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PHYSICS ( International AS level AQA OXFORD Physics ) (copy)
MECHANICS AND MATERIALS
SCALARS AND VECTORS
SCALARS AND VECTORS MEANINGS AND EXAMPLES
SCALAR: A scalar is a quantity that has only magnitude and no direction. It is described by a single numerical value.
VECTOR: A vector is a quantity that has both magnitude and direction.
MAGNITUDE: the size or quantity of physical quantity such as length, mass or force, without considering its direction.
EXAMPLES:
COMBINING VECTORS
PARALLEL AND ANTI-PARALLEL
Add the forces if they are in the same direction.
Subtract the forces if they are different directions.
ADDING PERPENDICULAR VECTORS
Vectors are represented by an arrow.
o The arrowhead indicates the direction of the vector.
o The length of the arrow represents the magnitude.
Vectors can be combined by adding them to produce the resultant vector.
o The resultant vector is sometimes known as the “net“ vector.
There are two methods that can be used to add vectors.
o Calculations - if the vectors are perpendicular.
o Scale drawing- if the vectors are not perpendicular.
Resultant vector is found using Pythagoras theorem.
-Magnitude of the resultant vector, ‘R’
Add two vectors. a & b
Link the vectors.
Form the resultant vector from Linking the tail of ‘a‘ to the head of ‘b‘
Calculate ‘R‘ using Pythagoras’ theorem.
RESOLVING VECTORS
Vector Resolution: The process of breaking down a vector into its horizontal and vertical components.
Vertical component: F sinθ
Horizontal component: F cosθ
ADDING NON PERPENDICULAR VECTORS
Arrange the vectors tips to the tail and draw the result.
Calculate using the sine and cosine rules.
3.Calculation by vector resolution.
EQUILIBRIUM OF FORCES
DIFFERENT FORCES ACTING ON A BODY:
WEIGHT: The gravitational force acting on object through its centre of mass.
FRICTION: The force that rises when two surfaces rub against each other.
DRAG/AIR RESISTANCE: The resistive force on an object travelling through a fluid.
TENSION: The force within a stretched cable or rope
UPTHRUST: An upwards buoyancy force acting on an object when it is immersed in a fluid
NORMAL CONTACT/REACTION FORCE: A force arising when one object rests against another object.
Buoyancy is the tendency of an object to float in a fluid.
EQUILIBRIUM: a state of balance between opposing forces or actions that is either static or dynamic.
NO RESULTANT FORCE
For forces to be in equilibrium, the sum of the forces acting on the object is zero.
NET FORCE: the sum of all the forces acting on an object.
The vector addition triangle will return back to the initial starting point when there is no resultant force.
EXAMPLE: If a spider is hanging in equilibrium, then when the forces are arranged as a scale diagram, they should form an enclosed shape.
QUESTION: A child of weight, W, on a swing is at rest due to the swing seat being pulled to the side by a horizontal force, F1. The rope supporting the seat is at angle, θ. Assume the child at equilibrium and the seat has no weight. Write an equation for θ.
Step 1: Identify forces are either horizontal or vertical. Step 2: Identify forces at angle then resolve into horizontal and vertical components. T is in both horizontal and vertical.
Step 3: Equate the horizontal and vertical components.
NO RESULTANT MOMENT
Resultant moment: the sum of all the moments acting on a body (both positive and negative)
Concurrent forces are forces whose lines of action intersect at a single point.
MOMENTS
MOMENT: The rotational effect of a force on a body about a fixed point or axis.
Moment = force * perpendicular distance to line of action of force from the point.
A couple is a pair of coplanar forces (meaning they are forces within the same plane), where the two forces are equal in magnitude but act in opposite directions.
To find the moment of a couple, you multiply the one of the forces by the perpendicular distance between the lines of action of the forces.
Moment of a couple = force * perpendicular distance between the lines of action of forces.
The principle of moments states that for an object in equilibrium, the sum of anticlockwise moments about a pivot is equal to the sum of clockwise moments.
CLOCKWISE MOMENTS = ANTICLOCKWISE MOMENTS
(2×50) + (3×35) = (3.5 * F)
100 + 105 = 3.5 * F
205 / 3.5 = 3.5 * F / 3.5
58.6N = F
The centre of mass of an object is the point at which an object’s mass acts.
If an object is described as uniform, it’s centre of mas will exactly at it’s centre.
A uniform beam of 6m and weight 2000N rest on two supports at its ends.
A man of weight 700N stands 1.8m from the right support.
Add arrows to this diagram to show the forces acting on the beam and calculate and their magnitudes.
R1 + R2 = 2700
R2 × 6 = (2000×3) + (700×4.2)
R2 × 6 = 8940
R2 = 1490Nm
R1 + 1490 = 2700
R1 = 1210Nm
A 3m drawbridge consists of a uniform plank of weight 1000N. It is supported in the position shown by a cable and a friction hinge. There are 3 forces acting on the bridge. Draw arrows on this diagram to show the direction of these forces. Calculate the tension in the cable.
1000 cos30 × 1.5 = T cos20 × 2
231.4 = T cos20 × 2
231.4 / cos20 × 2 = T
T = 283.5 N
MOTION ALONG A STRAIGHT LINE
Instantaneous velocity: the velocity of an object under motion at a specific point of time.
Instantaneous speed: a measurement of how fast an object is moving at that particular moment.
This is a displacement time graph for the motion of an object.
The red line shows a tangent drawn at t = 20s.
a) What is the average velocity of the object in the 50s shown?
b) What is the velocity of the object at t = 20s
a) average velocity = 75/50 = 1.5m/s
b) (75-30) / 20 = 2.25
for ALL velocity time graphs:
gradient : acceleration
Area under the line/curve = displacement
This graph shows a velocity time graph for a train journey in straight line.
