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Monday, 23 April 2012

Forces and Motion (In Upper Secondary Years)

"Forces and Motion" - An Overview:

1        Kinematics – the study of motion without considering the force acting on the object. Force F will not appear in any of the equations of motion. So, a convenient starting point in kinematics would be the study of linear motion and linear motion graphs.

1.1        In Linear Motion, you will learn about:
1.1.1  Distance and Displacement
1.1.2  Speed and Velocity
1.1.3  Acceleration and deceleration
1.1.4  Ticker-Timer
1.1.5  Equations of Linear Motion
         a) a = (v - u) / t
         b) v = u + at
         c) t = (v - u)/a
         d) s = 1/2 (u + v)t (average velocity x time)
         e) s = ut + 1/2(at^2) (i.e. the area under the Velocity-Time graph)
            (or, s = 1/2(g)(t^2), free fall of a stationary object: 2005 P1 Q4, pg 3)
         f) v^2 = u^2 + 2as

1.2        In Linear Motion Graphs, you will learn about:
1.2.1  Displacement-Time (s-t) Graphs
         (2007 P1 Q4 @ pg 92 /
          2010 P1 Q5 @ pg 231)
1.2.2  Velocity-Time (v-t) Graphs
          (2005 P1 Q10 @ pg 5 / 
           2007 P1 Q5 @ pg 92 /
           2008 P1 Q4 @ pg 140 /
           2009 P1 Q4 @ pg 186 / 2009 P2 Q5 @ pg 251
           2010 P1 Q3 @ pg 230 /
           2011 P1 Q10 @ pg 282)

1.2.3  The Use of Ticker-Tapes to Plot the Graphs
         (2008 P3 Q 2 @ pg 180 /  
          2011 P2 Q1 @ pg 294)

2        Inertia and Newton’s 1st Law of Motion (a.k.a. the Law of Inertia):

2.1        Inertia refers the tendency of an object to resist change in its state of motion: If at rests, it tends to remain at rest; if in motion, it tends to move in straight line at the same speed unless acted on by an external net force.
         (2005 P1 Q6 & Q8 @ pg 4)

Different words for the same thing - Inertia may also be referred to as:

I         An object’s natural resistance to change its current state of equilibrium – if in static equilibrium (at rest) or in dynamic equilibrium (motion with uniform velocity), it tends to remain so.

II      An object’s natural resistance to change its current velocity, including zero velocity (at rest).

III   An object’s natural resistance to change its current momentum including zero momentum – if at rest with zero momentum, it tends to remain at rest with zero momentum; if in motion in a straight line with a constant speed and therefore a constant momentum (= mass x velocity), it tends to continue to remain so with the same momentum.

2.2        To overcome an object’s inertia (i.e. resistance to change its current state of motion), an external force - i.e a non-zero net force - is needed
2.2.1  to make a stationary object moves and/or change position;
2.2.2  to make a moving object moves in the same direction with greater or lower speed (including stop with zero speed);
2.2.3  to make a moving object moves with changed direction.

2.3        Inertia increases as:
2.3.1  the mass of a stationary object (say, a trolley with different amount things loaded) increases and the converse is also true; or
2.3.2  the momentum of moving object increases and the converse is true too.

2.4        Inertia - although related to mass and momentum - has no unit of measurement because it merely describes the natural tendency of all objects to resist change in their states of motion.

2.5   An object’s inertia depends solely on its mass only if the object is stationary; But if the objects is moving (say, a person sprinting as opposed to walking), its inertia (i.e. resistance to change its state of motion) does not depend on mass alone (since his mass is the same in both situations) – inertia of a moving object depends on momentum, i.e. on both its mass and velocity.

