Mechanics 105

Kinematics – answers the question “how?”

Statics and dynamics answer the question “why?”

Force

Newton’s 1st law (object at rest/motion stays that way)

Inertial mass

Newton’s 2nd law (F=ma)

Gravity

Newton’s 3rd law (Action-reaction)

The laws of motion (chapter four)

Mechanics 105

Force

Intuitive concept of force

Contact and field forces

Most of the forces we experience are due to gravitational or electromagnetic

Vector nature of forces – acceleration will be in same direction as net force

Notation: F12 is the force exerted by object 1 on object 2

Mechanics 105

Newton’s 1st law

“In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line).”

In other words, objects accelerate only if there is a net force on themConcepTest

Mechanics 105

Inertial frames & inertial mass

Inertial frame of reference = frame in which 1st law is valid– at rest or at constant velocity

Inertial mass is the resistance of an object to a change in motion in response to an external force (basically it’s defined by F=ma)

SI unit: kg not the same as weight (=force due to gravitational attraction of

earth)

Mechanics 105

Newton’s 2nd law

“The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.”

i.e. the net (vector) force on an object is proportional to the (vector) acceleration of the object

Units: “Newton” (N) = kgm/s2

amF

Mechanics 105

Example

An object of mass m slides down a frictionless inclined plane (angle with respect to the horizontal). What is its acceleration?

m

Mechanics 105

Example

An object of mass m slides down a frictionless inclined plane (angle with respect to the horizontal). What is its acceleration?

N

Fg

Step one - draw the force vectors acting on the object

Mechanics 105

Example

An object of mass m slides down a frictionless inclined plane (angle with respect to the horizontal). What is its acceleration?

N

Fg

Component along direction of motion = Fgsin

Along this axis Fgsin=ma, or a= Fgsin/m

what happens along the direction perpendicular to the direction of motion?

Mechanics 105

2nd law - ConcepTests

Mechanics 105

Gravitational force and weight

What we call weight is the force of gravity on an object at sea level, it is proportional to the mass of the object

Note: this equation is valid even for no acceleration – so the gravitational mass and the inertial mass don’t have to be the same, but they are.

gmFg

Mechanics 105

A neutron walks into a bar; he asks the bartender, 'How much for a beer?' The bartender looks at him, and says 'For you, no charge.‘

"We cannot learn without pain."-Aristotle

"The only thing that interferes with my learning is my education."-Albert Einstein

"Do not worry about your problems with mathematics, I assure you mine are far greater."-Albert Einstein

Mechanics 105

Newton’s 3rd law (action-reaction)

If two objects interact, the force F12 exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force F21 exerted by object 2 on object 1, i.e.,

Conceptually, a little hard to grasp. Even for an accelerating object, there are always equal and opposite forces. Even when it seems like there is only a single object, i.e., an object in space, there must be a corresponding object which “receives” the reaction force.

2112 FF

Mechanics 105

Applications of Newton’s laws

Statics Block on the incline from earlier, now being held up by a force F If the block doesn’t move, the acceleration must be zero

y

x

Free body diagram

T

T

W

N

W

N

Mechanics 105

Applications of Newton’s laws

After making the free body diagram write out components of force vectors, then apply Newton’s second law along each axis

0cos

:

0sin

:

)ˆcosˆ(sin

ˆ

ˆ

yyy

xxx

y

x

mamgNF

y

maTmgF

x

jimgW

jNN

iTT

Mechanics 105

Applications of Newton’s laws

Atwood machine

assume m2>m1

m1

m2

+y

m1m2

T

T

gm

1 gm

2

gmm

mmT

gmm

mma

agmagmT

amgmT

m

amgmT

m

)(

2

givesback ngsubstituti

)(

)( gives

)()(

Tfor both Solving

:

:

21

21

12

12

21

22

2

11

1

Mechanics 105

Applications of Newton’s laws

One block pushing others

F m1m2

m3

Mechanics 105

Applications of Newton’s laws

Pulley’s

m2m1

Mechanics 105

Applications of Newton’s laws

Combination with circular motion

m1

m2