Chapter 4 - Newton’s Laws of motion

4.1 force and interactions

• A force is a push or a pull. It is an interaction between two bodies or between a body and its environment.

• Force as a vector quantity

For simplicity sake, all forces (interactions) between objects can be placed into two broad categories

Contact forces Long range force

Contact force• Contact forces are those types of forces which

result when the two interacting objects are perceived to be physically contacting each other.

• Examples of contact forces include • frictional forces tensional forces normal forces,

Action-at-a-distance forces• The types of forces which result even when the

two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite their physical separation.

• Examples of action-at-a-distance forces include – gravitational forces. The sun and planets exert a

gravitational pull on each other despite their large spatial separation. Even when your feet leave the earth and you are no longer in physical contact with the earth, there is a gravitational pull between you and the Earth.

– Electric forces are action-at-a-distance forces. For example, the protons in the nucleus of an atom and the electrons outside the nucleus experience an electrical pull towards each other despite their small spatial separation.

Contact Forces Action-at-a-Distance Forces

Frictional Force Gravitational Force

Tension Force Electrical Force

Normal Force Magnetic Force

Air Resistance Force

Applied Force

Spring Force

The unit of force• Force is a quantity which is measured using the

standard metric unit known as the Newton. • A Newton is abbreviated by a "N." To say "10.0

N" means 10.0 Newton of force. • One Newton is the amount of force required to

give a 1-kg mass an acceleration of 1 m/s/s. Thus, the following unit equivalency can be stated:

A force is a vector quantity• A vector quantity is a quantity which has both

magnitude and direction. To fully describe the force acting upon an object, you must describe both the magnitude (size or numerical value) and the direction. Thus, 10 Newton is not a full description of the force acting upon an object. In contrast, 10 Newton, downwards is a complete description of the force acting upon an object; both the magnitude (10 Newton) and the direction (downwards) are given.

Superposition of forces• When there are more than one force acting on a

body, the effect on the body’s motion is the same as if a single force R were acting equal to the vector sum of the original forces:

• R = F1 + F2.

• It’s more convenient to describe a force F in terms of its x and y components Fx and Fy

• Our coordinate axes does not have to be vertical and horizontal.

Note: we draw a wiggly line through the force vector F to show that we have replaced it by its x and y components.

• To fine the vector sum of all the forces (net force) acting on a body:

Example 4.1 superposition of forces• Three professional wrestlers are fighting over the same

champion's belt. As viewed from above, they apply the three horizontal forces the belt that are shown in the figure. The magnitudes of the three forces are F1 = 250 N, F2 = 50 N, F3 = 120 N. Find the x and y components of the net force on the belt, and find the magnitude and direction of the net force.

Test your understanding 4.1• The figure shows a force F acting on a crate. With the x- and

y-axes shown in the figure, which statement about the components of the gravitational force that the each exerts on the crate (the crate’s weight) is correct?

1. The x- and y-components are both positive2. The x-components is zero and the y-component is positive3. The x-component is negative and the y-component is positive4. The x- and y-components are both negative5. The x-component is zero and the y-component is negative6. The x-component is positive and the y-component is negative.

4.2 Newton’s 1st Law• Newton’s first law of motion: a body acted on by

no net force moves with constant velocity (which may be zero) and zero acceleration.

• The tendency of a body to keep moving once it s set in motion or the tendency of a body at rest to remain at rest is due to inertia.

• It’s important to note that Newton’s first law is under the condition that the net force on the object is zero.

Newton’s 1st Law – a body in equilibrium

Example 4.2 – zero net force means constant velocity

• In the classic 1950 science fiction film X-M, a spaceship is moving in the vacuum of outer space, far from an planet, when its engine dies. As a result, the spaceship slows down and stops, what does Newton's first law say about this event?

According the Newton’s 1st Law, an object in motion will remain in motion. The spaceship will not slow down, it will travel at constant velocity forever.

Example 4.3: constant velocity means zero net force

You are driving a Porsche Carrera GT on a straight testing track at a constant speed of 150 km/h. you pass a 1971 Volkswagen beetle doing a constant 75 km/h. For which car is the net force greater?

Since both cars are in equilibrium because their velocities are constant, therefore the net force on each car is zero.

• Suppose you are in a bus that is traveling on a straight road and speeding up. If you could stand in the aisle on roller skates, you would start moving backward relative to the bus as the bus gains speed. If instead the bus was slowing to a stop, you would start moving forward down the aisle. In either case, it looks as though Newton’s first law is not obeyed; there is no net force acting on you, yet your velocity change. What wrong?

Inertial frame of reference• The bus in accelerating with respect to the earth

and in not a suitable frame of reference for Newton’s 1st law. A frame of reference in which Newton's 1st law is valid is called an inertial frame of reference. The earth is at least approximately an inertial frame of reference, but the bus is not.

• Because Newton’s first law is used to determine what we mean by an inertial frame of reference, it is sometimes called the law of inertia.