2.6   Situations Involving Inertia:

A.     Things Lurching Backward When Carrier Starts to Move Forward;
B.     Things Lurching Forward When Carrier Suddenly Slows Down or Stops;
C.     Person Lurch Forward When Skateboard Suddenly Stopped by a Stone;
D.     Coin Drops Straight (into a glass) When Cardboard Is Pulled Away Quickly;
E.      Book Pulled Out Quickly from Middle of a Pile Will Not Cause Books Above It to be Pulled Along;
F.      Raw and Hard Boiled Eggs Spinning Experiment;
G.     Massive Objects Show Difficulty to Start Moving, Change Direction or Stop Moving: Difficult for a massive charging bull to charge in zigzag manner or for a massive supertanker to change direction or stop.
H.     Fine Thread of Freely Hung Weight Snaps When Weight is Raised and Released
I.    Upper Thread of Freely Hung Weight Snaps Only When Lower Thread is Gradually Pulled But When the Lower Thread is Yanked, It Is Lower Thread Which Snaps 

3        Momentum (a measure of a moving object's inertia - a vector quantity):

3.1        Momentum = Mass x Velocity of an object.

3.2        The higher the momentum of a moving object, the higher its inertia and therefore the higher its resistance to change its speed and direction of motion.

3.3        Newton’s Law of Inertia (1st Law of Motion), where it is applicable on a moving object, states that "unless an external net force acts on a moving object, the object would move in a straight line with constant speed (and therefore with the same momentum – herein lies the Principle of Conservation of Momentum).

3.4        Principle of Conservation of Momentum states that “the total momentum of a system is constant, if no external force acts on the system”.

3.5        In a collision, the total momentum of objects before collision equals to that after collision. The total energy is conserved too. However, kinetic energy is only conserved in an elastic collision where the colliding objects move apart after collision. In inelastic collision where the colliding objects stop moving or move together with a common velocity, kinetic energy is not conserved.

         (2011 P1 Q5 @ pg 280)

3.6        In an explosion, the sum of momentum remains zero after the explosion. Explosive situations include:
                                                              I.      Gun-powder triggered explosions
                                                           II.      Jet of Air Explosions
a.       Deflating balloon
b.      Jet engine
c.       Rocket

                              III  Spring-triggered explosions

4        The Effects of a Force (a Push or a Pull) on Motion and Newton’s 2nd Law of Motion:

4.1        Evidence in nature and in science lab. shows that F = ma or a = F/m:
4.1.1  The acceleration a of an object is directly proportional to the force applied F if the mass of the object m remains constant
4.1.2  The acceleration a of an object is inversely proportional to the mass m of the body (say, a trolley with different amount of things loaded) if the force applied F remains constant.

4.2        Combining the 2 results, we have:
4.2.1  Force F = Mass, m x Acceleration, a (F = ma)
4.2.2  Newton’s 2nd Law of Motion states that “the acceleration of a body is directly proportional to the net force acting on it (and inversely proportional to the mass of the body)

4.3        Balanced Forces - Static Equilibrium (at rest) and Dynamic Equilibrium (constant velocity):
         4.3.1 General Principle of Balanced Forces:
         When 2 or more forces act on an object and balance each other, the net or resultant force acting on the object is zero - the object, if at rest, will remain at rest (i.e. remain in static equilibrium); or, if in motion, will continue to move with the same velocity (i.e. in a straight line with the same speed with no acceleration) and thus with the same momentum (i.e remain in dynamic equilibrium).

         4.3.2 Newton's Third Law of Motion:

         Newton's Third Law of Motion states that: 
                                       "To every action there is an equal but opposite reaction."
         Action (Resultant Force) = Opposite Reaction means Balanced Forces Resulting in
                     Object at rest (static equilibrium
                     Object moving with uniform velocity (dynamic equilibrium)
         Action Not Equal to Opposite Reaction means Unbalanced Forces Resulting in
                     Object moving with acceleration

          4.3.3 Situations of Balanced Forces with Static Equilibrium

         Book Resting on a Horizontal Table Surface
         Box Resting on an Inclined Plane
         Object Hung by Two Ropes Fixed to Two Points in the Ceiling/Wall
         Object in Stationary Lift