Calculate the distance travelled and the acceleration for each of the 3 stages of the journey.
d = t x v
a = v / t
1) d = 100 × 20 x (1/2) a = 20 / 100
d= 1000m a = 0.2m/s²
2) d = 150 × 20 a = 20 / 100
d= 3000m a =0
1) d = 30 × 20 x (1/2) a = 20 / 30
d= 300m a = -0.67m/s²
A ball is dropped from a height of 0.7m. This graph show velocity against time for the ball as it bounces.
a) What is the gradient at x?
gradient at x = -9.8 m/s²
b) At what point does the ball reach its maximum height?
maximum height at y
c) What feature of this graph is equal to the height the ball was dropped from?
Drop height = area of triangle
d) What evidence is there that the collisions with the floor are inelastic?
The triangle area are getting smaller. The times between bounces are getting smaller.
PROJECTILE MOTION
Projectile Motion: An object moves along a curved path under the influence of gravity.
The acceleration of the object is always equal to g. The acceleration only affects the vertical motion of the object.
The horizontal velocity of the object is constant because the acceleration of the object does not have a horizontal component.
The motions in the horizontal and vertical directions are independent of each other.
VERTICAL PROJECTION
an object that moves vertically as it has horizontal motion. its acceleration is 9.8 m/s2 downwards. Using the direction code (+) is upwards, (-) is downwards, its displacement, y, and velocity, v, after time (t) are given by v = u - gt
y = ut - ½ gt²
u: initial velocity in the upward direction.
t: time
v: velocity
y: displacement
HORIZONTAL PROJECTION
A stone thrown from a cliff top follows a curved path downwards before it its the water. if its initial projection is horizontal:
Its path through the air becomes steeper and steeper as it drops.
the faster it is projected, the further away it will fall into the sea.
the time taken for it to fall into sea does not depend on how fast it is projected.
Two balls are released at the same time above a level floor such that one ball drops vertically and the other is projected horizontally. Which one hits the floor first?
Both of the balls are falling vertically under the influence of gravity. They both fall from the same height. Therefore, they will hit the ground at the same time. The fact that one is moving horizontally is irrelevant – remember that the x and y motions are completely independent.
NOTE:
The shape of the projectile is important because it affects the drag force and may also cause a lift force in the same way as the cross-sectional shape of an aircraft wing creates a lift force. This happens if the shape of the projectile causes the air to flow faster over the top of the object than underneath it. As a result, the pressure of the air on the top surface is less than that on the bottom surface. This pressure difference causes a lift force on the object.
air resistance increases with speed.
The frequency of air molecules colliding with an object increases with speed.
PROJECTILE PATH
Projectiles move an in a curved path towards the ground.
Their horizontal velocity remains constant.
Their vertical velocity increases according to free fall.
The increase in the vertical velocity with time is the cause of the curved path of the projectile.
VECTOR REPRESENTATION
At any time in a projectiles motion we can find the actual velocity of the projectile.
We resolve the actual velocity into horizontal and vertical components.
Using a vector triangle we see that the actual velocity is given by Pythagoras’ Theorem.
Example: A bullet is fired horizontally at a speed of 300ms−1. What is its actual Velocity 3 seconds after being fired? What angle is it making to horizontal at this time.
Vx = u = 300ms−1
Vy = u + at
u = 0
t = 3
a = 9.81ms-2
Vy = 0+9.81 × 3 = 29.43ms−1
FIRING UPWARDS
projectiles do not always start with purely horizontal motion.
projectiles that are fired at an angle to the horizontal follow a parabolic path.
in this case, the initial velocity can be separated into horizontal and vertical components.
NEWTON’S LAWS OF MOTION
Newtons first law
Every object will remain at rest or in uniform motion unless it is acted on by a resultant force.
Newtons second law
The rate of change of momentum of an object is proportional to the resultant force on it.
Newtons third law
When two objects interact, they exert equal and opposite forces on one another.
Newtons law of gravitation
The magnitude of the gravitational force between two masses is directly proportional to the product of the masses, and is inversely proportional to the square of the distance between them, (where the distance is measured between the two centres of the masses).
F: Force
G: Gravitational constant
r: distance between centres of the masses
MOMENTUM
Momentum: mass × velocity. The unit of momentum kgms-1
Momentum is a vector quantity - it has both a magnitude and a direction
This means it can have a negative or positive value
If an object travelling to the right has positive momentum, an object travelling to the left has a negative momentum.
The principle of conservation of linear momentum is that momentum is always conserved in any interaction where no external force acts, which means the momentum before an event (e.g collision) is equal to the momentum after.
momentum before = momentum after
EXAMPLE: A car with the mass of 500 kg, and velocity of 4 ms-1, collides with a stationary truck with the mass of 1500 kg. The two vehicles join together and move with the velocity ‘V‘. Find the value of V.
( m1*u1 ) + ( m2 * u2 ) = ( m1 + m2 ) v
Newtons second law states that F=ma, therefore, F=∆(mv)/∆t as a=∆v/t. From this you can see that force is the rate of change of momentum.
The principle of conservation of linear momentum can also be expressed by saying that the sum of the change in momentum of the object is zero.
Using Newtons third law, states: for each experienced by an object, the object exerts an equal and opposite force. so, for two objects in a collision, the force exerted by 1 onto 2 is equal and the opposite t the force exerted by 2 onto 1.
Basically: F1 = -F2
Applying News second law and a=∆v/∆t
multiply both sides by change of time (this will be equal in a collision between the two objects).
m1v1 = m2v2
p1 = -p2
p1 + p2 =0