Physical quantity: consist of a numerical magnitude and a unit of measure

Transcription

1 DEFINITIONS LIST Chapter 1: Measurement Physical quantity: consist of a numerical magnitude and a unit of measure Base quantity: quantity that is chosen and arbitrarily defined rather than being derived from a combination of other physical quantities and has a base unit from which all other units can be defined. Derived quantity: quantity that is defined based on combination of base quantities and has a derived unit that can be expressed as the product or quotient of these base units. *Random error: error that occurs without a fixed pattern resulting in a scatter of readings about a mean value. *Systematic error: error that occurs with a fixed pattern resulting in a consistent overestimation or under-estimation of the actual value. *A set of precise measurements is one that has a small spread or scatter of readings. *An accurate measurement is one that is close to the actual value *A vector quantity consists of a magnitude and a direction. *A scalar quantity consists of a magnitude only. Chapter 2: Kinematics *Displacement: The linear distance in a given direction from a reference point. *Speed: Rate of change of distance travelled. *Velocity: Rate of change of displacement. *Acceleration: Rate of change of velocity. By Stella Seah Page 1 of 13

2 DEFINITIONS LIST Chapter 18: Quantum Physics *One electron volt is defined as the amount of energy an electron gains when it is accelerated through a potential difference of 1V. Photon is a quantum of electromagnetic radiation. *Photoelectric effect is a phenomenon where electrons are liberated from the surface of a metal when the metallic surface is irradiated with electromagnetic radiation of high enough frequency. *Threshold frequency is the frequency of the incident electromagnetic radiation below which no electrons can be liberated from the metal surface. *Work function of a metal is the minimum energy required to liberate an electron from the surface of the metal. *Ionisation energy is the minimum energy required to remove an electron from an atom in its ground state. A potential barrier is a region within which the potential energy of the particle is much higher than if the particle were to be outside the barrier. *Quantum tunneling is a phenomenon where a particle can appear outside a potential barrier, even though the total energy of the particle is lower than the barrier energy (barrier height). *Transmission coefficient T is defined as the probability that a particle tunnels through the potential barrier *Reflection coefficient R is the probability that a particle is reflected by the barrier. By Stella Seah Page 11 of 13

3 PLANNING 1. Structure of Answer The outline for planning is typically structured as follows: 1. Define the Variables [2 mark] 2. Methods of Data Collection (a) Diagram (b) Details on how to vary & measure the independent variable (c) Details on how to measure the dependent Variable [6 marks] 3. Method of Analysis [1 mark] 4. Safety considerations [1 mark] 5. Precautions to Improve Accuracy/Reliability [2 marks] By Stella Seah Page 1 of 41

4 PLANNING 5. Apparatus List Apparatus Image Details Metre rule Vernier calipers Micrometer screw gauge Travelling microscope G-clamp Drawn & labeled in diagram to show how it can be used to measure length. Precision to 1 mm. Use with set square to ensure metre rule is perpendicular to a surface. External jaws and internal jaws to measure external and internal diameter. Precision to 0.1 mm. To measure diameter of a wire/cylinder/sphere, always measure at 3 different positions and find the average. Used to measure thickness or diameter of wires. Precision to 0.01 mm. To measure diameter of a wire/cylinder/sphere, always measure at 3 different positions and find the average. Measures length with a resolution in the order of 0.01mm. The instrument comprises a microscope mounted on two rails fixed to a very rigid bed. The position of the microscope can be varied coarsely by sliding along the rails, or finely by turning a screw. The eyepiece is fitted with fine crosshairs to fix a precise position, which is then read off the vernier scale. Used to secure set up to a table-top. As reliability & safety considerations, use a G-clamp to secure retort stands, oscillators, or apparatus to the bench top to ensure they do not topple or move during the experiment. By Stella Seah Page 5 of 41

