Forces - And Newton’s laws of motion

Causes of Motion & the concept of forceAristotle, (384 - 322), in an attempt to explain the causes of motion suggested that “motion ismaintained by forces”. Galileo Galilei about 2000 years later, through his thought experimentwhile attending mass at a church service, realized the importance of distinguishing betweenhorizontal motion and vertical motion in a gravitational field. This was after he observed apendulum tending to return to its original position at each swing. He reasoned (in his mind) thatif the path of the pendulum was not curved, the ball would continue indefinitely looking for theupward curve. Newton extended Galileo’s work and made it more scientific by framing threerules (the Newton’s 3 laws of motion) that govern the motion of objects. Although Newton’slaws of motion are very nearly correct under all common circumstances, they break down undercertain conditions: e.g as velocity approaches the speed of light.

In this section we introduce the concept of force. We discuss Newton’s laws, which describethe way a body responds to a net force. We discuss frictional forces and the way they can bemathematically represented. We study several applications of Newton’s laws.

In classical mechanics, Force is the cause of motion . Classical mechanics deals with systemsof size� 10−10 m (atomic dimensions) and velocity� 3.0 x 108 m/s (the speed of light).

Note:

• Force is a vector.

• There are two kinds of forces:

– Contact Forces - involve physical contact between objects. Examples: the forceinvolved in kicking a ball, pulling a wagon, compressing a spring, etc.

– Field forces - don’t involve physical contact between objects. Examples: thegravitational force and the electromagnetic force.

Newton’s First LawBoth Aristotle and Galileo realized that an understanding of forces and their interactions waskey to the understanding of motion. Here on Earth, if friction forces were absent, pushing anobject would cause it to move indefinitely This is the basis of Newtons first low of motion.

Newton’s First Law States: an object continues in its state of rest or uniform motion in a straightline if there is no net external force between the object and the environment. In other words, anobject at rest stays at rest, an object in motion stays in motion with a constant velocity unlessmade to change by the total force acting on it. Newton’s first law of motion defines for us whata force is, or rather what it does - a force is something that causes acceleration.

In mathematical form we can write:

Σ~F = 0⇒~a = 0. (1)

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This means if a body has a number of forces F1, F2, ...Fn, the sum of the forces F1 to Fn is equalto zero. This can be calculated separately for all the horizontal and vertical forces. Effectsof the horizontally acting forces are completely independent of those for all vertically actingforces.

The first step in solving problems involving forces and motion of objects, is to identify allthe forces acting on the object and then draw a force diagram (called the free body diagram)showing all the forces. Since force is a vector quantity with both magnitude and direction, weuse arrows to represent this. For solid objects we generally draw a force as acting through asingle point e.g, weight is usually represented as acting through the center of gravity. For ameter rule balanced at its midpoint, we would draw a free body diagram with an arrow pointingup and another pointing down. We think of the weight as acting through the midpoint. Similardefinition can be said of the center of mass as the point at which all the object’s mass maybeconsidered to be concentrated.

Example: Consider a man pushing a wheelbarrow, draw a free body diagram. I will show thison the black board!.

Note: one of the forces is the drag force (friction) that is due to the interactions of thewheelbarrow with the environment.

Friction

Friction originates from forces between atoms and molecules when surfaces are in contact. Forexample, friction occurs when a body moves on a rough surface or through a fluid medium(water, air, etc.). There are two types of friction:

1. The Static force of friction ( fs) is the force of friction between two objects when thereis no motion.

2. The Kinetic force of friction ( fk) is the force of friction between two objects when thereis motion.

Consider a block on a rough surface. Apply an external force Fext to the block.

∗ if Fext < fs(max) the block won’t move

∗ as Fext increases, fs will increase until it reaches its maximum value. When Fext = fs(max)the block will start to move (this is called the point of slipping).

∗ Once the block starts to move, the force of friction is given by fk .

Experimental facts about friction

1. fs ≤ µsFN where µs is the coefficient of static friction and FN is the magnitude of thenormal force. Equality holds when the object is on the point of slipping: fs(max) = µsFN .

2. fk = µkFN where µk is the coefficient of kinetic friction and is approximately constantfor any given pair of materials.

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3. Values of µs and µk depend on the nature of the surfaces that are in contact. Usuallyµk < µs . Examples: rubber on concrete µs = 1.0, µk = 0.8; waxed wood on wet snowµs = 0.14, µk = 0.10.

4. The direction of the force of friction is opposite to the direction the object wants to move.

5. µk and µs are nearly independent of the area of contact between the two surfaces.

6. µk is nearly independent of the velocity of the object under consideration.

