Chapter 2

Forces and Vectors

Vector & Scalar Quantities

Vector Quantities• Vectors are physical quantities that have

both magnitude and direction.

• Magnitude = amount and units.

• Direction can be stated as up/down, left/right, N/E/S/W or 35o S of E.

• Eg. of vectors: displacement, velocity, acceleration, force, and momentum.

• Vectors are sometimes represented by a line and arrow drawn on the line.

• The length of the line represents magnitude of the vector quantity.

• Arrow on the line represents direction.

• When asked to specify a vector quantity, state both its magnitude (size and units) as well as its direction.

• More about Vectors in Chapter 4!!

• Scalar quantities are physical quantities that have only magnitude.

• Scalars do not require direction in space when specifying them.

• Eg: distance, speed, mass, time, temperature and energy.

Scalar Quantities:

§2.1: Forces

The physical universe is made of objects (particles) that interact with each other. The interaction may define or change the “behavior” (temperature, motion) of the interacting objects.

Effects of these interactions are explained in different ways (models) such as force, momentum exchange, energy, etc.

We will first use force as a means of understanding some of these interactions.

Force: = Push or pull one object exerts on another.

• Forces come in pairs, isolated forces do not exist in physical interactions.

• Eg. When you push the door, the door pushes back on you.

• The SI unit of force is the Newton (N).

• 1N = 1 kg-m/s2.

• In the US, force is measured in pounds.

• 1.00 lb = 4.448 N and 1.00 N = 0.2248 lb.

• Robert Hooke (1635 – 1703) found that when a spring is pulled by a force F, it extends proportionally by an amount x.

• Hooke’s law: F = -kx

• k = spring constant, indicates how stiff the spring is.

•Weight = force of gravity pulling on objects.

•Extension of the spring is proportional to the weight (force). F x.

•Weight (force) can be measured using calibrated spring scale.

Measuring Forces •F = -kx

Examples of Forces:

• Contact forces – eg. Applied force of push/pull, force of tension in strings, friction, normal force, spring force.

• Long-range, action-at-a-distance (non-contact) forces. Eg. Gravitational (between earth-moon), Electric, magnetic.

• Weak nuclear force.

• Strong nuclear force.

All forces fall under 4 fundamental categories:– Gravitational (always attractive).– Electromagnetic (all contact forces).– Strong Nuclear (holds protons and

neutrons together.– Weak Nuclear (occurs in some

forms of radioactivity and thermonuclear reactions in the sun).

Newton’s Laws of Motion

Sir Isaac Newton (1642-1727)

• An object at rest will remain at rest and an object in motion will continue to move with constant speed and direction unless acted upon by a net force.

Other ways of stating it:• If no net force acts on an object at rest, it will

remain at rest but if the object is already moving, it will continue to move without change in its speed and its direction.

• If the sum of all forces acting on an object is zero, then its speed and direction will not change.

Newton’s First Law (Also called Law of Inertia)

“net force” = vector sum of all forces. A net force is needed to make an object

at rest start moving. A net force is needed to make a moving

object change direction of motion. A net force is needed to stop a moving

object. A force is not needed to keep an object in

motion if there is no force opposing its motion.

§2.4: Net Force and Vector Addition

• Net force = vector sum of all the forces acting on an object.

• Vectors are added in a special way.

• Co-linear vectors – 2 or more vectors parallel or antiparallel.

Inertia

• Inertia = resistance to change in motion.• Mass = amount of inertia of an object.• A larger mass has more resistance to change

in its motion than a smaller mass.• An object at rest wants to stay at rest, an

object in motion along a straight line wants to keep moving that way unless acted on by a net force. [inertia = resistance to change in motion] Newton’s Law one Law of inertia.

• Seatbelts are worn because of inertia.• Newton’s first law – closely related to the

reason why seatbelts are worn by motorists.

A force of 15 N is applied to the end of a spring, and it stretches 9 cm. How much further will it stretch if an additional 5.0 N of force is applied?

(A) 3.0 cm(B)1.67 cm(C)10.67 cm(D)15 cm(E) 5.0 cm

If the net force acting on a moving object suddenly becomes zero, the object will(A) continue moving but with non

zero acceleration.(B) stop abruptly.(C) continue moving at constant

velocity.(D) slow down gradually.

