AS Physics Unit 1 Revision Notes PDF

1 Unit 1: Mechanics and Materials Mechanics MOTION Distance: The length of a path and is a scalar quantity (SI unit: m) Displacement: The shortest distance between two points and is...

1 Unit 1: Mechanics and Materials Mechanics MOTION Distance: The length of a path and is a scalar quantity (SI unit: m) Displacement: The shortest distance between two points and is a vector (SI unit: m). Speed: Rate of change of distance (SI unit: ms-1). Velocity: Rate of change of displacement (SI unit: ms-1). Acceleration: Rate of change of velocity (SI unit: ms-2). An object undergoes acceleration: - When there is a change in the speed or - When there is a change in direction or - When there is a change in direction and speed. If the speed of an object is constant and if its velocity changes, the object is accelerating (Objects undergoing constant circular motion). MOTION GRAPHS Constant gradient (Straight Line) 𝑦2 − 𝑦1 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑥2 − 𝑥1 y y y y Constant gradient (negative) Gradient = infinite Gradient = Constant gradient zero (positive) x x x x © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  2 Varying gradient (Curves) To find the gradient of the curve at any point, draw a tangent to the curve at that point and calculate the gradient of the tangent drawn. y y Gradient is positive and Gradient is positive and decreasing increasing x x y y Gradient is negative and increasing Gradient is negative and decreasing x x Displacement – time graph - Gradient = Velocity. Velocity – time graph - Gradient = Acceleration - Area under the graph = Displacement Acceleration – time graph - Area under the graph = Velocity © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  3 1 An object moving with Constant Velocity Displacement Velocity Acceleration Time Time Time 2 An object moving with constant acceleration Displacement Velocity Acceleration Time Time Time 3 An object moving with constant deceleration Displacement Velocity Acceleration Time Time Time © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  4 4 An object dropped from a height (Upward direction is assumed to be positive) Displacement Velocity Acceleration Time Time -9.81 Time 5 An object thrown upwards (Upward direction is assumed to be positive) Displacement Velocity Acceleration Time Time Time -9.81ms-2 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  5 6 A bouncing ball (Combining graphs 4 and 5) (a) Ignoring time of contact of the ball with ground Displacement t1 > t 2 t2 = t3, t4 = t5 0 Time t1 t2 t3 t4 t5 Velocity v1 > v2 v2 v2 > v3 v3 Since K.E is lost when the B C ball hits surface 0 Time A v3 All graphs are parallel because gradient is v2 constant (g = 9.81 ms-2) v1 Area A = Height from which ball is dropped Acceleration Area B = Height to which it bounces back 0 Time -9.81ms-2 Acceleration is always negative since gravitational force on the ball is always downwards © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  6 (b) Considering time of contact of the ball with ground Displacement Time Velocity t1 t3 t2 t4 Time Acceleration Time -9.81ms-2 Time of contact increases after each bounce since force decreases.(t2>t2) Time for which the ball is in air decreases after each bounce (t3>t4) Positive values of acceleration show the acceleration given by the ground on the ball which is upwards. Force of the ground on the ball decreases after each bounce. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  7 VECTORS - Quantities which have both magnitude and direction. Example: displacement, force, velocity, etc. - Vector quantities are represented using straight lines. - Length and the direction represent the magnitude and direction of the vector. - Resultant Vector: Resultant of two or more vectors is the single vector which produces the same effect in both the magnitude and direction Addition of Vectors 1 Vectors in the same line - The resultant vector can be found by adding their magnitudes. - Direction of the resultant vector would be the direction of the given vectors 2 Vectors in opposite line - The resultant vector can be found by subtracting their magnitudes. - Direction of their resultant would be in the direction of the biggest vector 3 Vectors acting perpendicular to each other - Arrange them in order and use Pythagoras’ theorem. 𝑅 = √(402 + 302 ) 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 ( ) 𝑎𝑛𝑑 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 30 tan 𝜃 = 40 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  8 4 Vectors non-perpendicular to each other Triangle law - If the 2 vectors acting at a point are represented in magnitude and direction by two sides of a triangle, drawn one after the other, their resultant is the third side of the triangle. - A scale drawing of the vector triangle gives the size and direction of the resultant. Resultant 400 450 Parallelogram law - If 2 vectors at a point are represented in magnitude and direction by a parallelogram, then their resultant is the diagonal of the parallelogram 5 Vector addition of more than 2 vectors Polygon law - Used to find the resultant of more than 2 vectors. - If given vectors are represented by the sides of a polygon in same order, the resultant is represented by the closing side of the polygon © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  9 - If the vectors arranged in order already forms a polygon, then their resultant is zero Resolving Vectors - A vector inclined at an angle with horizontal or vertical can be resolved into 2 perpendicular components. - Consider a vector R lying in the xy plane and making an angle θ with x-axis 𝑥 𝑦 𝐶𝑜𝑠 𝜃 = 𝑎𝑛𝑑 𝑆𝑖𝑛 𝜃 = 𝑅 𝑅 𝑥 = 𝑅 𝐶𝑜𝑠 𝜃 𝑎𝑛𝑑 𝑦 = 𝑅 𝑆𝑖𝑛 𝜃 𝑥 𝑖𝑠 𝑡ℎ𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑅 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑅 𝑦 𝑇𝑎𝑛 𝜃 = 𝑥 MOMENTS - The moment of a force is a measure of the tendency of the force to produce rotation of a body about a pivot. Perpendicular distance © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  10 𝑀𝑜𝑚𝑒𝑛𝑡 = 𝐹𝑜𝑟𝑐𝑒 × 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑝𝑖𝑣𝑜𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑜𝑓 𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑜𝑟𝑐𝑒 𝑀𝑜𝑚𝑒𝑛𝑡 = 𝐹𝑥 - SI unit of moment is Nm. Principle of Moments If an object is in equilibrium, the total clockwise moment will be equal to the total anticlockwise moment. - In order to calculate the sum of moments in either direction, each individual moment must be calculated first and these individual moments can then be added together. - The weights and/or distances cannot be added together. Conditions for a body to be in equilibrium: - Forces should be balanced (resultant force = 0). - Sum of clockwise moment = Sum of anticlockwise moment. Centre of Mass / Centre of Gravity - Weight of an object is caused by the gravitational attraction between the Earth and each of the small particles in the object. - The sum of these tiny weight forces appears to act from a single point, called the centre of gravity. - The centre of gravity of symmetrical objects lies at the intersection of all lines of symmetry. - The centre of gravity of an irregularly shaped object still follows the rule that it is the point at which its weight appears to act on the object. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  11 EQUATIONS OF MOTION 𝑣 = 𝑢 + 𝑎𝑡 𝑣 = 𝑓𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 2 2 𝑣 = 𝑢 + 2𝑎𝑠 𝑢 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 1 𝑡 = 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑠 = 𝑢𝑡 + 𝑎𝑡 2 𝑎 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 2 𝑠 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑢+𝑣 𝑠= 2 Conditions to use equations of motion: - Acceleration of the object should be constant - Motion should be in a straight line. PROJECTILE MOTION - Projectile motion is motion is motion under gravity. 1 Object thrown vertically - Motion is only vertical. So, there is no horizontal component. - Object is thrown with an initial maximum velocity (u). - The velocity of the object at the highest point is zero. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  12 2 Object projected horizontally Horizontal velocity remains the same u 9.81 ms-2 height Vertical velocity is range increasing as the object moves down - The object undergoes both horizontal and vertical motion. - Hence, in doing calculations, horizontal and vertical plane is taken separately. - The only force acting on the projectile is gravity, which is a vertical force. Hence, the horizontal motion remains constant. - Horizontal and vertical motion are independent of each other. Horizontal plane Vertical plane - Initial velocity = u - Initial velocity = 0 - Acceleration (a) = 0 - Acceleration (a) = +9.81 ms-2 - Horizontal displacement (range) = ut - Vertical displacement (height) = ½ at2 Finding the final velocity © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  13 3 Objects launched at an inclined angle - The object undergoes both horizontal and vertical motion. - Hence, in doing calculations, horizontal and vertical plane is taken separately. - The only force acting on the projectile is gravity, which is a vertical force. Hence, the horizontal motion remains constant. - Horizontal and vertical motion are independent of each other. - Initial velocity has both horizontal and vertical components. Horizontal plane Vertical plane - Acceleration (a) = 0 - Acceleration (a) = - 9.81 ms-2 (upwards) and +9.81 ms-2 (downwards) - Initial velocity = u Cosθ - Initial velocity = u Sinθ - Displacement (range) = u Cosθ  t - Displacement (height) = u Sinθ - ½ gt2 FORCES Contact Force: The force which arises when two objects are in contact 1 Friction - A force exerted by a surface as an object moves across it. - Frictional force always opposes motion - The two types of frictional forces are sliding friction and static friction - Friction depends on the nature of the two surfaces and the degree to which they are pressed together © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  14 2 Tension - Force which is transmitted through a string, rope, cable or wire when it is pulled tight by force from opposite end - Tension force is directed along the length of the wire and pulls the material equally on the other side 3 Normal contact force - Support force exerted on an object which is in contact with another stable object 4 Drag force - A force which opposes motion always - Examples: air resistance, up thrust, viscous drag Non-contact forces - They occur even when the two interacting objects are not in direct contact with each other. - These forces act at a distance. 1 Gravitational force - When two bodies are exerting force on each other without being in contact. gravitation acts at a distance. - Example: the sun and planets exert a gravitational pull on each other despite their large spatial separation. - When the mass of the bodies increase, the force of gravity also increases, but when the distance between the bodies increase, the force decreases. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  15 2 Magnetic force - A force by which magnets attract magnetic materials towards it. 3 Electrostatic force - A force which attracts charge carrying objects towards itself. Free Body Force Diagram - It is a diagram which shows all the forces acting on an object. - When drawing free body force diagrams arrows are used to represent the magnitude and direction of the forces. Newton’s first law of motion Every object remains in a state of rest or uniform motion, unless an external resultant force acts on it Significance: - If all forces acting are balanced, the object is in equilibrium. - The first law of motion is sometimes called the law of inertia. - Inertia: Inability of an object to change its velocity or motion. - Mass affects inertia: mass is directly proportional to inertia. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  16 Newton’s second law of motion Acceleration of a body of constant mass is directly proportional to the resultant force and takes place in the direction of the force Significance: - Definition of forces (F = ma) - Definition of 1N (the force needed to accelerate 1 Kg of an object by 1ms-2) Newton’s third law of motion If a body A exerts a force on a body B, then body B will also exert an equal but opposite force on A Significance: - Forces always occur in pairs - Action and reaction are together called as Newton’s third law pairs Properties of Newton’s third law pair of forces. Similarities - They are equal in magnitude. - They are same type of forces. - They act along the same line. - They act at the same time. Differences - They have opposite directions. - They act on different bodies (so they never cancel each other). © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  17 WORK DONE 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝐹𝑜𝑟𝑐𝑒 × 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑟𝑐𝑒 If the motion is not in the direction of the force: Work done = distance moved × component of force in that direction Work done = d  F cosθ where θ is the angle between the direction of the force and the direction of motion If the object moves at right angles to the force: - If you are dragging a box along the floor using a rope, then there are other forces on the box besides you pulling it: Normal contact force Forward push Friction Weight - No work is done by the weight and normal reaction forces, because they are perpendicular to the direction of motion. (W = Fd cosθ, since in this case θ is 90°, and cos90° = 0). © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  18 ENERGY - Energy is defined as the ability to do work. 1 - Kinetic energy is the energy an object has due to its motion (𝐾𝐸 = 2 𝑚𝑣 2 ) - Potential energy is the energy an object has due to its position or the arrangement of its particles. - Gravitational potential energy is defined as the amount of work which needs to be done in order to move the object to the height it is at (𝐺𝑃𝐸 = 𝑚𝑔ℎ) - Energy transferred is equal to the amount of work done. Law of Conservation of Energy - Energy cannot be created or destroyed. It can only be transformed from one form into another one. OR - The total energy of an isolated system (the Universe is an example of such a system) is constant. POWER - Power is defined as the rate of work done. - Power can also be defined as the rate of transfer of energy. 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝐹×𝑠 - 𝑃𝑜𝑤𝑒𝑟 = = =𝐹×𝑣 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑡 EFFICIENCY 𝑢𝑠𝑒𝑓𝑢𝑙 𝑜𝑢𝑡𝑝𝑢𝑡 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = × 100 𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑝𝑢𝑡 - Efficiency can be expressed as a percentage or as a decimal. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  19 MOMENTUM - The momentum of a body is defined as the product of its mass and its velocity. 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑚𝑎𝑠𝑠 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 - Momentum is a vector quantity. - Direction of momentum is always the same as that of its velocity. - According to Newton’s Second Law: The rate of change of (linear) momentum of an object is directly proportional to the resultant force acting on it. - If the force acting on a body is constant, then Newton’s second law can be written as 𝐹𝑖𝑛𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝐹𝑜𝑟𝑐𝑒 = 𝑡𝑖𝑚𝑒 𝑚𝑣 − 𝑚𝑢 𝐹= 𝑡 - If the force acting is not constant, then we have: 𝐹𝑖𝑛𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑓𝑜𝑟𝑐𝑒 = 𝑡𝑖𝑚𝑒 Impulse The impulse of a constant force is given by: 𝐼𝑚𝑝𝑢𝑙𝑠𝑒 = 𝐹𝑜𝑟𝑐𝑒 × 𝑡𝑖𝑚𝑒 𝑓𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑓𝑜𝑟𝑐𝑒 𝑎𝑐𝑡𝑠 𝐼𝑚𝑝𝑢𝑙𝑠𝑒 = 𝐹 × 𝑡 𝐼𝑚𝑝𝑢𝑙𝑠𝑒 = (𝑚𝑣 − 𝑚𝑢) 𝐼𝑚𝑝𝑢𝑙𝑠𝑒 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  20 Law of Conservation of Momentum The total momentum of a system before collision is equal to the total momentum after collision provided no external forces are acting on the system. 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑚1 𝑢1 + 𝑚2 𝑢2 𝐹𝑖𝑛𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑚1 𝑣1 + 𝑚2 𝑣2 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝐹𝑖𝑛𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 (𝑚1 𝑢1 + 𝑚2 𝑢2 ) = (𝑚1 𝑣1 + 𝑚2 𝑣2 ) - Total momentum is only conserved when no external forces (such as friction) act on the system. - In a collision in which two objects join together to become one and move off together, they are often said to coalesce. In such collision masses are added. - Explosion is a situation in which a stationary object (or system of joined objects) separates into component parts, which move off at different velocities. Momentum must be conserved in explosions. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  21 FLUIDS - Fluid is any substance that can flow. It includes gases and liquids. Density - It is an important property of materials. - It is defined as the mass per unit volume (one cubic metre). 