Vectors and scalars Scalar has magnitude only e.g. 10 mph Vector has magnitude and direction e.g. 10 mph in a NE direction Can be represented by a line with and arrow on it drawn to scale 10 mph

Vectors and scalars Vectors Scalars Distance and displacement Displacement is distance travelled in a definite direction

Nottingham (N) London (L) Speed and velocity distance speed time displacement velocity time

Units distance (metres) time (sec) speed (m/s) x s t

Average velocity total displacement x average velocity time t final velocity initial velocity Average velocity 2 Distance-time graphs

Steady speed gradient = a/b Sign of gradient indicates direction of motion x a b

time Velocity time graphs Uniform velocity vel acceleration a= change in velocity time uniform acceleration gradient = a/b Sign of gradient indicates

acceleration or deceleration a b time Area under a velocity time graph Area under graph

indicates the displacement (distance travelled) vel v u t

time Acceleration-time graphs Gradient at any point gives rate of change of acceleration at that time Area under graph represents change in velocity experienced

accel a t time Equations of motion v u at (u v)t

x 2 1 2 x ut at 2 2 2 v u 2ax Where u = initial velocity (m/s)

v = final velocity (m/s) x = displacement (m) t = time (s) a = acceleration (m/s2) Acceleration under gravity Sign convention Acceleration is always downwards Accel (a) is +ve when moving down and a is ve when moving up Accel (a) = g (the accel due to gravity, in

magnitude) Projectile motion Horizontal motion is independent of vertical motion Constant velocity for horizontal component of motion Constant acceleration for vertical component of motion Equations of projectile motion

An object in freefall: moves at a constant horizontal (x) velocity ax = 0 moves at a constant vertical (y) acceleration.

ay = g The following equations can therefore be applied. Can you see how they have been derived? x = vxt constant x velocity vy = uy + gt y = uyt + gt

vy2 = uy2 + 2gy 2 y= uy + vy 2 t suvat equations

for uy and vy with a = g Height of a projectile A tennis player hits a volley just above ground level, in a direction perpendicular to the net. The ball leaves her racquet at 8.2 ms-1 at an angle of 34 to the horizontal. Will the ball clear the net if it is 2.3 m away and 95 cm high at this point? What assumptions should you

make to solve this problem? no air resistance no spin

initial height is zero. Height of a projectile We need to calculate the value of y at x = 2.3 m and determine whether or not it is greater than 0.95 m. What are the relevant equations of motion? 8.2 ms-1 x = vXt

y = uyt + gt2 0.95 m 34 2.3 m First, use the x equation to calculate t when x is 2.3. 2.3 = 8.2 cos34 t t = 0.34 s

Height of a projectile So the ball reaches x = 2.3 m when t = 0.34 s. Now substitute this value of t into the y equation to find y, and determine whether or not it is greater than 0.95 m. y = uyt + gt2 8.2 ms-1

0.95 m 35 2.3 m y = ((8.2 sin34) 0.34) + ( -9.81 0.342) y = 0.99 m So y is greater than 0.95 and the ball clears the net! Vector addition and resolution Vectors have Magnitude and direction

Can be represented by lines drown to scale with arrows on them Can be added to find the resultant Resultant can be found by drawing or calculation Resultant of 2 vectors 1N

Resultant = 4N 3N + = Resultant = -2N + =

Vector addition and resolution Vector addition Resultant can be found by drawing or calculation or Pythagoras Theorem

Resolving vectors Resolving vectors Resolving vectors Resolving vectors The vector F can be resolved into 2

components at right angles to each other X component = F cos Y component = F sin Tan = F sin F cos F sin F

F cos Types of Force Forces acting at a distance Gravitational Electrostatic

Electromagnetic Nuclear Contact forces Frictional forces Reaction forces Forces Thrust Weight

Lift, Thrust, Drag Reaction and weight Action and Reaction Equal and opposite Introduction to turning forces Forces can make things accelerate. They can also make things rotate.

Whats wrong with these pictures? too short! too short! wrong place! We know instinctively that we need to apply a force at a large

distance from the pivot for it to be effective. Moments and torque A moment is the turning effect of a force. It can also be called a torque. Torque is given the symbol (the Greek the Greek letter tau). Its units are newton metres (the Greek Nm). pivot d F

F the force applied in newtons (N). d the perpendicular distance (in m) between the pivot and the line of action of the force. = F d Moments for non-perpendicular distance Couples and torques A couple is a pair of forces acting on a body that are of

equal magnitude and opposite direction, acting parallel to one another, but not along the same line. Forces acting in this way produce a turning force or moment. The torque of a couple is the rotation force or moment produced. F d F The forces on this beam are

a couple, producing a moment or torque, which will cause the beam to rotate. The torque of a couple There is a formula specifically for finding the torque of a couple. A point P is chosen arbitrarily. Take moments about P. F P

x dx d F total moment = Fx + F(d x) = Fx + Fd Fx = Fd

perpendicular distance torque of a couple = force between lines of action of the forces Centres of mass and gravity The centre of gravity of an object is a point where the entire weight of the object seems to act. The centre of mass of an object is a point where the entire mass of the object seems to be concentrated. In a uniform gravitational field the centre of mass is in the same place as the centre of gravity.

An alternative definition is that the centre of mass or centre of gravity of an object is the point through which a single force has no turning effect on the body. Equilibrium A body persists in equilibrium if no net force or moment acts on it. Forces and moments are balanced. Newtons first law states that a body persists in its state of rest or of uniform motion unless acted upon by an external unbalanced force. Bodies in equilibrium are therefore bodies that are at rest or

moving at constant velocity (uniform motion). F1 F2 F2 F1 equilibrium Balanced moments If the total clockwise moment on an object is balanced by the total anticlockwise moment, then the object will not rotate.

Provided that there are no other unbalanced forces on it, the object will be in equilibrium, like the beam below: 4m 3N 2m 6N total anticlockwise moments = total clockwise moments 34=62 12 Nm = 12 Nm

The principle of moments The principle of moments states that (for a body in equilibrium): total clockwise moments = total anticlockwise moments

This principle can be used in calculations: 5m What is d? d 4 5 = 6d 4N

6N 20 = 6d d = 20 / 6 d = 3.3 m Can you make the beam balance? Human forearm The principle of moments can

be used to find out the force, F, that the biceps need to apply to the forearm in order to carry a certain weight. When the weight is held static, the system is in equilibrium. Taking moments about the elbow joint: 4F = (16 20) + (35 60) 4F = 2420 F = 605 N

weight of arm = 20 N 60 N 4 cm F schematic diagram

16 cm 20 N 35 cm 60 N Centre of gravity and equilibrium