Velocity (v)• Velocity (v) in simple terms is speed

and direction. (Better definition later).

• If a nonzero net force (Fnet) is applied to an object, its velocity will change.

• Thus, if the net force acting on an object is zero, there will be no change in its speed, no change in its direction.

• Net force (resultant force) = vector sum of all the forces acting on an object.

Acceleration (a)• Change in velocity gives rise to acceleration

(a). If an object moves with changing velocity, we say the object moves with an acceleration. Acceleration is rate of change of velocity. ie a = v/t (“” means “change”).

• Change in velocity could mean(a) change in speed only - while direction

stays constant.(b)change in direction only while speed stays

constant. (c) change in speed and direction of motion

simultaneously. Fnet = ma

Newton’s Second Law:

• The greater the net force, the greater the acceleration, ie, a Fnet

• The greater the mass of the object, the less acceleration, ie, a 1/m.

• The direction of the acceleration is the same as the direction of the net force.

• Thus, acceleration

a Fnet/m or a = Fnet/m. • From Newton’s second law of motion,

we have the relation

Fnet = ma

A box is pushed on a a floor when a horizontal force of 250 N is applied against a frictional force of 180 N. If the box moves with an acceleration of 1.20 m/s2, what is the mass of the box?

If a net force of 1 N acts on a 200 g-book, what is the acceleration of the book?

If the net force on an object is zero, it could either:

• Be at rest – static equilibrium.

• Be moving with zero acceleration ie no change in velocity – constant velocity – dynamic (or translational) equilibrium.

• If the net force acting on an object is not zero, the object will move with changing velocity (acceleration).

Static and Dynamic Equilibrium

Newton’s Third Law of Motion• In an interaction between two objects,

the forces that each exerts on the other are equal in magnitude but opposite in direction.

“To every action, there is an equal an opposite reaction”

Note that the two equal and opposite forces are not acting on the same object!!

Action/Reaction Scenarios:• A person throws a package out of a boat

at rest. Boat starts to move in opposite direction.

• Ice skater pushes against railing and moves in opposite direction.

• Rocket exerts strong force expelling gases. Gases exerts equal force in opposite direction, propelling the rocket forward.

Two people pull on a rope in a tag-of-war.

Each pulls with a force of 100 N. The tension in the rope is

(A)200 N

(B) 100 N

(C) 0 N

(D)Diffeent at different points in the rope

(E)50 N

y

x

+

+

-

-

Force Laws1. Gravitational Forces:

• Newton’s law of universal gravitation states that any two objects of masses m1 and m2 separated by a distance r will exert a gravitational force on each other. This gravitational force is attractive force and is directly proportional to the product of the masses (F m1m2) and inversely proportional to r2 (F 1/r2).

• F m1m2 and F 1/r2 combine to give F = Gm1m2/r2

• G = Universal Gravitational constant = 6.673 x 10-11 N.m2/kg2

• Objects near the surface of the earth, gravitational force is called weight, W.

• An object of mass m near the surface of the earth has weight W = mg

g = acceleration due to gravity = 9.8 m/s2.

rm1

m2

Gravitational force between m1 and m2 is

F = Gm1m2/r2

•Gravitational force between m and ME is F = GmME/RE

2

• This force on m is the weight of the mass m.

• Weight of m = mg

• mg = GmME/RE2

• Thus g = GME/RE2 = 9.8 m/s2

• Weight of an object of mass m is W = mg

m

RE

ME

W = mg = GMEm/RE2 or

g = GME/RE2

• Acceleration due to gravity, g, is directed downwards, towards the center of the earth.

• Far away from the surface of the earth, (r = RE + h), the magnitude of g (and therefore the weight of an object at that location), decreases:

g´ = GME/r2 = GME/(RE + h)2

m

RE

ME

m

RE

ME

h

Near the earth’s surface: g = GME/RE2

Far from surface: g´ = GME/r2 = GME/(RE + h)2

Weight (W)

• Is the force of gravity due to the pull of the earth.

• g = Acceleration due to gravity = 9.8 m/s2.