         4.3.4 Situations of Balanced Forces with Dynamic Equilibrium

         Object (Parachutist, etc) Falling at Terminal Velocity
         Object Rolling Down Friction-Compensated Incline at Constant Velocity
         Plane Cruising Horizontally at Constant Velocity
         Car Moving Horizontally at Constant Velocity
         Object in Lift Moving (Upwards or Downwards) at Constant Velocity

4.4        Unbalanced Forces - Object Moving with Acceleration:

         4.4.1 General Principle of Unbalanced Forces:
         When 2 or more forces act on a object of mass m with a net resultant force F, then the object will experience an acceleration a (= F/m) and a change in momentum or impulse (= mv – mu =  Ft, where v = final velocity; u = initial velocity and t = time during which the force F acts) in the direction of the net force.  (2005 P1 Q9 @ pg 4 / 2011 P1 Q6 @ pg 281)

         4.4.2 Situations of Unbalanced Forces:

         Unbalanced Parallel Forces
                              A) Lift Moving (Upwards or Downwards) with Acceleration
                              B) Rocket Lifting Off Vertically with Acceleration (Force > Its Weight + Friction)
                              C) Car Moving Horizontally with Acceleration (Force > Static Friction + Drag)
                              D) Plane Moving Horizontally with Acceleration (Force > Drag)

         Unbalanced Non-Parallel Forces

                              A) Object Sliding Down Inclined Slope with Acceleration
                              B) Mowing Machine Moving with Horizontal Acceleration When Pushed
                              C) Object Moving with Horizontal Acceleration When Towed 

5.      Impulse and Impulsive Force:

5.1        Impulse is defined as the product of a force F acting on an object of mass m for time t: Impulse = Ft = mv-mu = change in momentum
              Impulse = Ft (in N s)
          = mat (since F = ma)
          = mt (v – u)/t
          = m(v – u)
         = mv – mu (change in momentum) (in kg m s^-1)

5.2        Impulsive Force is defined as the rate of change in momentum:
Impulsive Force F = Change in Momentum / Time Taken
F = (mv – mu)/t

(Effects of time on impulsive force F if the change in momentum is constant as in collision or explosion: If time t is small, then impulsive F is large)

(2005 P1 Q7 @ pg 4 / 2011 P1 Q4 @ pg 280)

6.      Safety Features in Vehicles That Seek to Reduce Impulsive Force on Collision
6.1        Padded Dashboard
6.2        Automatic Airbag
6.3        Headrest
6.4        Safety Seat Belt
6.5        Shatter-Proof Wind Screen
6.6        Rubber Bumper
6.7        Strong Steel Struts

7.      Gravity

7.1        Gravity refers to the force that attracts a body towards the centre of the earth or towards the centre of any object that has matter/mass.

7.2        A gravitational field is the space around a mass in which another mass would experience a force of attraction towards the centre of the object with mass. The more matter/mass that an object has, the stronger the gravitational field in the space around it.

7.3        Gravitational field strength g at a point in the gravitational field is a measure of the gravitational force F acting on per unit mass of a body (say, of mass m):
g = F / m (g is the ratio of Weight / Mass)
F = mg (Gravitational F = Weight of the body of mass m)

7.4  Thus, at the same point in the gravitational field, a body of higher mass will experience a higher gravitational force on it (F = mg); and a higher weight (because gravitational force = its weight) (F = mg = Weight).

7.5  If a body of mass m is allowed to free-fall at a point with gravitational field strength g, it would experience a gravitational force F = mg; and its acceleration a due to gravitation force F would be the same as g (because F = mg = ma; therefore a = g)

7.6  The closer to the centre of earth, the stronger the gravitational field strength g and therefore the higher the gravitational acceleration. Thus, with the earth being somewhat flattened at the polar regions, g at the polar regions (of maximum latitude 90`N / 90`S) is higher at 9.832 m s^-2 as they are nearer to the centre of the earth as compared to locations of lower latitudes.

7.7  Thus, at the equator (of lowest latitude of 0`), g is lower at 9.780 m s^-2.

7.8  Since the gravitational field force acting on a body of mass m is a measure of the weight of the body (Weight = mg), at polar region, a body would weigh more (since g is higher) and lower latitude such as at equator, the same body would weigh less (g is lower at equator). We weigh less on the surface of the Moon because g is lower at the surface of the Moon as compared to that on Earth because Moon has less mass than our earth.

