Mechanics

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Mechanics

• Topic 2.2 Forces and Dynamics

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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

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• The last three of these can be summarised by
  stating that a force produces a change in
  velocity
• Or an acceleration

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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

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Example 1
A block resting on a worktop
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Example 2
A car moving with a constant velocity
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Example 3
A plane accelerating horizontally
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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?

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Answer

• First we need to draw a free-body diagram

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• We can then resolve the force into the 2
  components

Vertical 50 sin 30o
Horizontal 50 cos 30o
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• Therefore
• Vertical 50 sin 30o 25N
• Horizontal 50 cos 30o 43.3 43N

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Determining the Resultant Force

• Two forces act on a body P as shown in the
  diagram
• Find the resultant force on the body.

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Solution

• Resolve the forces into the vertical and
  horizontal components (where applicable)

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• Add horizontal components and add vertical
  components.

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• Now combine these 2 components

R2 252 13.32 R 28.3 28N
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Finally to Find the Angle
tan ? 25/13.3 ? 61.987 ? 62o
The answer is therefore 28N at 62o upwards from
the horizontal to the right
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Springs

• The extension of a spring which obeys Hookes 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

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Spring Diagram
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Newtons 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

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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

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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.

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Newtons Laws

• The Second Law
• There are 2 versions of this law

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Newtons 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 F ?? t t

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Newtons 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

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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.

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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

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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.

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Deriving This Law

• To derive this law we apply Newtons 2nd law to
  each body and Newtons 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

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• 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

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• Therefore mAvA mAuA -(mBvB
  mBuB) t t
• Therefore mAvA mAuA mBuB mBvB
• Rearranging mAvA mBvB mAuA mBuB
• Total Momentum after
• Total Momentum before

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Newtons 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.

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Example of Newtons 3rd

• Q. According to Newtons 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!

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Newtons 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.

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Important

• The equal and opposite forces do not act on the
  same body!