4.1 | SCALARS & VECTORS |
4.2 | MOTION |
4.3 | ENERGY |
4.4 | MATERIALS |
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.
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 / s^{2})
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 + ˝ at^{2}
^{ }
^{ } s = ˝ (u + v) t v^{2} = u^{2} + 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.
Momentum
momentum = mv (a vector, with units kg m/s or Ns)
Law of conservation : TOTAL MOMENTUM before collision = TOTAL
MOMENTUM after collision provided NO EXTERNAL FORCES ACT.
Elastic collision one where ke is conserved
Inelastic collision where ke is converted to other forms of energy.
NEWTON2 : force = rate of change of momentum,
F = Dmv / Dt , F = (mv – mu) / Dt , Impulse F Dt = (mv – mu)
So, FDt = Dmv , so the area under a F / t graph is the change in momentum.
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 v^{2} pe = mgh
v Efficiency = useful output power / total input power
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 / m^{2} 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 / m^{2} 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 / m^{3} )