𝑀𝑎𝑠𝑠 𝑚 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 = 𝑜𝑟 𝜌= 𝑉𝑜𝑙𝑢𝑚𝑒 𝑉 - The density of water is 1000 kgm-3. This means that 1m3 water has a mass of 1000 Kg. Forces in Fluids 1 Upthrust - When an object is submerged in a fluid, it feels an upward force that is called upthrust. - This is the force that keeps ships and boats floating. Archimedes’ Principle - Greek scientist Archimedes found that the size of the upthrust force is equal to the weight of the fluid that has been displaced by the object. This is called the Archimedes’ Principle. - It can be written in mathematical form as 𝐹 = 𝑚𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 𝑓𝑙𝑢𝑖𝑑 𝑔 = 𝜌𝑓 𝑉𝑔 (In the above equation 𝜌𝑓 is the density of the fluid. Do not mistake it as the density of the object!) - If the upthrust is smaller than its weight, then the object will sink. - If the upthrust is just equal to its weight, then the object will be able to stay anywhere in the fluid - If the upthrust is larger than the object’s weight, then the object will move upwards. When it gets to the surface of the fluid, part of its body will get out of the fluid and thus reduce the volume submerged in the fluid. So, the upthrust on the object will decrease to a value that is equal to the weight of the object. This state is called floating. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  22 2 Viscous drag - When an object is moving in a fluid, it will experience a drag force (resistance) that resists the motion of the object. - Drag force is caused by the viscosity of the fluid. Viscosity - Viscosity is an intrinsic property of fluid. - The factor coefficient of viscosity η (or simply viscosity) describes how large the viscosity of the fluid is. - The flow rate (volume of fluid passing through in unit time) of a fluid is inversely proportional to viscosity. - The viscous drag is larger while moving in a fluid with larger coefficient of viscosity. - A ball moving in honey will experience a much larger resistance than moving in water. Viscosity and Temperature - Viscosity, like density, is a property of material. - Viscosity is dependent on temperature. - For liquid, viscosity decreases with temperature. For gases, viscosity increases with temperature. Stokes’ law The viscous drag for a small sphere moving at low speed in a fluid is given by 𝐹 = 6𝜋𝜂𝑟𝑣 Where η is the viscosity of the fluid (Pa s), r is the radius of the sphere (m), v is the speed (ms-1) Note: Stokes’ law is only valid for small sphere moving at low speed. In this case the fluid flow is laminar flow. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  23 Terminal velocity - An object in a fluid moves under the influence of three forces, Weight, Upthrust and Viscous Drag. - The weight and upthrust are constant forces, but the viscous drag increases with the increase of speed. - At the moment the object is released, the speed is still zero; the viscous drag is also zero. The object will accelerate due to the gravitational force and upthrust. - However, when the speed increases, the viscous drag increases accordingly, causing the net force to decrease. Although the acceleration is decreasing, the speed is increasing because there is acceleration. - This process continues until when the speed reaches a certain value so that the viscous drag is large enough to balance the weight and upthrust. Then the net force becomes zero and so do the acceleration. - The speed will not change anymore. This velocity is called the terminal velocity At terminal velocity, weight = upthrust + viscous drag For a small sphere, the above equation can be written as 𝑚𝑠 𝑔 = 𝜌𝑓 𝑉𝑔 + 6𝜋𝜂𝑟𝑣𝑡𝑒𝑟𝑚 Where 𝑚𝑠 , V , r and 𝑣𝑡𝑒𝑟𝑚 are the mass, volume, radius and terminal velocity of the sphere and 𝜌𝑓 and η are the density and viscosity of the fluid respectively. 4 4 Substituting 𝑉 = 3 𝜋𝑟 3 and 𝑚𝑠 = 𝜌𝑠 𝑉 = 𝜌𝑠 3 𝜋𝑟 3 (𝜌𝑠 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑒) into the above equation gives; 4 4 𝜌𝑠 𝜋𝑟 3 𝑔 = 𝜌𝑓 𝜋𝑟 3 𝑔 + 6𝜋𝜂𝑟𝑣𝑡𝑒𝑟𝑚 3 3 Solving it for 𝑣𝑡𝑒𝑟𝑚 gives: 2𝑟 2 𝑔(𝜌𝑠 − 𝜌𝑓 ) 𝑣𝑡𝑒𝑟𝑚 = 9𝜂 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  24 - We can see from the above expression that the terminal velocity is dependent on the radius of the sphere, the density difference between the sphere and the fluid, as well as the viscosity of the fluid. 𝑣𝑡𝑒𝑟𝑚 is directly proportional to r2. Types of fluid flow Laminar and turbulent flows Laminar Flow Turbulent Flow - At any given point the velocity is - Velocity at any given point changes constant with time - No sudden change in velocity of flow - There are sudden changes in velocity of flow - Layers do not mix/cross OR layers are - Mixing/crossing of layers occurs. So parallel. So no eddies are formed. eddies are formed. Example: Slow water flow along a Example: Air flow in a storm smooth pipe © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  25 SOLID MATERIALS Hooke’s Law - The stretching force F is directly proportional to the extension (or compression) Δx. 𝐹 = −𝑘𝑥 where k is a constant called spring constant, Hooke’s constant or stiffness of the spring. - The minus sign here only indicates that the force is in the opposite direction to the extension. You do not have to include the minus sign in your calculation. - The gradient of the graph is the Stiffness. Elastic Strain Energy - When a spring is stretched or compressed, energy is stored in the spring. This form of energy is called elastic potential energy or elastic strain energy. - The amount of elastic strain energy stored in a spring is equal to the work done by the force exerted on it. - Thus, it can be obtained by calculating the area under the force-extension graph, which is 1 1 𝐸𝑒𝑙 = 𝐹∆𝑥 = 𝑘∆𝑥 2 2 2 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  26 Force - extension graph Proportionality limit: The point beyond which force is no longer proportional to extension is called the proportionality limit. (The point upto which a material obeys Hooke's law) Elastic limit: The point upto which a material return to its original length, when the force is removed is called elastic limit. During elastic deformation, the bonds between molecules in the material are stretched but come back to original length once the forces are removed. Plastic deformation: After the elastic limit, further force produces permanent deformation, which means that after removing the force they can’t return to their original lengths. This is called plastic deformation. During plastic deformation molecular bonds are strained to the point of fracture, making it not possible to return to the same state. Stress, Strain and Young Modulus - Stiffness is dependent on the material the spring is made of and the dimensions of the spring. - Young Modulus is the ratio of stress to strain. - Young Modulus is a property of the material and it is independent on its geometrical dimensions. Stress (σ): It is defined as the force divided by cross-sectional area. 