8.      Forces in Equilibrium

8.1        When forces acting on an object are in equilibrium, the net force or resultant force on the object is zero. The object will either be:
8.1.1  at rest (static equilibrium); or
8.1.2  in motion with constant velocity (dynamic equilibrium).

8.2        When 2 forces acting on an object are in equilibrium, they must be acting in opposite direction with equal magnitude. For examples:
8.2.1  A book resting on a table exerts a downward force which equals to its weight (F = mg) and the table reacts with an equal but opposite reaction force in accordance with Newton’s 3rd Law of Motion: The 2 opposite forces of equal magnitude balance each other out;
8.2.2  A skydiver descending at constant velocity exerts a downward force equals to his weight (F = mg) and the air reacts with an equal but opposite air resistance force.

8.3     When 3 forces are in equilibrium, it means either:
8.3.1  that the resultant force of 2 of the forces is equal but opposite to the third force; or
8.3.2  that if 1 of the 3 forces is resolved into 2 forces in the direction parallel to the other 2 forces, the 2 resolved forces balance out the other 2 forces; or
8.3.3  that if all 3 forces are resolved into 2 perpendicular axis such as x and y axis, the resultant forces along the 2 axis are zero.

8.4        The resultant force is the single force that will produce the same effect as the 2 or more combined forces that it replaces – the resultant force can be determined by graphical method by way of geometric construction of:
8.4.1  parallelogram; or
8.4.2  triangle; or
8.4.3  rectangle or square if the forces are perpendicular

8.5        Resolution of Force: A force can be resolved into any number of component forces in any direction provided the combined effect of these component forces is the original force. Normally, a force is resolved into 2 component forces that are perpendicular to each other.
9.      Work, Energy, Power and Efficiency

9.1     Work:
a)   In lower secondary, you have learnt about work done W as the product of the force, F and the distance, s in the direction of the force:
W = F x s

a)      In upper secondary: If the distance moved, s is not in the direction of the force but at an acute angle A` to the direction of the force, then work done W is the product of the component of force in the direction of displacement, FcosA` and s:
W = FcosA` x s

b)      If displacement s is perpendicular to the direction of the force F, then work done W is zero:
W = F x 0 (since displacement in direction of force = 0)
W = 0

c)      Work done, Wg, against gravity: The displacement s is always the vertical height h because gravitational force, mg (mass x gravitational acceleration), always acts vertically downwards and work done against gravity is always in the opposite direction. Thus:
Wg = mg x h (even if the object is lifted upwards along an inclined slope)

9.2        Energy – The Capacity to Do Work (Energy, E = Work Done, W)

a)      Energy can exist in different forms:
1)      mechanical energy (kinetic energy + potential energy)
2)      heat energy
3)      electrical energy
4)      chemical energy
5)      sound energy
6)      light energy
7)      nuclear energy

b)      When work is done, energy is transferred from one object to another. For examples:
1)      When a weightlifter lifts a weight, chemical energy (stored in food) of the weightlifter is transferred to the weight as gravitational potential energy;
2)      When a pitcher throws a ball, chemical energy (stored in food) of the pitcher is transferred to the ball as kinetic energy.
(2011 P1 Q9 @ pg 282)

c)      Potential energy, Ep, of an object is defined as the energy stored in the object because of its position (gravitational potential energy = mgh) or its state (elastic potential energy = ½ kx^2).
      (2005 P1 Q5 @ pg3)

d)      Kinetic energy, Ek, is the energy possessed by an object due to its motion (Ek = ½ m(v^2 – u^2) (or, = ½ mv^2, if u = 0)

e)      Principle of Conservation of Energy states that: “energy cannot be created or destroyed. It can however be transformed from one form to another – the total energy in a system is constant”

      (2011 P1 Q8 @ pg 281)

