Mechanics L2 NCEA Achievement Standard 2.4 Text Book reference: Chapters 2-8

Scalars and Vectors A scalar quantity is one that has a size (or magnitude) only Eg. Mass, energy, time A vector quantity is one that has a size and a direction Eg. Force, velocity, momentum

Motion Distance(scalar) How far in total the object has moved Symbol: d Unit: m Displacement(vector) How far the object ends up from it’s starting position Symbol:d or s Unit: m

Motion Speed(Scalar) How fast an object is travelling Symbol:s Unit: ms -1 Velocity(Vector) The speed and direction that an object is travelling Symbol: v Unit: ms -1

Motion Velocity is calculated by Where d is displacement, t is time and  means “the change in”. Velocity may refer to either average velocity or instantaneous velocity. Constant velocity means that neither the speed nor the direction of the objects motion is changing.

Motion Acceleration can be calculated by Acceleration(Vector) The rate at which the velocity of an object is changing Symbol: a Unit: ms -2

Motion Acceleration is used to describe motion where the object slows down as well as when it speeds up. Sometimes the word deceleration is used. Acceleration is given a negative value when the object is slowing down. Objects are accelerating when their direction changes, even though their speed may remain constant. (Do Page 27Questions 2A)

Graphs of Motion Distance / diplacement versus time… A=Constant velocity (slow) B=Constant velocity (faster) C=Stopped D=Constant velocity (backward) E=Constant velocity (backward past starting point) A B C D E Time (s) Displacement(m)

Graphs of Motion Speed / velocity versus time… A=Constant acceleration (low) B=Constant acceleration (high) C=Constant velocity D=Constant deceleration to stop E=Constant acceleration in opposite direction A B C D E Time (s) Velocity (ms -1 ) (Do Page 32 Questions2B)

Kinematic Equations To solve problems involving objects moving in straight lines with constant acceleration. Terms used: d=distance/displacement (m) v i =initial velocity (ms -1 ) v f =final velocity (ms -1 ) a=acceleration (ms -2 ) t=time (s)

Kinematic Equations (Do Page 33 Questions 2C)

Vectors A vector is drawn as a straight, arrowed line. The arrow points in the direction of the vector The length of the line represents the size of the quantity

Vectors Vectors can be multiplied or divided by a scalar This will change the length of the vector A negative scalar will reverse the direction Eg Force F= So 2F= 1/2F=-3F=

Vectors Vectors can be added together. This is done by drawing them “head to tail”. The result is a vector called a resultant. The resultant has the same effect as the 2 vectors combined. The order in which they are added does not matter. Eg d 1 +d 2 d1d1 d2d2 d 1 + d 2

Vectors Vectors can be subtracted. This is done by adding a negative vector Order does matter. Eg. v 1 - v 2 v1v1 v2v2 v 1 - v 2 -v 2

Vectors Vectors can be resolved into components. This is done using SOHCAHTOA and/or a 2 +b 2 =c 2 Horizontal component Vertical Component F 40° (Do Page 43 Questions 3A)

Relative Velocity The velocity of one object in relation to another object. The velocity an object appears to move at may change if the object measuring is also moving. The velocity of B relative to A can be calculated by doing this vector subtraction…. (Do Page 49 Questions 3B)

Forces A force causes the motion or shape of an object to change. Force is a vector quantity so must have both a size and a direction Force is measured in Newtons N. A resultant (or net) force is produced when 2 or more forces act on an object. These forces can be added to find the resultant.

Forces Newtons First Law Of Motion: An object will remain in it’s current state of motion until a force acts to change it. Newton’s Second Law Of Motion: The acceleration of an object is proportional to the net force applied. Law 2 can be written like this for short:

Forces Newton’s Third Law Of Motion: For every action there is an equal and opposite reaction.

Forces Friction: Friction occurs when two surfaces move past each other. One of these surfaces could be air – eg air resistance is a frictional force. Friction is a force that always opposes the direction of the motion. Friction is sometimes called: drag, water resistance, air resistance or the retarding force.

Forces Tension: This is the force that occurs in connecting strings and ropes Tension pulls in both directions along the string or rope. Weight: This is the force of gravity pulling downwards on an object. Weight can be calculated by: g is acceleration due to gravity and has a value of 10ms -2 on Earth (Do Page 55 Questions 4A)

Torque Torque causes things to spin. Symbol:  (Greek letter Tau) Units: Nm The size of a torque depends on the size of the force and the perpendicular distance from the pivot to where the force is applied.