Initially, you and the vehicle stay at rest. Your body tends to stay at rest as the vehicle accelerating around you.

Initially, you and the vehicle move at constant velocity. Your body tends to move at constant velocity as the vehicle slows down you.

• As the vehicle round a corner at constant speed, your body tend to continue moving in a straight line.

• In each case, an observer in the vehicle’s frame of reference might be tempted to conclude that there is a net force acting on the passenger, since the passenger’s velocity relative the vehicle changes in each case. This conclusion is simply wrong because the vehicle is not an inertial frame and Newton’s law isn’t valid.

• We know the earth’s surface is one inertial frame of reference. But there are many inertial frames. Any frame of reference that moves at constant velocity to the earth is also an inertial frame of reference. Viewed from this light, the state of rest and the state of constant velocity are not very different.

Test your understanding 4.2• In which of the following situations is there zero

net force on the body?

1. An airplane flying due north at a steady 120 m/s and at a constant altitude.

2. A car driving straight up a hill with a 3o slope at a constant 90 km/h

3. A hawk circling at a constant 20 km/h at a constant height of 15 m above an open field

4. A box with slick, frictionless surfaces in the back of a truck as the truck accelerates forward on a level road at 5 m/s2.

4.3 Newton’s 2nd lawNewton’s 1st law tells us the when a body is acted on by zero net force, it moves with constant (including zero) velocity and zero acceleration.

But what happens when the net force is not zero?

Newton’s 2nd law - A net force acting on a body causes the body to accelerate in the same direction as the net force.

If the magnitude of the net force is constant, then the magnitude of the acceleration is also constant. In fact, the magnitude of the acceleration is directly proportional to the magnitude of the net force acting on the body.

These conclusions about net force and acceleration also apply to a body moving along a curved path.

Mass and Force• The ratio of the magnitude |∑F| of the net

force to the magnitude of a = |a| of the acceleration is constant, regardless of the magnitude of the net force.

• We call this ratio the inertial mass, or simply the mass, of the body and denote it by m.

• Mass is a quantitative measure of inertia, the greater its mass, the more a body “resists” being accelerated.

• The SI unit of mass is the kilogram.

One Newton is the amount of net force that gives an acceleration of 1 m/s/s to a body with a mass of 1 kilogram.

Since

• Suppose we apply a constant net force ∑F to a body having a known mass m1 and we find an acceleration of magnitude a1. We then apply the same force to another body having an unknown mass m2, and we find an acceleration of magnitude a2.

For the same net force, the ratio of the masses of two bodies is the inverse of the ratio of their acceleration.

• When two bodies with masses m1 and m2 are fastened together, we find that the mass of the composite body is always m1 + m2.

• Mass is the quantity of matter in a body.

Stating Newton's 2nd law• If a net external force acts on a body the body

accelerates. The direction of acceleration is the same as the directions of the net force. The mass of the body times the acceleration of the body equals the net force vector.

Using Newton's second law

• Since the equation is a vector equation, we will use it in component form, with a separate equation for each component of force and the corresponding acceleration:

Note:

1. ∑F means the sum of all external forces.

2. the equation is only valid if mass is constant.

3. the equation is only valid in the inertial frame of reference

caution

• ma is not a force.

• The vector ma is equal to the vector sum of all the forces acting on the body (∑F)

Example 4.4 determining acceleration from force

• A worker applies a constant horizontal force with magnitude 20 N to a box with mass 40 kg resting on a level floor with negligible friction. What is the accelerations of the box?

Example 4.5 determining force from acceleration

• A waitress shoves a ketchup bottle with mass 0.45 kg to the right along a smooth, level lunch counter. The bottle leaves her hand moving at 2.8 m/s, then slows down as it slides because of the constant horizontal friction force exerted on it by the counter top. It slides a distance of 1.0 m before coming to rest. What are the magnitude and direction of the friction force acting on the bottle?

• Find acceleration:

Some notes on units

1 dyne = 10-5 N

1 pound = 4.48 N

Test your understanding• Rank the following situations in order of the

magnitude of the object’s acceleration, from lowest to highest. Are there any cases that have the same magnitude of acceleration?

a. A 2.0 kg object acted on by a 2.0 N net force;

b. A 2.0 kg object acted on by an 8.0 N net force.

c. An 8.0 kg object acted on by a 2.0 N net force.

d. An 8.0 kg object acted on by a 8.0 N net force.

4.4 mass and weight• Mass characterizes the inertial properties

of a body. The greater the mass, the greater the force needed to cause a given acceleration; ∑F = ma

• Weight is a force exerted on a body the pull of the earth. Bodies having large mass also have large weight.

• The force that makes the body accelerate downward at 9.8 m/s2 is its weight.

• A body with mass m has weight of magnitude of w

w = mg– The magnitude w of a body’s weight is directly

proportional to its mass m.– The weight of a body is a force, a vector

quantity,w = mg

Caution:

A body’s weight acts at all times

Example 4.6 Net force and acceleration in free fall

• A one-euro coin was dropped from rest from the Leaning Tower of Pisa. If the coin falls freely, so that the effects of the air are negligible, how does the net force on the coin vary as it falls?