𝐹𝑜𝑟𝑐𝑒 𝐹 𝑆𝑡𝑟𝑒𝑠𝑠 = 𝜎= 𝑈𝑛𝑖𝑡: 𝑁𝑚−2 𝑜𝑟 𝑃𝑎 𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝐴 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  27 Strain (ε) : It is defined as the ratio of change in length to the original length of the material. 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑙𝑒𝑛𝑔𝑡ℎ ∆𝑥 𝑆𝑡𝑟𝑎𝑖𝑛 = 𝜀= 𝑆𝑡𝑟𝑎𝑖𝑛 ℎ𝑎𝑠 𝑁𝑂 𝑈𝑁𝐼𝑇 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑥 Young Modulus (E): It is defined as the ratio of stress and strain. 𝑆𝑡𝑟𝑒𝑠𝑠 𝜎 𝑌𝑜𝑢𝑛𝑔 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 = 𝐸= 𝑆𝑡𝑟𝑎𝑖𝑛 𝜀 𝐹/𝐴 𝐸= ∆𝑥/𝑥 𝐹𝑥 𝐸= 𝐴∆𝑥 - Young Modulus is a quantity that is similar to the spring constant (stiffness) of a spring as it also measures how difficult it is to produce an extension. - The difference is that Young Modulus is a property of material and is not dependent on the dimensions of a particular sample. Anything made from the same material have the same Young Modulus, regardless of their shape or length. Stress Gradient = = Young Modulus (E) Strain 1 Area under graph = × Stress × Strain 2 1 F ∆x = ×( )×( ) 2 A x 1 𝐹∆𝑥 = × 2 𝐴𝑥 Energy stored = = Energy density Volume © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  28 Types of Stress and Strain Tensile Stress: Stretching force / Area of cross-section Compressive Stress: Compressive force / Area of cross-section Tensile Strain: Increase in length / Original length Compressive Strain: Decrease in length / Original length Stress - strain graph - A stress—strain graph is a very convenient tool to describe characteristics of a solid - The gradient of the linear part of the stress-strain graph is equal to Young Modulus of the material. - Proportionality limit: before this point, stress α strain; after this point, stress - strain graph is no longer linear. - Elastic limit: before this point, the material behaviors elastically, which means that it will return to its original shape if the stress is removed; after this point, the material behaves plastically, which means that it can’t return to its original shape if the stress is removed. - Yield point: The point at which a material behavior changes from elastic to plastic. - Ultimate Tensile Strength (Stress) (UTS): The maximum stress a material can withstand before breaking. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  29 - Linear Region: The region before the proportionality limit. - Elastic Region: The region before the elastic limit. - Plastic Region: The region after the elastic limit. Describing materials - Strength of a material is defined as the stress (the force per unit cross-sectional area) that it can withstand. Strong materials will have high Ultimate tensile Strength. - Hard materials resist plastic deformation by denting or scratching. Hard materials usually will be strong. Eg: diamond. - Tough materials can withstand impact forces and absorb a lot of energy before breaking, by undergoing plastic deformation. Eg: rubber, Kevlar (a type of material for fabricating bullet proof vest) Brittle Stress Tough Strain - Brittle: Materials which break/shatter/snap with little or no plastic deformation. Eg: Ceramics, biscuit. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  30 - Stiffness is a measure of a material's resistance to deformation. Stiffness is determined by calculating its Young’s Modulus. Stiff materials will have high Young Modulus. - Malleable materials show large plastic deformation under compression. They can be beaten into sheets. Eg: iron, gold, tin. - Ductile materials show large plastic deformation under tension. These materials can be pulled into wires. Eg. Copper. Stress/Pa Strong, Stiff Malleable, ductile Strain © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  31 Elastic Hysteresis When rubber is stretched the stiffness gradually decreases and it becomes relatively easy to stretch as the chains uncoil; once this has happened the rubber is much stiffer. The stress–strain graph for rubber shows that the behavior as a load is removed is not the same as that when the load is being increased. This is called hysteresis and the curves are said to make a hysteresis loop. Rubber absorbs more energy during loading than it releases in unloading. The difference is represented by the area of the hysteresis loop, shown shaded in the stress-strain graph. If you repeatedly stretch and release a rubber band, you can feel the effect of heating caused by hysteresis. Combinations of Springs 1 Springs in Series - Consider two springs with force constants k1 and k2 connected in series supporting a load, F = mg. - Let the force constant of the combination be represented by k - For the combination, supporting the load F = mg: 𝐹 𝐹 = 𝑘𝑥 (𝑤ℎ𝑒𝑟𝑒 𝑥 = 𝑡𝑜𝑡𝑎𝑙 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛) 𝑎𝑛𝑑 𝑥 = 𝑘 K1 - For each spring, the stretching force is same (F = mg) 𝐹 𝐹 = 𝑘1 𝑥1 𝑜𝑟 𝑥1 = 𝑘1 K2 𝐹 𝐹 = 𝑘2 𝑥2 𝑜𝑟 𝑥2 = 𝑘2 F=mg - The total extension, 𝐹 𝐹 𝐹 𝑥 = 𝑥1 + 𝑥2 𝑜𝑟 = + 𝑘 𝑘1 𝑘2 1 1 1 = + 𝑘 𝑘1 𝑘2 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  32 2 Springs in Parallel - Consider two springs with force constants k1 and k2 connected in parallel supporting a load F = mg. - Let the force constant of the combination be represented by k 𝐹 = 𝑘𝑥 (𝑤ℎ𝑒𝑟𝑒 𝑥 = 𝑡𝑜𝑡𝑎𝑙 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛) K1 K2 - The two individual springs both stretch by x, but share the load (F = F1+F2) F1 F2 𝐹1 = 𝑘1 𝑥 𝑤ℎ𝑖𝑙𝑒 𝐹2 = 𝑘2 𝑥 F=mg - Thus, the total force is 𝐹 = 𝐹1 + 𝐹2 𝑜𝑟 𝑘𝑥 = 𝑘1 𝑥 + 𝑘2 𝑥 𝑘 = 𝑘1 + 𝑘2 IMPORTANT DEFINITIONS AND KEY POINTS Displacement (Vector): The shortest distance between the starting point and the finishing point. Distance (Scalar): The length of a path. Velocity (Vector): Rate of change of displacement. Average velocity (Vector): Since the velocity changes at a constant rate from the beginning to the end, we can calculate the average velocity by adding the velocities and dividing by two. 