9.3        Power is defined as:
a)      The rate at which work is done:
Power = Work Done / Time taken;

(2011 P1 Q7 @ pg 281)

b)      The rate at which energy is transformed:
Power = Energy Transformed / Time Taken

c)      The product of force, F and velocity v (in the direction of the force):
Power = Work Done / Time
           = Force x Displacement / Time
           = Force x Velocity (since Velocity = Displacement / Time)
        P = F x v

9.4        Efficiency:
a)      Efficiency = Useful Energy Output, Eo / Energy Input, Ei x 100%; or,
b)      Efficiency = Useful Power Output, Po / Power Input, Pi x 100%.
c)      An engine is a device which converts one form of energy (say, heat or lectricity) into another (usually mechanical) to do work for us.
d)      No engine is 100% efficient because there will be energy loss as noise, heat, etc.

10.  Maximising Efficiency – Means:
10.1    Minimising Energy Wastage
10.2    Prolonging the Availability of Existing Non-Renewable Energy Resources (Fossil Fuels, etc.)
10.3    Minimising Sound and Other Environmental Pollutions

11.  Elasticity, Hooke’s Law and Elastic Potential Energy
11.1    Elasticity is the ability to return to the original 1) shape and 2) size after being deformed - stretched (extended), squashed (compressed) or bent - by a force:
    1. Materials that can do so are elastic materials.
    2. Materials that cannot do so are inelastic.

11.2  A spring is elastic.
a.       At its natural length, there is no net force stretching or squashing it.
b.      When subjected to a force to stretch it, a spring will lengthen and the amount of lengthening is known as the extension.
c.       Conversely, when subjected to a force to squash it, a spring will shorten and the amount of shortening is known as the compression.

11.3    There is a relationship between the force applied and the amount of extension or compression of a spring; and this relationship is summed up by Hooke’s Law which states that:
“The force, F, applied to a spring is directly proportional to the spring’s extension or compression, x.”
F = kx, where:
k is the spring’s constant which indicates the stiffness of the spring.
(2011 P1 Q15 @ pg 283)
F-x graph is a linear graph which passes through the origin;
k is the gradient of the linear F-x graph.

(2005 P1 Q3 @ pg 3 - on unit of measurement for k constant;
 2005 P1 Q11 @ pg 5 / 2011 P1 Q17 @ pg 284 - comparison on extension of identical springs in different   arrangements)

11.4    If the force F applied is progressively increased, there will come a point (1st point) when the spring's extension or compression x is no longer directly proportional to the force applied F but will still return to its original shape and size if the force applied is removed - that point is call limit of proportionality. Shortly after this point, there will come another point (2nd point) when further force is applied, the spring becomes permanently lengthened or shortened and will not return to its original shape and size:
a.       the 2nd. point is known as the elastic limit of the spring;
b.      the 1st point is the point beyond which the F-x graph deviates from being a linear graph.
c.       a spring that obeys Hooke’s Law is known as an ideal spring - it only behaves so up to the 1st point, the limit of proportionality.

11.5    When a force is applied from zero to F to lengthen or shorten a spring by x (metres):
a.       the average amount of force moves over the distance of x is: F / 2 (newtons, N)
b.      thus, the work done W by the force to bring about extension or compression x is:
W = (F/2) (x) (which is the triangular area under the F-x linear graph)
c.   the work done W is converted into elastic potential energy Ep. Thus,
W = Ep = (1/2) F x; or
Ep = (1/2) (Kx) x (since F = kx, from Hooke’s Law)
Ep = (1/2) k x^2

11.6    Nitinol is a special elastic material: When deformed, Nitinol has the ability to return to its original shape through heating – it has shape memory! Its possible uses include: springs in car suspension systems, frames and panels of car body. Nitinol is an alloy that contains nickel and titanium metal.

11.7    Applications of Spring:
a.       As shock absorber and vibration damper in vehicles when moving over uneven surface;
b.      As automatic door-closing device: After the force that opens the door is released, the elastic potential energy built-up in the spring will push and close the door.

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