Equilibrium An object is at equilibrium if it is at rest or moving uniformly (First Law) Two conditions apply: All the forces acting on the object must add to zero, All the torques acting on the object must add to zero. (Do page 63 Questions 4B)

Momentum The amount of “ooomph” an object has. Momentum depends on the mass of an object and it’s velocity. Symbol: p Unit: kgms -1 Momentum is a vector. Momenum can be calculated using:

Momentum If a force acts on an object, it’s momentum will change. The change in momentum can be calculated by subtracting vectors. Change in momentum =final momentum – initial momentum.

Impulse When a resultant force acts on an object, the amount it changes the object’s momentum by depends on how long the force acts for. The force multiplied by the time it acts for is called impulse. Units: Ns Impulse equals the change in momentum. (Do Page 73 Questions 5A)

Conservation of Momentum The conservation of momentum principle states: Momentum is conserved in collisions and explosions as long as there is no net external force acting. This means the momentum before equals the momentum after.

Conservation of Momentum The same principle applies in 2 dimensions. The vector representing the sum of the momentums before must be the same vector as the one representing the sums after. Do Page 79 Questions 5B

Motion due to Gravity All objects accelerate towards the ground at (-) 10ms -2 because of gravity when dropped. This acceleration is fairly constant at the Earth’s surface, but varies at great altitudes or on other planets. Gravity is always an attractive force unlike magnetism or electric forces. Do Page 86 Questions 6A

Projectile Motion Projectile motion is a parabolic shaped motion experienced by moving objects that have only the force due to gravity acting on them. Eg. Bullets,shotputs,netballs, water jets

Projectile Motion When dealing with projectiles, the horizontal and vertical components are treated separately. The horizontal motion is constant velocity (as there are no forces acting in this direction). The vertical motion is constant acceleration of 10ms -2 due to the force of gravity. Kinematic Equations are used to solve problems. Do Page 89 Questions 6B

Circular Motion Period of Rotation T - time it takes to make one rotation (revolution, cycle) Measured in seconds s. Frequency f – number of rotations completed per second. Measured in Hertz Hz or s -1 T and f are inverses of each other.

Circular Motion Circumference – distance travelled in one rotation (m) The speed of an object moving in a circle can be calculated by:

Circular Motion An object moving in a circle may be travelling at constant speed, but because its direction is always changing, its velocity is changing…. If velocity is changing, the object is accelerating…. If an object is accelerating, there must be a net force acting on it….

Circular Motion The force acting on an object in circular motion is in towards the centre of the circle, changing the objects direction but not its speed. This is called centripetal force. This force causes a centripetal acceleration towards the centre of the circle.

Circular Motion Centripetal force and acceleration can be calculated using the following formulae: v=speed(ms -1 ) r=radius of motion Do Page 98 Questions 7A

Energy The three kinds of mechanical energy are: kinetic, gravitational and elastic. Energy E is measured in Joules J Work W is the process of transforming energy from one kind to another. d is the distance moved in the direction of the force.

Energy If an object is lifted against gravity, work is done transforming chemical energy(muscles) into gravitational potential energy. The force needed is the weight force of the object, the distance moved is the change in height:

Energy Any moving object has kinetic energy. Doubling the speed increases the energy by four. (Squared relationship) When moving objects stop, this energy is transformed into other forms, eg sound, heat

Power Power P is the rate at which work is done. Measured in Watts W (or Js -1 )

Conservation of Energy The conservation of energy principle states: Energy cannot be created or destroyed, only transformed from one kind to another.

Energy Efficiency Often some of the forms it is transformed into are not useful. The energy is “lost” to us The efficiency of an object is a measure of the ratio of input energy to useful output energy

Springs Energy can be stored in a spring as elastic potential energy. Hookes Law: F=kx F=force k=spring constant (Nm -1 ) – a measure of how stiff or soft a spring is. x=extension (m) – the amount a spring is stretched or compressed when the force is applied.

Springs Hooke’s Law as a graph: Gradient = k Extension (m) Force (N) Area under graph = energy stored in spring

Springs Elastic potential energy can be found by calculating the area under a Hooke’s Law graph. Do Page 107 Questions 8A

A person sits in a car with a suspension of spring constant 10 4 Nm -1. If the suspension is compressed 1.0 cm how much energy is stored in the springs?

E p =1/2kx 2 =0.5 x 10 4 x (0.01) 2 =0.5 x 10 4 x (0.01) 2 =0.5 J =0.5 J