W 2 =B N 0 N=W 2 - (B+W 2 ) = -B

40 5.8: Newton s Third Law The minus sign means that these two forces are in opposite directions

41 5.8: Newton s Third Law Varying Anywhere Just do not try this Newton s Third Law: When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction. The forces between two interacting bodies are called a third-law force pair.

42 Think Deep Newton used the third law to derive the law of conservation of momentum p=mv (p and v are vectors); However from a deeper perspective, conservation of momentum is the more fundamental idea It connects with Galilean invariance: the laws of motion are the same in all inertial frames. Conservation of momentum holds in cases where Newton's third law appears to fail, for instance in quantum mechanics.

43 5.9: Applying Newton s Laws Sample problem Figure 5-13 shows a block S (the sliding block) with mass M =3.3 kg. The block is free to move along a horizontal frictionless surface and connected, by a cord that wraps over a frictionless pulley, to a second block H (the hanging block), with mass m 2.1 kg. The cord and pulley have negligible masses compared to the blocks (they are massless ). The hanging block H falls as the sliding block S accelerates to the right. Find: (a) the acceleration of block S (b) the acceleration of block H (c) the tension in the cord

44 5.9: Applying Newton s Laws Sample problem, cont. Key Ideas: (1) Forces, masses, and accelerations are involved,! and they should! suggest Newton s second law of motion:! F = ma! (2) The expression F = ma is a vector equation, so we can write it as three component equations. (3) Identify the forces acting on each of the bodies and draw free body diagrams like this:

45 5.9: Applying Newton s Laws Sample problem, cont. (1) From the free body diagrams, write Newton s Second Law in the unit vector form, (2) Assuming a direction of acceleration for the system. (3) Identify the net forces for the sliding and the hanging blocks:

46 5.9: Applying Newton s Laws Sample problem, cont. For the sliding block, S, which does not accelerate vertically. Also, for S, in the x direction, there is only one force component, which is T. Combine (1) and (2), we have: Ma-mg = -ma Ma+ma = mg (M+m)a = mg Divide both sides by (M+m), (1) For the hanging block, because the acceleration is along the y axis, Insert this a into (1) to get: Since F gh =mg, so this can be written as: (2)

47 February 25 rd (fraction operation) a b ± c d log a b a b c d d + a b + c e f g + h a b c d

48 The greatest common factor (GCF) 4 4

49 The least common multiple of the denominators (the lowest common denominator)

50 5.9: Applying Newton s Laws Sample problem In the figure to the right, a cord pulls on a box of sea biscuits up along a frictionless plane inclined at θ= 30.The box has mass m =5.00 kg, and the force from the cord has magnitude T =25.0 N. Q: What is the box s acceleration component a along the inclined plane?

51 5.9: Applying Newton s Laws Sample problem (1) Choose a coordinate system (2) Draw a free-body diagram (3) The positive direction of the x axis is up the plane. (4) Force analysis: Force from the cord & gravitational force. (5) Projection to x- and y-axis: using θ

52 5.9: Applying Newton s Laws Sample problem Using Newton s Second Law, we have : That is, a = (T - mg sinθ)/m We were given θ = 30, the box mass m =5.00 kg, and the force from the cord has magnitude T =25.0 N. which gives: a = ( )/5.0 = m/s 2. The positive result indicates that the box accelerates up the plane.

53 5.9: Applying Newton s Laws Sample problem We don t specify the motion: It can be with any velocity or acceleration: v(t), a(t)!

54 5.9: Applying Newton s Laws Sample problem The free-body diagram Reference system Attention: We can use Newton s 2 nd Law only in an inertial frame. If the cab accelerates, then it is not an inertial frame.

55 5.9: Applying Newton s Laws Sample problem Let s choose the ground to be our inertial frame and make any measure of the passenger s acceleration relative to it. Relative to the ground!

56 5.8: Applying Newton s Laws Sample problem, cont.

57 5.8: Applying Newton s Laws Sample problem, cont. Pay attention to +/- sign for the accelerations!

58 5.8: Applying Newton s Laws Sample problem, cont. 1. F g of the gravitational force does not depend on the motion. So, from Question (b) Fg = 708 N in the cab frame. 2. From Question (c), the normal force is upward, Fn= 939 N as one can read in the cab frame. 3. So, the net force on the passenger in the cab frame is: F net = F N -F g = = 231 N.

59 5.8: Applying Newton s Laws Sample problem, cont. But the passenger s acceleration a p,cab relative to the frame of the cab is ZERO! F net ma p,cab Newton s second law is valid only in inertial frames!

60 Where we are now