• The value of g varies from point to point on the earth’s surface, from about 9.78 to 9.82 m/s2, because the earth is not perfectly spherical and because of effects due to its rotation and orbital motion.

• At a point where g = 9.80 m/s2, the weight of a standard kilogram is w = 9.80 N. At a different point, where g = 9.78 m/s2, the weight is w = 9.78 N but the mass is still 1 kg. the weight of a body varies from one location to another; the mass does not.

Variation of g with location

If we take a standard kilogram to the surface of the moon, its weight is 1.62 N, but is mass is still 1 kg.

An 80.0 kg astronaut weighs 784 N on Earth, but weighs only 130 N on the moon.

Measuring mass and weight• The easiest way to measure

the mass of a body is to measure its weight, often by comparing with a standard.

• Two bodies that have the same weight at a particular location also have the same mass.

• The equal-arm balance can determine with great precision when the weights of two bodies are equal and hence their masses are equal.

• In outer space, we can compute the mass as the ratio of force to acceleration.

Example 4.7 mass and weight

Test your understanding of section 4.4

• Suppose an astronaut landed on a planet where g = 19.6 m/s2. Compare to Earth, would it be easier, harder, or just as each for her to walk around?

• Would it be easier, harder, or just as easy for her to catch a ball that is moving horizontally at 12 m/s (assume that the astronaut’s spacesuit is a light-weight model that doesn’t impede her movements in any way.)

4.5 Newton’s Third Law• A force acting on a body is always the result of its

interaction with another body, so forces always comes in pairs.

• Newton’s 3rd law of motion: if body A exerts a force on body B (an “action”), then body B exerts a force on body A 9a “reaction”). These to forces have the same magnitude but are opposite in direction. Theses two forces act on different bodies.

• In the statement of Newton’s third law, “action” and “reaction” are the two opposite forces; we sometimes refer to them as an action-reaction pair. This is not meant to imply any cause-and effect relationship; we can consider either force as the “action” and the other as the “reaction.”

CAUTION: the two forces in an action-reaction pair act on different bodies. Unlike in Newton's 1st or 2nd Law, which involve the forces act on one body.

The action and reaction forces can be contact forces or long-range forces.

When you drop a ball, both the ball and the earth accelerate toward each other. The net force on each body ahs the same magnitude, but the earth’s acceleration is microscopically small because its mass is so great. Nevertheless it does move!

Example 4.8 which force is greater?

• After your sports car breaks down, you start to push it to the nearest repair shop. While the car is starting to move, how does the force you exert on the car compare to the force the car exerts on you? How do these forces compare when your are pushing the car along at a constant speed?

In both cases, the force you exert on the car is equal in magnitude and opposite in direction to the force the car exerts on you.

Example 4.9 applying Newton's 3rd law – object at rest

An apple sits on a table in equilibrium. What forces act on it? What is the reaction force to each of the forces acting on the apple? What are the action-reaction pairs?

Example 4.10 applying Newton's 3rd law – object in motion

• A stonemason drags a marble block across a floor by pulling on a rope attached to the block. The block may or may not be in equilibrium. How are the various force related? What are the action-reaction pairs?

Action: Man on Rope

Reaction: Rope on Man

Action: Rope on Box

Reaction: Box on Rope

Man on Rope

Box on Rope

Man on Rope

Rope on Box

Action-Reaction Pair

Not Action-Reaction Pair

Example 4.11 We saw in example 4.10 that the stonemason pulls as hard on the rope-block combination as that combination pulls back on him. Why, then, does the block move while the stonemason remains stationary?

Check your understanding 4.5• You are driving your car on a country road when a

mosquito splatters itself on the windshield. Which has the greater magnitude:

1. The force that the car exerted on the mosquito

2. The force that the mosquito exerted on the car?

Whyis the mosquito splattered while the car is undamaged?

They are the same in magnitude and opposite in direction.

The car’s mass is much bigger than the mosquito, therefore, its acceleration a = Fnet / mass is much smaller than that of the mosquito.

4.6 free-body diagrams1. Newton’s first and seconds laws apply to a specific

body.• Newton’s 1st law, ∑F = 0, (equilibrium)• Newton’s 2nd law, ∑F = ma, (non equilibrium)

2. Only forces acting on the body matter.

3. Free-body diagram are essential to help identify the relevant forces. “free” of its surroundings, with vectors drawn to show the magnitudes and directions of all the forces applied to the body by the various other bodies that interact with it.

4. Label all the forces acting on the body.

Test your understanding 4.6• The buoyancy force shown in the figure is one half

of an action-reaction pair. What force is the other half of this pair?

1. The weight of the swimmer;2. The forward thrust force3. The backward drag force4. The downward force that the swimmer exerts on the

water5. The backward force that the swimmer exerts on the

water by kicking.