Newton’s Laws

1st Law: A body acted on by no net force moves with constant velocity (which may be zero) 2st Law: The acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the net force acting on the object.3rd Law: For every action there is an equal, but opposite reaction

amFi

The First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. It may be seen as a statement about inertia, that objects will remain in their state of motion unless a force acts to change the motion.

Aristotle: a natural state of an object is at rest; a force is necessary to keep an object in motion. It follows from common sense.

Galileo: was able to identify a hidden force of friction behind common-sense experiments, abstracted from it a fundamental law of inertia and the principle of relativity

1564-1642

384-322 B.C.

Galileo Galilei

1564-1642

Laws of physics look the same for all observers who move with a constant velocity with respect to each other, i.e. in all inertial frames of reference.

Galilean principle of relativity (Galileo’s ship!)

First Law is identical to Galilean Principle of relativity

Indeed, if no force in needed to keep the body in motion with constant velocity, all such states of motion are equivalent. For different inertial observers the object will appear moving with different but constant velocity (which may be zero).

The first law is valid only with respect to an inertial observer, i.e. in inertial frames of reference. It is violated in accelerated reference frames.

The First Law contains implications about the fundamental symmetry of the universe in that a state of motion in a straight line must be just as "natural" as being at rest. If an object is at rest in one frame of reference, it will appear to be moving in a straight line to an observer in a reference frame which is moving by the object. There is no way to say which reference frame is "special", so all constant velocity reference frames must be equivalent.

2nd Law

From experiments we know:1. A force is needed to change the state of motion2. Force is a vector; obeys superposition principle: the net

force is a vector sum of all forces acting on an object3. The direction of acceleration vector is the same as the

direction of the force vector4. The magnitude of the force and acceleration are related

by a constant which intuitively is a “quantity of matter”. This is the inertial mass.

Newton’s 2nd Law of Motion

2. The acceleration a of a body is inversely proportional to its mass m, directly proportional to the net force F, and in the same direction as the net force.

a = F/m F = m a 1 N = 1 kg m/s2

Newton’s 3rd Law of Motion

3. To every action, there is an equal and opposite reaction.

The same force that is accelerating the boy forward, is accelerating the skateboard backward.

Clockwork universe

Types of forces

• Gravity and weight

• Normal force

• Friction

• Tension and spring force

All are manifestation of four Fundamental “forces”

Contact versus long-range forces

•Gravity•Electromagnetic•Strong•Weak

Gravity is a strange force. It has a unique property:

M

m

R

2R

mMGF

2R

MG

m

Fa

All bodies in the same point in space experience the same acceleration!

Galileo, about 1600

2R

MmGam g

i

!!!gi mm

22

m/s8.9; gmgR

mMGF

E

Weight, “apparent weight”, the force of gravity, and the normal force

• Riding an elevator

Units of Force

2/111 smkgNNewton

British system:

kgslug 59.141 units of mass:

units of force: Nsftsluglbpound 448.4/111 2

One pound is 0.4536 kg

One pound is the weight of 0.4536 kg on the Earth

Box on an inclined planeA box with mass m is placed on top of a frictionless incline

with angle and height H and is allowed to slide down.

a) What is the normal force?

b) What is the acceleration of the box?

c) What is the velocity of the box when it reaches the bottom?

Friction

Two types of friction:1. Kinetic: The friction force

that slows things down2. Static: The force that

makes it hard to even get things moving

Refrigerator

• If you push a refrigerator when there is no friction what happens?

• In the real world what happens? Especially when it’s fully loaded and on a sticky kitchen floor?–When does static friction kick in?

–When does kinetic friction kick in?

Friction

There is some maximum value the friction force can achieve, and once we apply a force greater than this maximum there is a net force on the object, so it accelerates.

The maximum of the force of friction varied linearly with the amount that the block pushes on the table.

NFfriction

- coefficient of friction, is the vertical force exerted by

the block on the table N

The friction force only exists when there is another force trying to move an object

Kinetic Friction

• For kinetic friction, it turns out that the larger the Normal Force the larger the friction. We can write

• FFriction = KineticFNormal

Here is a constant• Warning:

– THIS IS NOT A VECTOR EQUATION!

Static Friction

• This is more complicated• For static friction, the friction force can vary

FFriction StaticFNormal

Example of the refrigerator: – If I don’t push, what is the static friction

force?– What if I push a little?

Is it better to push or pull a sled?

You can pull or push a sled with the same force magnitude, FP, and angle , as shown in the figures.Assuming the sled doesn’t leave the ground and has a constant coefficient of friction, , which is better?

FP

FP

A Recipe for Solving Problems1. Sketch Isolate the body, draw a free-body diagram (only

external forces but not forces that one part of the object exert on another part)

2. Write down 2nd Newton’s law

amF

Choose a coordinate system Write 2nd Newton’s law in component form:

yyxx

yxyx

maFmaF

jmaimajFiFF

,

3. Solve for acceleration

Pulling Against FrictionA box of mass m is on a surface with coefficient of kinetic and

static friction . You pull with constant force FP at angle The box does not leave the surface.

1. Find the minimum force you need to apply in order to move the block

2. What is the magnitude of the acceleration? 3. What angle maximizes the acceleration?

Box on an inclined plane with frictionA box with mass m is placed on an incline with

angle and is allowed to slide down.

a) What is the acceleration of the box?

Tension and pulleys

Massless, unstretchable string; massless, frictionless pulley

Force of tension

The advantage of a pulley

What minimum force F is needed to lift the piano of mass M?

Conical pendulum

A ball of mass m is swung around a circle at the end of a string of length L. The string will break if the tension in it exceeds a critical value, Tc. What is the largest constant angular velocity the ball can have without breaking the string?What is the largest period the ball can have without the string becoming slack?

A mass m1 is going around in a circle on a string on a frictionless table and the string goes through a hole where it is attached to a hanging mass m2. If the mass m1 is going around with constant , what must the distance from the mass m1 to the hole be if the mass m2 is to remain at rest?

0

m1

m2

Playing with weight:

• A car on an arched bridge

• Your weight in a rotating space station or on the rotating Earth

A race track designer wants to have the cars able to maintain a speed vmax without skidding on a circular track. If the track is flat with a coefficient of friction what does the radius have to be?

A race track designer wants to have the cars able to maintain a speed vmax without skidding. At what angle must the track of radius R be banked assuming no friction? Assuming a coefficient of friction ?

A satellite of mass m is attracted to the Earth of mass M with a force of gravity proportional to the inverse square of the distance to the Earth center, r:

rir

MmF

2

Find the velocity of a satellite on circular orbit of radius r.

Find the radius of the orbit for a geostationary satellite

is a gravitational constant