• Hence for an object of mass m, the weight is W = mg

• Direction of W is always straight downward - ie. Toward the center of the earth.

A man travels to a planet that has the same mass as the earth, but twice the radius of the earth. How will his weight on earth (WE) compared to his weight on this planet (WP)?

(A) WE = WP

(B) WE < WP

(C) WE > WP

(D) It could be any of the above, depending on his mass.

W = mg

g = GME/RE2

A man travels to a planet that has the same radius as the earth, but twice the mass of the earth. How will his weight on earth (WE) compared to his weight on this planet (WP)?

(A) WE = WP

(B) WE < WP

(C) WE > WP

(D) It could be any of the above, depending on his mass.

W = mg

g = GME/RE2

2. Spring Force • Spring or elastic string stretched or compressed by distance x.

• The force that restores the spring (string) to its original length is given by the expression F = -kx [Hooke’s Law].• Negative sign is because direction of F is always opposite to the direction of x.

x

x

3. Normal Force (N)

mg

N

Consider a book of mass m at rest on a table. By Newton’s law, since the book is at

rest, the net force on it must be zero.

Hence the table must be exerting an

upward force on the book to cancel out

the force of gravity. In this case, N = mg.[It is not always the case that N = mg!!]

• Normal force is the force on an object when it is in contact with a surface.

• It is always directed perpendicularly away from the surface, ie “normally.”

mg

N

mg

N 10.0 N

mg

N 3.0 N

m = 2.0 kg

g = 9.8 m/s2

(a) (b)

(c)

Friction is a contact force between an object and a surface, and directed parallel to the surface. There are two types of friction:

Static friction and Kinetic friction.

4. Friction

(a) Static Friction: (fs)

• Is the frictional force that exists when there is no sliding or skidding between an object and a surface.

• Is the force that keeps an object at rest against the tendency for it to slide on a surface.

• Increases to a maximum value fs(max) when the object starts to slide against the surface. 0 fs ffmax

• Maximum static friction ff(max) = sN.

s = coefficient of static friction.

(b) Kinetic (sliding) Friction:

• Is the frictional force that exists when an object slides against a surface.

• Is the force that opposes the sliding movement of an object on a surface.

• fk = kN,

where k is coefficient of kinetic friction.

• Usually, k s so static friction > kinetic friction.

Box sliding at constant velocity. Find:

(a) Mass of the box.

(b) Normal force acting on the box.

(c)Coefficient of kinetic friction for the box-floor.

450 N

W = 750 N

N

fkW = mg

fk = kN

A box of weight 50 N is at rest on a floor where s = 0.3.

A rope is attached to the box and pulled horizontally with tension T = 30 N. Will the box move?

50NT

Free Body Diagram• A sketch drawing to help find net force

acting on an isolated (free) body.

• Draw the object. May be represented by just a dot.

• Draw all forces acting on the object. The length of the line and arrows should represent the forces as closely as possible. Do not include forces acting on other objects.

• The net force is obtained by performing vector addition of all the forces drawn.

Identify all forces acting on:

1. A wooden block sliding down an incline plane.

2. A wooden block sliding up an incline plane.

3. One of the tires of a car skidding on a flat road.

4. One of the tires of a car moving normally on a flat road.

5. A stone in mid air going upward.

6. A stone in mid air coming downward.

Object A is moving with constant velocity. Object B is at rest. What does A and B have in common?

(A) Acceleration not zero but constant.

(B) Acceleration is zero.

(C) A non-zero net force acts on them.

(D) Same mass and weight.

The forces acting on a plane are:

Lift L = 14 kN up,Weight W = -14 kN down,

Thrust T = 0.8 kN east, andDrag D = 1.2 kN west.

What is the net force acting on it?

The moon:

Radius = 1.74 x 106 m

Mass = 7.35 x 1022 kg

What would be the magnitude of g acting on a mass m placed near the moon’s surface?

F = Gm1m2/r2, Weight W = mg

G = 6.673 x 10-11 N.m2/kg2

A force of 10 N is applied to the end of a spring, and it stretches 5 cm. How much further will it stretch if an additional 5 N of force is applied?

Hooke’s law: F = -kx