Newton’s Second LawHaving established a connection between force and acceleration, which is qualitative, Newtonwent further to find a quantitative connection between the two. He reasoned that if Σ~F 6= 0 (i.e.there is a net external force acting on an object). Newton determined a relationship betweenF , m and a and found that (i) F ∝ a i.e as F increases by a factor x, so does acceleration, (ii)a ∝ 1/m i.e. as the mass is increased by some factor, the acceleration produced decreases bythe same factor.

a ∝ Fa ∝ 1/m

}⇒ a ∝ F/m or F ∝ ma

another way to express this is:F = kma

where k is a constantIf we define the unit of newton in such a way that one unit of force accelerates a mass of onekg at a rate of one meter per second per second, then the constant in the equation must have thevalue of one, so that

F = ma

This is Newton’s second law of motion. More generally is it expressed as:

Σ~F = m~a (2)

By SI units for our measurements of mass and acceleration, units of force are kg ms−2. TheSI unit of force is the Newton (N) defined as the force that produces an acceleration of 1 m/s2

when acting on a 1 kg mass. In the cgs system: 1 dyne = 1 g cm/s2 = 10−5N. In the Britishengineering system: 1 pound (lb) = 4.448 N.

Note:

• ~a is inversely proportional to m . This means that, for the same force, a smaller mass willhave a larger acceleration.

• Newton’s second law is a vector equation which contains three scalar equations (in threedimensions): ΣFx = max, ΣFy = may, ΣFz = maz.

• The first law is a special case of the second law.

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Worked Example: An aeroplane lands with a velocity of 50 m/s, the reverse thrusts from theengines are used to slow it to a velocity of 25 m/s in a distance of 240 m.If the mass of theaeroplane is 3x104 kg, what is the size of the reverse thrust supplied by the engines? We willdo this on the black board!

Newton’s second law of motion is useful in considering what we mean by the term force butcan do more.

The tendency of an object to resist any attempt to change its state of motion is called Inertia.The definition of mass is very difficult. The common idea that mass is the measure of theamount of matter in an object - is not false, but it is not the whole truth either. The satisfactorydefinition of mass uses the idea of inertia. e.g Mass can be stated as the force required per unitof acceleration produced and is a measure of inertia. Mass is a scalar quantity and has SI unitsof kilograms (kg). Example: If a bowling ball and a golf ball are hit with a bat, the bowlingball would be much harder to get moving since it has greater mass and thus greater inertia. Weoften use the term weight even when we mean mass. An objects weight is a force acting onit. We have already noted how Newton set up a theory of how masses attract each other- thegravitational theory. All masses have a gravitational field around them. Field is the physicsterm used to explain action at a distance. If another mass is brought into the field of one, itexperiences a force which pulls it towards the mass. The size of the force varies with strengthof the field. Also the size varies with the position of the mass in the field.

We define Weight (~w) as the force exerted on an object by a gravitational field. From Newton’ssecond law,

w = mg. (3)

Note:

• Weight is a vector with direction towards the earth’s center, or perpendicular to the earth’ssurface.

• The weight of an object is different on the earth and on the moon since the strength ofthe gravitational field is different (gearth 6= gmoon).

• The value of g varies with distance from the center of the earth (more on this in unit 4).As a consequence:

– Since the earth isn’t a perfect sphere, the weight of an object varies slightly fromplace to place on the earth’s surface.

– The weight of an object varies slightly with altitude above the earth’s surface.

• In comparison, mass is a scalar with a value independent of location. Notice howeverthat, in the approximation that g is constant, mass is proportional to the magnitude of theweight and the two quantities can be used interchangeably. This is called the equivalenceprinciple.

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Newton’s Third LawIdea: Forces in nature always exist in pairs. Newton’s third law states: For every action, thereis an equal and opposite reaction. When two bodies interact:

~F2 on 1 =−~F1 on 2. (4)

Where ~F2 on 1 is the force exerted on body 1 by body 2 and ~F1 on 2 is the force exerted on body2 by body 1.

For Example: When an object falls towards the earth, the earth exerts a force on it that causesit to accelerate towards the earth. According to Newton’s third law, the object exerts a forceon the earth as well, and the earth accelerates towards the object. Why don’t we feel the earthaccelerate?

Solution:2nd Law → meae = ~Fobj on earth

3rd Law → ~Fobj on earth = - ~Fearth on obj ≡ - ~w⇒~ae = - ~w/me

⇒ |~ae| =(

mobj

me

)g� g.

Conclusion: the acceleration of the earth is too small to detect because the mass of the earth ismuch larger than the mass of the object.

Applications of Newton’s Laws

Assumptions:

- We treat objects as point particles (no rotational motion).

- We neglect masses of ropes and springs. One consequence of this assumption is that theforce exerted along a rope is the same at all points in the rope.

Note: In problems with several bodies, apply Newton’s 2nd law to one body at a time.

Problem Solving Strategy

• Draw a picture of the situation and a force diagram of all the forces for each body (a freebody diagram).

– In the force diagram for each object, include only the forces acting on that object.

– The force exerted by a rope is called the tension and usually denoted ~T .

– The contact force exerted by a surface is called the normal force (FN) and alwaysacts perpendicular to the surface.

• Set up a coordinate system and apply Newton’s second law:

ΣFx = max, ΣFy = may

• If necessary, use the kinematic equations of motion to solve for the desired quantities.

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