Linear Motion

 Kinematics - Linear Motion

1.1  Linear motion and Non-Linear Motion:

·        Linear motion is motion in a straight line – For examples: A passenger is carried by an escalator; or an athlete running a 100 m race; whereas

·        Non-linear motion is motion not in a straight line - A top spinning; or the earth orbiting the Sun.

1.2  Kinematics and Dynamics:

·        Kinematics is the study of the motion of an object without considering the force acting on it. Therefore, the equations of motion do not have force F as a variable in them.

·        Dynamics is the study of motion and the forces acting on the object.

1.3  The physical quantities involved in linear motion:

·        Distance and Displacement

·        Speed and Velocity

·        Acceleration and Deceleration

·        Time

The Physical Quantities Explained:

Suppose an object at point O moves east 100 metres (m) to point A in 14 seconds (s); then it immediately moves west from point A to point B 40 m away in 8 s:

·        Distance:

o       The total distance moved by the object = (100 + 40) m = 140 m;

o       Distance is the total length of the path travelled by an object in motion from 1 position to another position;

o       Distance is a scalar quantity – it does not take into account the direction of motion.

o       In linear motion: Distance = Magnitude of Displacement (since direction of motion remains constant)

·        Displacement:

o       The total displacement of the object from point O = (100 – 40) m = 60 m east of point O;

o       Displacement is a measure of how far and the direction in which an object has been displaced from a reference point (e.g. original position) due to the motion

o       Displacement measures the straight-line distance and the direction between the initial position and the final position of an object due to the motion.

o       Displacement is a vector quantity because it has both the magnitude and the direction.

·        Speed:

o       The object’s Average Speed from O to B = 140 m / 22 s = 6.36 m s^-1:

§         Its Av. Speed from O to A = 100 m / 14 s = 7.14 m/s

§         Its Av. Speed from A to B = 40 m / 8 s = 5 m/s

o       Thus, Average Speed, v, = Total Distance Travelled, s (m) / Time Taken, t (s).

o       Constant Speed: If an object moves equal distances in equal time intervals, then it is moving with constant speed – otherwise, it is moving with non-uniform speed.

o       Speed is a measure of how fast an object moves – the rate of distance travelled in the motion.

o       Speed is a scalar quantity – it measures only the magnitude of distance moved over time with no regard to direction of motion.

o       In linear motion: Speed = Magnitude of Velocity (since the direction of motion remains constant)

·        Velocity:

o       Its average velocity from O to B = (100 – 40) m / 22 s or 2.73 m/s  due east of O:

§         Its Av. Velocity from O to A = 100 m / 14 s = 7.14 m/s due east of O;

§         Its Av. Velocity from A to B = - 40 m / 8 s = - 5 m/s (west of A)

o       Thus, Average Velocity, v = Total Displacement, s (m) / Time Taken, t (s).

o       Constant Velocity v means equal displacements in equal time intervals.

o       Constant Velocity: If an object moves with equal displacement in equal time intervals, then it is moving with constant velocity – otherwise, it is moving with non-uniform velocity.

o       Velocity is a measure of how fast an object is displaced (the rate of displacement of the object) from a reference position (its initial position).

o       Velocity, unlike speed, is a vector quantity because it measures both the magnitude and direction of displacement due to motion.

o       A change in velocity means either a change in speed or a change in direction (or both) of motion.

·        Speed and Velocity in Linear Motion:

o       In a linear motion, the direction of motion remains unchanged: The speed of linear motion equals to the magnitude of velocity since the distance travelled per unit time in a linear motion is the same as the displacement per unit time along the straight line. Thus, in linear motion:

v = v = Total distance travelled, s (m) / Time taken, t (s)

o       In linear motion, a change in velocity can only means a change in the speed with no change in direction of motion and this change of speed can only be due to acceleration (increase in speed over time) or deceleration (decrease in speed over time).

·        Acceleration and Deceleration:

o       Acceleration is defined as the rate of change of velocity (or rate of change of speed, in linear motion):

Acceleration, a = Change in Velocity / Time Taken

                         = (Final Velocity, v – Initial Velocity, u) / Time, t

                      a = (v – u) / t …1st Equation of Motion

o       Acceleration is positive if initial velocity, u, increases with time, t, to final velocity, v: that is, if v > u, a is +ve. 

o       Conversely, acceleration is negative if velocity decreases to final velocity, v, from initial velocity, u: that is, if v < u, a is –ve, in which case, it is known as deceleration or retardation.

o       Constant (or uniform) acceleration:

·        If an object moves with equal change in velocity (or speed in linear motion) in equal time intervals, then it is moving with constant acceleration – otherwise, it is moving with non-uniform acceleration.