𝑢+𝑣 (𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = ) 2 Instantaneous velocity: When a body is accelerates its velocity is constantly changing. The displacement-time graph for this motion is therefore a curve. To find the instantaneous velocity from the graph, a tangent should be drawn to the curve and the gradient should be calculated. Speed (Scalar): Rate of change of distance. Acceleration (Vector): Rate of change of velocity. Vector quantity: A physical quantity which has a magnitude and a unit. Scalar quantity: A physical quantity which only has a magnitude. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  33 Acceleration of freefall / Acceleration due to gravity (g): When an object falls vertically down in the absence of air resistance, there is only gravitational force acting on it. Hence, it accelerates downwards with the constant acceleration of 9.81 ms-2. Systematic error: A constant or consistent error in measurement which may be attributed to a fault in the instrument or a consistent flaw in the measuring technique, and which cannot be eliminated by averaging. Random error: An error of variable magnitude in which the readings are scattered about the true value. This type of error is due to limitations on the part of observer or an inconsistency in measuring equipment. It can be eliminated by averaging. Precision: The smallest measurement which can be done by an instrument (least count). Accuracy: The degree of conformity of a measure to a standard or a true value. Projectile Motion: - The only force acting on the object is the gravitational force / weight of the object which is a vertical force. - Hence, it only affects the vertical motion. - There is no horizontal force acting on the object. - Therefore, the horizontal motion remains constant throughout the journey. - Horizontal and vertical motion are independent of each other. If air resistance is considered, as shown below: The difference between Path Q (if there was no air resistance) and Path P (if there was air resistance): For path P, less range (horizontal distance) is travelled - There is a backward horizontal force acting on the ball. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  34 - The ball’s horizontal velocity is decreasing / ball has a horizontal deceleration. - Ball is in the air for a shorter time, hence travels a short distance horizontally. For path P, maximum height reached is lower: - As the ball rises, there is a greater downward force. - Hence, the average vertical velocity is lower / the vertical deceleration is greater. Graphs - The gradient of a displacement-time graph is velocity. - The gradient of a velocity-time graph is acceleration. - The area under the graph is the total displacement. Graphical representation of motion Line A: A body that is not moving. Displacement is always the same. Velocity is zero. Acceleration is zero. Line B: A body that is travelling with a constant positive velocity. Displacement increases linearly with time. Velocity is a constant positive value. Acceleration is zero. Line C: A body that has a constant negative velocity. Displacement is decreasing linearly with time. Velocity is a constant negative value. Acceleration is zero. Line D: A body that is accelerating with constant acceleration. Displacement is increasing at a non-linear rate. The shape of this line is a parabola since 1 displacement is proportional to t2 (𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 ). © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  35 Velocity is increasing linearly with time. Acceleration is a constant positive value. Newton’s First Law: Examples: Mass on a string: If a mass is hanging at rest on the end of a string then N1 says the forces must be balanced. This means the upward force = downward force (T = mg). Car travelling at constant velocity: If the car is travelling at constant velocity, then N1 says the forces must be balanced. (Force up = Force down; R= mg Force left = Force right; Engine thrust = Drag) Newton’s Second Law Examples: Elevator accelerating upwards, Elevator accelerating downwards, Inclined plane Newton’s Third Law 3 Examples: A falling body: A body falling freely (if air resistance is ignored), has only one force acting on it, which is the weight. If the earth pulls the body down, then the body must pull the earth up with an equal and opposite force. We have seen that the gravitational force always acts on the centre of the body, so N3 implies that there must be a force equal to W acting upwards on the centre of earth. A box resting on the floor: There are two forces acting on the box. Normal contact force: The floor is pushing up on the box with a force of N. According to N3, the box must therefore push down on the floor with a force of magnitude N. Weight: The earth is pulling the box down with a force W. According to N3, the box must be pulling the earth up with a force of magnitude W. Recoil of a gun: When a gun is fired the velocity of the bullet changes. N1 implies that there must be an unbalanced force on the bullet; this force must come from the gun. N3 says that if the gun exerts a force on the bullet the bullet must exert an equal and opposite force on the gun. This is the force that makes the gun recoil or ‘kick back’. Rocket propulsion: When a rocket is started in space, hot gases are released in downward direction of the rocket. This results in an equal size upward force on the rocket that pushes the rocket ahead. Momentum (Vector): The product of mass and velocity. