P4- Mechanics and Materials

scalars and vectors

vector: any physical quantity which has direction as well as magnitude.( displacement, velocity , force , acceleration)

Scalar: a physical quantity with only magnitude. ( distance, speed , mass)

any vector can be represented an arrow. the length of the arrow represents the magnitude of the vector quantity. the direction of the arrow gives the direction of the vector.

Adding vectors

resolving vectors


this is where you are given the resultant vector and you have to split it up into horizontal and vertical components

Vx= Vcosθ


object on a slope


weight = resultant force acting vertically down



resolve to find the component parallel and perpendicular to the slope

force of an equilibrium

if an object is in equilibrium all the forces acting on it are balanced and therefore canceling each other out so that there is no resultant force ⇒, therefore, an object in equilibrium is either at rest or traveling at a constant velocity.


forcing acting on an object in equilibrium form a closed loop. when you draw them from tip-to-tail.

parallelogram rule 

forces f1,f2 andd f3 acting on P are in equilibrium so any two should give a resultant equal and opposite to the third 





moment- the turning effect if a force

the force multiplied by the perpendicular distance from the line of the action of the force

moment(Nm) = force(N) x perpendicular distance (m)

M= f x d

 moments can be increased by

  • increasing perpendicular distance
  • increasing size of the force

the principle of moments states that for a body to be in equilibrium the sum of the clockwise moments about axis point must equal the sum of the anticlockwise moment about the same point

Couple: is a pair of forces of equal size which acts parallel to each other but in opposite direction producing a turning effect called a torque

T(Nm) =2fd

the center of mass of the body is the point through which a single force on the body has no turning effect. for uniform regular solids this is at the center

motion along straight line

displacement: the distance moved in a stated direction(vector)

speed: the distance traveled per unit time(scalar)

velocity: displacement traveled per unit time (vector)

acceleration: change in speed per unit time (vector)

speed= distance / time

displacement time graph

velocity time graph

acceleration time graph


  • shows how acceleration changed over time
  • area under graph gives objects change in velocity
  • a=0 the object is moving at constant velocity
  • a negative acceleration is deacceleration


displacement -s-m

initial velocity -u-ms1

final velocity -v-ms1

acceleration -a-ms2

time taken -t-sec

free fall: when object is under the gravitational force only


projectile motion

a projectile: an object that has been projected. a force acts on it to make it move and it is subjected to constant force while it is moving.

the motion in horizontal and vertical direction are independent of each other.

the horizontal motion is not subjected to any acceleration and therefore velocity remains constant.

then use the equation to find unknown variable S/E

projection at angle:

the starting velocity can be split into horizontal and verticle components




horizontal remains constant                                     verticle motion

= ucosθ                                                                     =usinθ                a=-9.8

the key component that links verticle and horizontal motion is time.

effect of air resistance

  • the projectile moving through air experience a force that drags upon it because of resistance of air it  passes through.
  • the drag force is partly caused by the friction between the layers of air near the objects surface , where the air flows over it.
  • the drag force
    • act in opposite direction to the direction of motion of the projectile and it increases as the projectile speed increases.
    • has a horizontal component that reduces both the horizontal speed of the projectile and its range.
    • reduce the maximum height of the projectile if its initial direction is above the horizontal makes its decent then accert.

Newtons Laws of motion

Newton’s first law of motion

a moving object has kinetic energy. the energy will not lost unless force is applied to slow the object down.

“object moving in a space where no force acting on the them will continue to move in a straight line at a constant velocity until a force causes them to change speed or direction”

Newton’s second law of motion

the acceleration on an object is proportional to resultant force acting on it.

Newton”s Third law of motion

“If an object A exerts a force on object B, Object B will exert equal and opposite force on abject A.”


Momentum is always constant.

total momentum before = total momentum after

Impulse (Ns) = Force (N) × time (s)

= change in momentum

Elastic Collision

Total kinetic energy after = total kinetic energy before the collision

Inelastic collision

total kinetic energy after < total kinetic energy before collision

vehicle safety features like seat belts and airbags works to increase the amount of  time it takes to slow down.

F = m(v-u)/t

This reduce force of impact on the people.

Work energy and power

work done- energy transferred by means of force.

work done (J) = Force (N) * distance moved in direction of force (d)

W= f*d

Power- the rate of transfer of energy (rate of the work done)

Power = work done (J) / Time (s)  =  W/t   = d/t =fV

p= Δw/ Δt  = FV

if the distance is not in the direction of the force.

Effiency = (useful work output / total energy input) * 100

Conversation of energy

Energy cannot be created or destroyed only conserved.

potential energy (gravitational = mass (kg) 9.81 * height (m)

Ep = mgh

Kinetic energy (J) =1/2 * mass (kg) * (velocity (ms-1))2

Bulk properties of solid

Density-density of substance is defined as its mass per unit volume.

Hooke’s law – states that the extension of the spring is directly propotional to force applied to it.

Where k is spring constant.

a k increase the spring gets stronger.

k= f / ΔL = gradient on a force length graph

the energy stored by the spring is represented by the area under the graph.

  1. The limit of proportionality -a point beyond which behavior no longer confirms the hook’s law.
  2. he elastic limit- a point beyond which spring will not return to its original shape.

Tensile Stress– is defined as the forced applied per unit cross sectional area of a wire.

Tensile strength– is defined as the extension per unit length that the wire undergoes when tensile force is applied.

Young’s modulus – a measure of the stiffness of the material.

  – the ration of stress to strain

Stress-Strain graph

elastic deformation- deformation after which a material will return to its original shape.

Plastic deformation- deformation that is permanent after the removal of load.

Breaking stress-  The maximum stress experienced by a stressed sample of a material before the sample fracture.

also called the ultimate tensile strength. (vts)


  • linear stress- strain graph
  • don’t undergo plastic deformation
  • high uts
  • example :glass


  • undergo significant plastic deformation before fracture.
  • can be drawn into wires.
  • Example: copper


  • made from polymer
  • have non-linear stress-strain graph
  • Example: rubber, fishing lice