·        An object moves with constant acceleration, a, for time, t, will change the initial velocity, u, to final velocity, v, as follows:

v = u + at …2nd Equation of Motion; Therefore,

t = (v – u) / a …3rd. Equation of Motion 

o       Zero acceleration: If velocity is constant or uniform (or, speed is constant or uniform in the case of linear motion), the acceleration is zero since change in velocity (or, speed) is zero.

o       Acceleration or deceleration is a vector quantity with SI unit metre per second per second, m s^-2.

o       Equations of Motion are derived on the basis of constant or zero acceleration – please see later.

·        Time (Scalar Quantity in second (s)):

·        Time interval is an important quantity in the study of motion – it is measured by a stopwatch or by the use of “Ticker-Timer”.

·        There are 2 types of stopwatches depending of the accuracy needed:

Accuracy Needed              Type of Stopwatch Used

0.1 s ~ 0.2 s                       Analogue (mechanically-operated)

0.01 s                                 Digital (electronically-operated)

·        Apart from stopwatch, another device that we use to measure time interval of linear motion is the “ticker-timer” – please see the ensuing.

·        We know that an object that moves with constant acceleration, a, for time, t, will change its initial velocity, u, to final velocity, v:

v = u + at … 2nd Equation of Motion (as in foregoing);

         Therefore, time, t = (v – u) / a …3rd Equation of Motion

·        Ticker-Timer

Ticker-timer is a device that can be used to determine a number of quantities relating to linear motion of an object (including time interval), namely:

a)      Time Interval of the Motion

b)      Displacement of the Object

c)      Velocity of the Object

d)      Acceleration of the Object

e)      Type of Motion of the Object

·        The above quantities can be determined because ticker-timer has a metal strip with a pin that vibrates up and down at 50 Hz (which is the frequency of the 12 V or 6 V ac power supply). Each time the pin moves down at the interval of 1/50 seconds (or 0.02 s), it makes a dot on the pre-carbonated ticker tape which passes beneath it as the tape is pulled by the moving object to which the tape is attached. Thus,

a)      The time interval which elapses between successive dots is 1/50 second or 0.02 s (T = 1/f) – therefore, the time interval of the motion between any 2 points / dots can be determined by: Multiplying the number of dots after the 1^st point / dot until the other point / dot by 1/50 s; Therefore:

1 dot-space (1-tick space) = 1 x 0.02 s = 0.02 s

2 dot-space (2-tick of time) = 2 x 0.02 s = 0.04 s

5 dot-space (5-tick of time) = 5 x 0.02 s = 0.1 s

10 dot-space (10-tick space) = 10 x 0.02 s = 0.2 s

b)      Displacement of the object between any 2 points can also be determined by measuring the distance between the 2 points on the ticker tape

c)      Velocity:

·        The velocity of the objects between 2 points / dots is the displacement between the 2 points / dots over the number of tick-time over the 2 points / dots;

·        The average velocity between any 2 points is the total displacement between the 2 points of the tape over the total time intervals elapsed (i.e. the number of tick-time) between the 2 points.

·        Constant (or, Uniform) Velocity: When the ticker tape shows equal displacement over equal time intervals, the object is moving linearly with constant or uniform velocity.

d)      Acceleration or Deceleration:

·        The acceleration (or deceleration) of the objects between 2 successive intervals of motion on the ticker-tape is the change in velocities between the 2 intervals over the time between the mid-points of the 2 intervals. The velocities can be determined as in the foregoing.

·        The average acceleration (or deceleration) of a moving object between any 2 intervals of motion (note: intervals not points) can be determined by comparing the corresponding velocities for the 2 intervals over the time between the mid-points of the 2 intervals - increasing velocity means acceleration and decreasing velocity means deceleration.

·        Constant (or uniform) Acceleration: When the ticker tape shows equal increase (or decrease) in velocity over equal time interval, the object is moving linearly with constant acceleration (or constant deceleration, which ever is appliacable)

e)      From the foregoing, it is therefore clear: That the type of motion of an object – whether it is moving linearly at constant velocity (zero acceleration), at irregular acceleration or deceleration or at constant acceleration– can be seen and determined from the pattern of the dots on the ticker tape.

