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Chapter Four

Newton's Laws

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Newton's Laws In this chapter we will consider Newton's thre

e laws of motion. There is one consistent word in these three la

ws and that is "body" (newtonian body). We will define force through the motion it caus

e on mass.

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Newton's Laws (1) First Law: Every body of matter continue in a s

tate of rest or moves with constant velocity in a straight line unless compelled by a force to change state.

Second Law: When net unbalanced forces act on a body, they will produce a change in the momentum (mv) of that body proportional to the vector sum of the force. The direction of the change in momentum is that of the line of action of the resultant force.

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Third Law: Forces, arising from the interaction of particles, act in such a way that the force exerted by one particle on the second is equal and opposite to the force exerted by the second on the first and both are directed along the line joining the two particles. (Or, action and reaction are equal and opposite).

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Newton's Laws (2) The average force is defined as

Let a = dv/dt, the force is define as

The forms of Newton's law that we will use are

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Mass Let m0 be the standard kilogram. If we

exert a force on the mass with no other forces to interfere, we can measure an acceleration a0. If we apply this same force to a different mass m1, we measure a different acceleration a1. Then

and

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Force has units of mass length/time2 or kilogram-meter per second2 (newton, N).

A force of 1 N is that force which causes a mass of 1 kg to be accelerated at a rate of 1 m/sec2.

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Weight The rate of free fall of all objects in a vacuum a

t a given point on earth is the same. The downward acceleration at sea level is app

roximately the same at all locations, or g= 9.8 m/sec2.

Weight = mg.

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Applications of Newton's Laws--Example 4-2 A child pulls a toy boat through the water at

constant velocity by a string parallel to the surface of the water on which he exerts a force of 1 N. What is the force of resistance of the water to the motion of the boat? See Fig. 4-2.

Sol : Because constant velocity means zero acceleration,

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Applications of Newton's Laws-- Example 4-3 Two ropes attached to a ceiling at the angles s

hown in Fig. 4-3 support a block of weight 50 N. What are the tensions T1 and T2 in the ropes?

Sol : If we examine the newtonian body, we see that it i

s not accelerating in either the x or y directions. We have

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Thus,

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Substituting into second equation

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Applications of Newton's Laws-- Example 4-5 A block of mass 8 kg is released from rest on a

frictionless incline that is at an angle of 37o with the horizontal (Fig. 4-6a). What is its acceleration down the incline?

Sol: See Figure 4-6b.

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From Newton's second law, we have

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Two important points: Because the acceleration is independent

of the mass, all masses starting from rest at the same height on the same plane will have the same acceleration and, therefore, reach the bottom at the same time.

The acceleration is less than the acceleration of gravity because only a component of the force of gravity on the body is directed down the plane.

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Applications of Newton's Laws-- Example 4-6 Masses of 2 kg and 4 kg connected by a

cord are suspended over a frictionless pulley (Fig. 4-7a). What is their acceleration when released?

Sol: Three important facts: 1. Because the pulley is frictionless, the

tension in the rope is the same on both sides.

2. The tensions are not the same as in a static situation.

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3. There are two newtonian bodies and while m1 moves upward with a positive acceleration, m2 moves with an acceleration having the same magnitude but directed downward.

See Figure 4-7b. For body m1 we have

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For body m2 we have

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Friction There is a force equal and opposite to the force that w

e exert that resists the motion of the object. This resistive force is called the force of friction.

There are two types of friction, static and kinetic. The starting friction is called static. The friction of mot

ion is called kinetic. Static friction is larger than kinetic friction. We will onl

y consider kinetic friction. The force of friction is proportional to the normal forc

e (mg = N). See Figure 4-8. The force of friction is f = μN, where μ is called the coef

ficient of friction.

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Example 4-7 A force of 10 N is required to keep a box of mas

s 20 kg moving at a constant velocity across a level floor (Fig. 4-9). What is the coefficient of friction?

Sol: Since ax = 0 and ay = 0, we have

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and

But

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Example 4-8 A block is places on a plane inclines to the hori

zontal at 37o. The coefficient of friction between the plane and the block is μ = 0.4. When the block is released what is its acceleration down the plane? (See Fig. 4-10)

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Sol The forces along the plane are the force of

friction f upward and the component of the force of gravity FD downward. Choose the downward direction as positive and we have

since

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we have

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Homework 2. 4. 7. 9. 11. 14. 16. 19. 21. 24.