Transcript Topic 2.2 ppt

Mechanics
Topic 2.2 Forces and Dynamics
Forces and Free-body
Diagrams
To a physicist a force is recognised by
the effect or effects that it produces
A force is something that can cause an
object to




Deform (i.e. change its shape)
Speed up
Slow Down
Change direction
The last three of these can be
summarised by stating that a force
produces a change in velocity
Or an acceleration
Free-body Diagrams
A free-body diagram is a diagram in which
the forces acting on the body are represented
by lines with arrows.
The length of the lines represent the relative
magnitude of the forces.
The lines point in the direction of the force.
The forces act from the centre of mass of the
body
The arrows should come from the centre of
mass of the body
Example 1
Normal/Contact Force
A block resting on a worktop
Weight/Force due to Gravity
Example
2
A car moving with a constant velocity
Normal/Contact Force
Resistance
Motor Force
Weight/Force due to Gravity
Example
3
A plane accelerating horizontally
Upthrust/Lift
Air Resistance
Motor Force
Weight/Force due to Gravity
Resolving Forces
Q. A force of 50N is applied to a block
on a worktop at an angle of 30o to the
horizontal.
What are the vertical and horizontal
components of this force?
Answer
First we need to draw a free-body diagram
50N
30o
We can then resolve the force into the 2
components
Vertical = 50 sin 30o
50N
30o
Horizontal = 50 cos 30o
Therefore


Vertical = 50 sin 30o = 25N
Horizontal = 50 cos 30o = 43.3 = 43N
Determining the Resultant
Force
Two forces act on a body P as shown in
the diagram
Find the resultant force on the body.
50N
30N
30o
Solution
Resolve the forces into the vertical and
horizontal components (where
applicable)
50 sin 30o
30N
50N
30o
50 cos 30o
Add horizontal components and add
vertical components.
50 sin 30o = 25N
50 cos 30o – 30N = 13.3N
Now combine these 2 components
25N
R
13.3N
R2 = 252 + 13.32
R = 28.3 = 28N
Finally to Find the Angle
R
25N

13.3N
tan  = 25/13.3
 = 61.987
 = 62o
The answer is therefore 28N at 62o upwards from
the horizontal to the right
Springs
The extension of a spring which obeys
Hooke´s law is directly proportional to the
extending tension
A mass m attached to the end of a spring
exerts a downward tension mg on it and if it
is stretched by an amount x, then if k is the
tension required to produce unit extension
(called the spring constant and measured
in Nm-1) the stretching tension is also kx and
so
mg = kx
Spring Diagram
x
Newton´s Laws
The First Law
Every object continues in a state of
rest or uniform motion in a straight
line unless acted upon by an external
force
Examples
Any stationary object!
Difficult to find examples of moving
objects here on the earth due to friction
Possible example could be a puck on ice
where it is a near frictionless surface
Equilibrium
If a body is acted upon by a number of
coplanar forces and is in equilibrium ( i.e.
there is rest (static equilibrium) or
unaccelerated motion (dynamic
equilibrium)) then the following condition
must apply
The components of the forces in both of any
two directions (usually taken at right angles)
must balance.
Newton´s Laws
The Second Law
There are 2 versions of this law
Newton´s Second Law
1st version
The rate of change of momentum of a
body is proportional to the resultant
force and occurs in the direction of the
force.
F = mv – mu
t
F =
t
Newton´s Second Law
2nd version
The acceleration of a body is
proportional to the resultant force
and occurs in the direction of the
force.
F = ma
Linear Momentum
The momentum p of a body of constant mass
m moving with velocity v is, by definition mv
Momentum of a body is defined as the mass
of the body multiplied by its velocity
Momentum = mass x velocity
p = mv
It is a vector quantity
Its units are kg m s-1 or Ns
It is the property of a moving body.
Impulse
From Newtons second law
F = mv – mu
F =
t
t
Ft = mv – mu
This quantity Ft is called the impulse of the
force on the body and it is equal to the
change in momentum of a body.
It is a vector quantity
Its units are kg m s-1or Ns
Law of Conservation of Linear
Momentum
The law can be stated thus
When bodies in a system interact the
total momentum remains constant
provided no external force acts on the
system.
Deriving This Law
To derive this law we apply Newton´s 2nd law
to each body and Newton´s 3rd law to the
system
i.e. Imagine 2 bodies A and B interacting
If A has a mass of mA and B has a mass mB If
A has a velocity change of uA to vA and B has
a velocity change of uB to vB during the time
of the interaction t
Then the force on A given by Newton 2 is
FA = mAvA – mAuA
t
And the force on B is
FB = mBvB – mBuB
t
But Newton 3 says that these 2 forces are
equal and opposite in direction
Therefore
mAvA – mAuA =
t
-(mBvB – mBuB)
t
Therefore
mAvA – mAuA = mBuB – mBvB
Rearranging
mAvA + mBvB = mAuA + mBuB
Total Momentum after =
Total Momentum before
Newton´s Laws
The Third Law
When two bodies A and B
interact, the force that A exerts on
B is equal and opposite to the force
that B exerts on A.
Example of Newton´s 3rd
Q. According to Newton’s third Law
what is the opposite force to your
weight?
A. As your weight is the pull of the
Earth on you, then the opposite is the
pull of you on the Earth!
Newton´s 3rd Law
The law is stating that forces never occur
singularly but always in pairs as a result of
the interaction between two bodies.
For example, when you step forward from
rest, your foot pushes backwards on the
Earth and the Earth exerts an equal and
opposite force forward on you.
Two bodies and two forces are involved.
Important
The equal and
opposite forces
do not act on the
same body!