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  36 Conservation of linear momentum: The total momentum before collision is equal to the total momentum after collision, provided that there is no external force acting on it. Newton’s second law of motion states that the rate of change in momentum is directly proportional to the resultant applied force. Impulse (I): Change in momentum (mv-mu) - It is useful to sketch a simple diagram showing the masses and velocities of the bodies before and after the interaction. - Remember that momentum is a vector. If you assign positive values to left-to-right motion, the velocities and momentum in the right-to-left direction must be negative. Centre of gravity / Centre of mass: The point through which the weight of the whole body acts. Moment: Turning effect of a force about a point. Moment is force multiplied by the perpendicular distance of the force from the pivot. For an object to be in equilibrium: - All the forces acting on it should be balanced / net force should be equal to zero. - TCWM should be equal to TACWM / Net moment should be zero. 1N:The resultant force which, when acting on a mass of 1 Kg, produces an acceleration of 1 ms-2. Gravitation field strength (Vector): The amount of gravitational force acting per unit mass. Weight (Vector): The amount of gravitational force acting on a unit mass. Work done (Scalar): The product of the force times the distance moved in the direction of the applied force. Energy (Scalar): The ability to do work. Power (Scalar): Rate of work done. Gravitational potential energy (GPE): Energy stored in a body due to its position relative to the ground. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  37 Elastic potential energy (EPE) / Elastic strain energy: Energy stored in a stretched/compressed material. Kinetic energy (KE): Energy in moving objects. 𝑢𝑠𝑒𝑓𝑢𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑡𝑝𝑢𝑡 Efficiency (scalar): 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = × 100 𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑝𝑢𝑡 Density (Scalar): Mass per unit volume. Upthrust (Vector): The upward force that acts on a body wholly or partially immersed in a fluid. Upthrust is equal to the weight of the fluid displaced. Archimede’s Principle: It states that the upthrust on a body immersed in a fluid is equal to the weight of the fluid displaced. Rate of flow depends on: - The viscosity of the fluid - The diameter of the tube - The length of the tube - The pressure across its ends - Whether the flow is streamlined or turbulent Stoke’s law: Stoke’s law states that the viscous drag on a spherical object moving through a fluid is given by the equation 𝐹 = 6𝜋𝜂𝑟𝑣 Conditions for Stoke’s law: - Object should be small - Object should be spherical - Flow of the fluid relative to the body must be laminar/streamlined - Object should have lower speed Viscosity: The property of fluids which gives a measure of the resistance to the flow. It is related to the stickiness. Coefficient of viscosity: A quantity which indicates the viscous effect of fluid. Drag / Viscous drag / Air resistance / Friction (Vector): A force which opposes motion. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  38 Terminal velocity: - An object falling vertically would have weight acting downwards and upthrust and drag acting upwards. - As the speed increases, the drag on the object increases (weight and upthrust remains same). - When the upward forces (upthrust and drag) equals the downward force (weight), the resultant force becomes zero. - According to N2 (𝐹 = 𝑚𝑎), when the resultant force becomes zero, the acceleration becomes zero and the object travels with a constant velocity known as terminal velocity. Hooke’s law: Up to a certain limit (elastic limit), load is directly proportional to the extension. Limit of proportionality: The point up to which, the material obeys Hooke’s law. Elastic limit: The point up to which the material regains its original length when the load is removed. Yield point: The point after which, the material undergoes a large strain for a small stress. Breaking point: The point after which, if additional load is attached, the material breaks/fractures. Spring constant: The stiffness of the spring (the gradient of a F-x graph). Stress (Scalar): Force per unit area. Strain: The ratio of extension of a material to its original length. Young modulus: Young modulus is stress divided by strain (the gradient of the linear region of stress-strain graph). Ultimate Tensile Strength (UTS): The maximum stretching force/tension a material withstands before breaking. Explanation of Hysteresis in terms of elastic strain energy: - During loading, work is done on the rubber. This is stored as elastic strain energy and is represented by the area under the loading curve. