CORE SECTION 4: MECHANICS & MATERIALS

 

CONTENTS

4.1 SCALARS & VECTORS
4.2 MOTION
4.3 ENERGY
4.4 MATERIALS

 

4.1 SCALARS & VECTORS

v  Scalar : a quantity having size, but no specified direction

v  Vector : a quantity having size AND direction

  egs Displacement (vector) and Distance (scalar), velocity (vector) and speed (scalar), etc.

v  The RESULTANT of a set of vectors is the single vector that can replace the set to produce the same effect.

v  FINDING THE RESULTANT OF 2 OR MORE VECTORS:

    Parallel vectors: we add to find the resultant, Anti-parallel vectors we subtract to find the resultant:

     eg. here the resultant vector is in red:  

   

    For 2 vectors at 90°, you need to draw the rectangle as shown,

     the resultant vector R is then the diagonal of the rectangle:

     

v  You would usually be required to calculate both the size and the direction of the

     resultant force. The quickest way is to use Pythagoras to find the magnitude and

      then trigonometry to find the direction. But you may also be asked to solve it using a scale drawing instead.

v  You must make sure you can resolve a vector into its two perpendicular components:

     here the force F is resolved into Fsinθ acting vertically up, and Fcosθ acting horizontally:

            

v  when both RESULTANT FORCE and RESULTANT TURNING MOMENT

    acting on a body are zero it is in equilibrium

v  a body is in equilibrium if the 3 co-planar forces acting on it meet at a point

v  if a body is in equilibrium due to 3 co-planar forces acting on it, then the forces can

    be represented in size and direction by the sides of a triangle (triangle of forces rule)

 

v  MOMENT OF A FORCE (OR TORQUE) = size of force x perpendicular distance

     between line of action of force and pivot (units Nm) = Fd

v  LAW OF MOMENTS: for an object in equilibrium, the total of the clockwise moments about any point =

    the total of the anticlockwaise moments about that same point

v  A COUPLE is a pair of EQUAL and OPPOSITE, PARALLEL forces.

v  MOMENT OF A COUPLE = one of the parallel forces x the perpendicular

    distance between them. (units Nm)

v  CENTRE OF MASS this coincides with the centre of gravity and is the

     point through which the weight force due to the Earth acts on a body.

 

4.2 MOTION

v  DISPLACEMENT(s) = distance moved in a specified direction (a vector)

v  VELOCITY at any moment = the gradient of the displacement / time

     graph at that moment (also a vector)

v  ACCELERATION at any moment = gradient of the velocity / time graph

      at any point (units  m / s2)

                                                v = Ds / Dt  ,            a = Dv / Dt

 

v   Be familiar with s / t , v / t , and  a / t graphs

v   The area under a v/t graph gives the distance travelled

v   The area under an a/t graph gives the change in velocity

v   Displacement may be calculated from area too but beware of negative

                          values due to direction changes

          

 

                v = u + at                            s = ut + ˝ at2

 

                s = ˝ (u + v) t                     v2  =   u2  + 2as

 

      PRACTICAL 3: Determination of g by a freefall method

 

           horizontal components and then to use the equations above. Remember to

           NOT mix up the vertical data with the horizontal data! 

        

           NEWTON1 : a body continues in its state of rest or constant velocity unless

           acted on by a resultant force.

           NEWTON2 :    F = (mv – mu) / Dt , from which we get:   F = ma

           NEWTON3 : if body A exerts a force on body B, then body B will exert an

           equal and opposite force on body A.

 

v  WORK DONE = force x displacement,

     area under a Force / Displacement graph = work done / energy stored  

     W = Fs    also :  W = Fs cosq

v  POWER = work done / time taken ,  P = DW / Dt     (units Watts or J/s)

v  Also,  POWER = force x velocity    P = Fv

v  TOTAL ENERGY = A CONSTANT  (law of conservation of)

v  Energy = Power x Time,  E = Pt

v  ke = ˝ m v2                            pe = mgh

v  Efficiency = useful output power / total input power

 

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         4.4 MATERIALS

v  Density = Mass / Volume

v  HOOKE’S LAW : the force extending a material is proportional

    to the extension produced F µ x (up to the limit of proportionality)

    F = k ΔL, k is the Spring Constant and a measure of the stiffness.

v  An ELASTIC material is one which returns to its original shape

     after the applied force is removed. Stretched beyond its

     ELASTIC LIMIT a material will not return to its original length.

   NB the elastic limit and limit of proportionality do not necessarily coincide. 

v  An INELASTIC material is one which does not return to its original

    shape after the applied force is removed.

v  If a material is loaded beyond its elastic limit PLASTIC FLOW may

    occur where small increases in force can cause large increases in

    extension (molecular planes sliding over one another)

v  A BRITTLE material is one which shows little or no plastic flow,

   material A in the graph below (material B is a ductile material): 

 

 

v  TENSILE STRESS = force / cross-sectional area  (units are N / m2 or Pa )                                              

v  BREAKING STRESS = maximum stress a material can withstand

v  TENSILE STRAIN = extension / original length  (no units)

 

v    YOUNGS MODULUS = tensile stress / tensile strain

   Y.M =   F / A     =  the gradient of the stress / strain graph

                                         ΔL / L               

      

                            Units are N / m2 or Pa       

                                             

                       PRACTICAL 4: Determination of The Young Modulus by a simple method

v   NB STRESS / STRAIN graphs are useful since the gradient of the

     graph line = the Young Modulus of the material

v   ENERGY STORED = ˝ x Force x extension  (unit Joules)

                       NB AREA under the F / ΔL graph = work done in stretching = energy stored

v   ENERGY STORED PER METRE CUBED = ˝  X  STRESS  X  STRAIN (units J / m3 )

 

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