·        Equations of Motion

We have already learnt from the foregoing 3 basic equations of linear motion where the object moves with constant (or uniform) acceleration, a, with initial velocity, u, final velocity, v, for time, t:

1)      a = (v – u) / t …1^st  Equation of Motion

2)      v = u + at …2^nd Equation of Motion; and

3)      t = (v – u) / a …3^rd. Equation of Motion 3

The 4^th Equation of Motion is about displacement or distance travelled, s, of the object under the same state of motion:

4)      s = Average Velocity x Time Taken

      s = ½ (u + v)t … 4^th Equation of Motion

     

The 5^th and 6^th Equations of Motion are obtained from the 4^th Equation [s = ½ (u + v)t] by substituting the 2^nd Equation (v = u + at) and the 3^rd equation [t = (v - u)/a] respectively into the 4^th Equation. Thus,

5)   s = ½ (u + v)t…from 4^th Equation

s = ½ [u + (u + at)]t …(substitute 2^nd equation into the 4^th)

s = ½ [2u + at]t

s = ut + ½ at^2 …5^th Equation of Motion

(or, s = 1/2 (g) (t^2), free fall under gravity: 2005 P1 Q4. pg 3)

6)      s = ½ (u + v)t…from 4^th Equation

      s = ½ (u + v)(v – u)/a …(substitute 3^nd equation into the 4^th)

      s = ½ (v^2 – u^2)/a …(Form 3 algebraic expansion)

2as = v^2 – u^2

 v^2 = u^2 + 2as … 6^th Equation of Motion

Science is best learnt by understanding rather than by memorizing formulae (rote learnig). I believe, by understanding alone, you should be able to easily derive the first 4 equations of motion. In solving a kinematics problem, just ask yourself this: What is the variable that the question wants me to find the value?

·        If acceleration, a: then, use 1^st equation;

·        If final velocity, v: use 2^nd equation;

·        If time, t: use 3^rd equation;

·        If displacement, s: use 4^th equation.

At times, you need to find the value of another variable before you can solve your problem using the above simplified method – ample examples in my handouts.

Summary: SPM Past Year Questions on "Forces and Motion" (2005~2012)

Updated as at 4/8/2013:

SPM Physics Past Year Questions on “Forces & Motion” (2005 ~ 2012):

Year	Paper 1 (1hr 15min) (50 Qs – Do All)	Paper 2 (2hr 30min) Section A: (90 min, 60 pts) 8 Qs – Do All Section B: (30 min, 20 pts) 2 Qs – Do 1Q Section C: (30 min, 20 pts) 2 Qs – Do 1Q	Paper 3 (1hr 30min) Section A: (advice: 1 hr) 2 Qs (28 pts) Section B: (30 min, 12 pts) 2 Qs – Do 1Q
2012	8 Qs: Q4 – Q11 @ Pgs 332 ~ 334	2 Qs: Q5 @ pg 350 Q11 @ pg 363	NIL
2011	7Qs: Q4~10, 15, 17 Pgs 280~282	2 Qs: A.Q1 (Pg. 294) B.Q9 (Pg. 308)	NIL
2010	9 Qs: Q3~11 Pgs 230~233	2 Qs: A.Q5 (Pg 251) C.Q11 (Pg 262)	1 Q: B.Q1 (Pg.276)
2009	9 Qs: Q3~11 Pgs 186~188	1 Q: A. Q2 (Pg 201)	1 Q: B.Q3 (Pg 227)
2008	10 Qs: Q3~12 Pgs 140~143	2 Qs: A. Q5 (Pg. 158) B.Q9 (Pg. 165)	1 Q: A. Q2 (Pg 180)
2007	8 Qs: Q4~11 Pgs 92~94	3 Qs: A. Q4 (Pg.109) B. Q9(b)(c) (Pgs.117~118) C. Q11 (Pg 121)	1 Q: A. Q2 (Pg 132)
2006	7 Qs: Q2, Q4~10 Pgs 48~50	1 Q: A.Q6 (Pgs 68 ~69)	1 Q: B. Q3 (Pg 86)
2005	9 Qs: Q3~11 Pgs 3~5	1 Q: A.Q6 (Pg 22)	1 Q: A. Q1 (Pg 36)

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 (1^st 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 2^nd 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 2^nd 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:

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

          4.3.3 Situations of Balanced Forces with Static Equilibrium

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

         4.3.4 Situations of Balanced Forces with Dynamic Equilibrium

                  4.3.4.1 Object (Parachutist, etc) Falling at Terminal Velocity
                  4.3.4.2 Object Rolling Down Friction-Compensated Incline at Constant Velocity
                  4.3.4.3 Plane Cruising Horizontally at Constant Velocity
                  4.3.4.4 Car Moving Horizontally at Constant Velocity
                  4.3.4.5 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:

                  4.3.2.1 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)

                  4.3.2.2 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 3^rd 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.