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  39 - During unloading, the energy stored in the rubber does work by raising the load. This is represented by the area under the unloading curve. - More energy has been used in stretching the band than has been retrieved during unloading. By the law of conservation of energy this ‘lost’ energy must be transferred to another form. This will be in the form of internal energy within the rubber, which will increase the temperature of the band and then be dispersed as thermal energy to the surrounding. - Increase in internal energy within the rubber during loading-unloading cycle = area enclosed by the loop. Property Definition Example Opposite Definition Example Strong High breaking stress Steel Weak Low breaking stress Expanded polysterene Stiff Gradient of a force- Steel Flexible Low young modulus Natural extension graph rubber High young modulus (A measure of material’s resistance to deformation) Tough High energy density Mild Brittle Little or no plastic Glass, up to fracture: metal steel, deformation before fracture Ceramics that has a large copper, plastic region rubber tyres Elastic Regains original Steel in Plastic Extends extensively and Copper, dimensions when the Hooke’s irreversibly for a small plasticine deforming force is law increase in stress beyond the removed region, yield point rubber Hard Difficult to indent Diamond Soft Surface easily Foam the surface indented/scratched rubber, balsa wood © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  40 Ductile Can be readily Copper Hard, (see above) (see above) drawn into wires brittle Malleable Can be hammered Gold Hard, (see above) (see above) into thin sheets brittle Examiner tips: ‘suvat’ equations are used when bodies move in straight lines with uniform acceleration. Free-falling objects always have a constant acceleration of 9.81 ms-2. When analyzing d-t and v-t graphs, observe how the gradient is changing to visualize how the velocity is changing. When stating vector quantities, direction should be stated or shown on the diagram. Vector diagrams: Should always be straight lines and the direction must be shown with arrows. Free-body force diagrams: All the forces shown should act at a single point and be represented by straight lines (use ruler!), and the direction should be indicated using arrows. When using N2 (F = ma), always calculate the resultant force as the first thing. N3 cannot be applied to single bodies. When explaining about forces, always state which bodies the forces act upon and the direction of the forces. When describing a force, it should contain all details like type, direction and magnitude. Most questions involving objects to be lifted gives the mass. Remember to calculate the weight using W = mg. For energy conservation questions, it is a common error to use 𝑣 2 = 𝑢2 + 2𝑎𝑠, to find velocity. This may give the correct answer, but it is an error in physics. In ‘show that’ questions, answer should be given to at least one more significant figure than the approximate value shown in the question. In many upthrust calculations, mass of displaced fluid would be given, make sure that it is converted to weight. When drawing streamlines, use ruler and make sure that lines are continuous and never cross. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  41 For solids and liquids falling through air, upthrust is very small compared to other forces. So, upthrust can be omitted from calculations. In loading-unloading curves, arrows must be drawn. Stoke’s law is used in the study of raindrops, mist particles, aerosol droplets in the atmosphere and motion of sloid spheres in liquids. Always pay particular attention to diagrams, sketching graphs and calculations. Students often lose marks by failing to label diagrams properly, by not giving essential numerical data on sketch graphs and, in calculations, by not showing all the working or by omitting the units. Command terms State: a brief senetence giving the essential facts; no explanation is required (nor should you give one). Define: you can use a word equation; if you use symbols, you must state what each symbol represents. List: simply a series of words or terms, with no need to write full sentences. Outline: a logical series of bullet points or phrases will suffice. Describe: for an experiment, a diagram is essential, then give the main points concisely (bullet points can be used). Draw: diagrams should be drawn in section, neatly and fully labelled with all measurements clearly shown, but don’t waste time (remember it is not an art exam). Sketch: usually a graph, but graph paper is not necessary, although a grid is sometimes provided; axes must be labelled, including a scale if numerical data are given, the origin should be shown if appropriate, and the general shape of the expected line should be drawn. Explain: use correct physics terminology and principles; the depth of your answer should reflect the number of marks available. © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024  42 Show that: usually a value is given so that you can proceed with the next part; you should show all your working and give your answer to more significant figures than the value given (to prove that you have actually done the calculation). Calculate: show all your working and give units at every stage; the number of significant figures in your answer should reflect the given data, but you should keep each stage in your calculator to prevent excessive rounding. Determine: means you will probably have to extract some data, often from a graph, in order to perform a calculation. Estimate: a calculation in which you have to make a sensible assumption, possibly about the value of one of the quantities (think: does this give a reasonable answer). Suggest: there is often no single correct answer, credit is given for sensible reasoning based on correct physics. Discuss: you need to sustain an argument, giving evidence for and against, based on your knowledge of physics and possibly using appropriate data to justify your answer. Useful formulae not included in the formulae sheet ▪ 𝑃𝑜𝑤𝑒𝑟 = 𝐹𝑜𝑟𝑐𝑒 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑃 = 𝐹𝑣) 1 ▪ 𝐸𝑙𝑎𝑠𝑡𝑖𝑐 𝑆𝑡𝑟𝑎𝑖𝑛 𝐸𝑛𝑒𝑟𝑔𝑦 = 2 𝑘 𝑥 2 𝜋𝑑2 ▪ 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒, 𝐴 = 𝜋𝑟 2 𝑜𝑟 𝐴 = 4 4 ▪ 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑠𝑝ℎ𝑒𝑟𝑒, 𝑉 = 3 𝜋𝑟 3 ▪ 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟, 𝑉 = 𝜋𝑟 2 ℎ 𝑣2 ▪ Energy conservation, 𝑣 = √2𝑔∆h , ∆ℎ = 2𝑔 ▪ Change in momentum, , ∆𝑝 = 𝐹 × ∆𝑡 𝐹𝑥 ▪ Young Modulus, , 𝐸 = 𝐴 ∆𝑥 © GNAEC / REVISION NOTES / AS PHYSICS / UNIT 1/ 2024