5 PLANNING Sample 1 (Suggested Solution) Frequency of Sound in a Bell Jar (Modified from GCE A/AS LEVEL Specimen Paper 2007) Diagram : Pressure To vacuum pump gauge Thermometer Connecting wires microphone Rubber seal Retort stand Glass Jar Diagram Clearly drawn & labeled apparatus Relative positioning of apparatus must be sensible Indicate physical quantities & symbols (e.g. d) clearly in diagram, which would later be referred to in the write-up. Signal generator speaker d C.R.O. Variables: Independent variable : Dependent variable : Controlled variables : air pressure p frequency f (a) air temperature, (b) amplitude of sound wave at source, (c) distance of microphone to loudspeaker. (d) speaker to always face microphone Procedures: 1. Set up the apparatus as shown in the diagram. Comments: Indicate variables clearly & include symbols used Do not spend time describing the aim of the experiment, listing the apparatus and drawing results tables. (Type of apparatus used to measure/ vary the variables) 2. The air pressure p inside the jar can be changed using the vacuum pump and measured using a pressure gauge. 3. The sound waves are detected using a microphone connected to a calibrated cathoderay-oscilloscope (c.r.o.). The frequency of sound, f, can be measured by adjusting the calibrated time-base of the c.r.o. to a suitable scale such that a full wave form is frozen on the screen. 4. Calculate the period, T, of the wave by multiplying the interval L on screen with the time-base scale. Frequency, f = 1 T. L Do not assume that quantities are given describe how you can measure them. When using C.R.O., explain clearly how the C.R.O. is used to measure the frequency. The C.R.O. requires a sensor which changes the sound waves into electrical signals. By Stella Seah Page 16 of 41

6 1. MEASUREMENT MEASUREMENT Physical Quantities and SI Units Errors and Uncertainties Scalars and Vectors Learning Outcomes Candidates should be able to: (a) recall the following base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol). (b) express derived units as products or quotients of the base units and use the named units listed in Summary of Key Quantities, Symbols and Units as appropriate (c) show an understanding of and use the conventions for labelling graph axes and table columns as set out in the ASE publication SI units, Signs, Symbols and Abbreviations, except where these have been superseded by Signs, Symbols and Systematics (The ASE Companion to Science, 2000). (d) use the following prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units: pico (p), nano (n), micro (µ), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T). (e) make reasonable estimates of physical quantities included within the syllabus. (f) show an understanding of the distinction between systematic errors (including zero errors) and random errors. (g) show an understanding of the distinction between precision and accuracy. (h) assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties (a rigorous statistical treatment is not required) (i) distinguish between scalar and vector quantities, and give examples of each. (j) add and subtract coplanar vectors. (k) represent a vector as two perpendicular components. By Stella Seah Page 1 of 15

7 1. MEASUREMENT 1. Physical Quantities & SI units A physical quantity defines some measurable feature of many different items. All physical quantities consist of a numerical magnitude and a unit of measure. Example Mass of an elephant M, Physical quantity M = 2700 kg Numerical magnitude Unit of measure There are 2 types of physical quantities: (1) Base quantities (2) Derived quantities 1.1 Base Quantities A base quantity is chosen and arbitrarily defined rather than being derived from a combination of other physical quantities and has a base unit from which all other units can be defined. SI base units is the standards assigned to each of these basic quantities and to no others. It is not derived from other units, i.e. independent of other units. The seven base quantities and their SI base units are: 1.2 Derived Quantities A derived quantity is defined based on combination of base quantities and has a derived unit that can be expressed as the product or quotient of these base units. Example Force (N) = mass (kg) x acceleration (m s -2 ) N kg m s -2 Energy (J) = force (kg m s -2 ) x distance (m) J kg m 2 s -2 By Stella Seah Page 2 of 15

8 1. MEASUREMENT Worked Example 1 The table below shows some commonly used quantities and their SI units. Complete the table for the rest of the derived quantities. 1.3 Homogeneity of Equations For any physical equation, every term of the equation should be dimensionally consistent, i.e. each term of the equation should have the same units and only quantities of the same units can be added, subtracted or equated in an equation. Worked Example 2 The experimental measurement of the specific heat capacity c of a solid as a function of temperature T is to be fitted to the expression c = at + bt 3. What are the units of a and b expressed in SI base units? Worked Example 3 (RVSH/I/1) Which of the following could be the correct expression for the speed v of sound in a gas of density ρ and at a pressure P? (γ is a dimensionless constant.) A v =! "# B v=!" # C v=!# " D v= $%& By Stella Seah Page 3 of 15

9 1. MEASUREMENT The resultant or sum of 2 vectors A and B, denoted by A + B is obtaining by placing the starting point of B on the ending point of A and then joining the starting point of A to the ending point of B, resulting in a vector triangle as shown in the figure. 5.3 Subtraction of Vectors When one vector is subtracted from another vector, say A B, this can be viewed as A + ( B) i.e. the addition of A and the negative vector B. Useful Mathematical Formulae Worked Example 9 A car travels 100 km due west and then 120 km due 50.0o east of north. Determine the magnitude and direction of the car s resultant displacement. By Stella Seah Page 11 of 15

10 2. KINEMATICS KINEMATICS Rectilinear motion Non-linear motion Learning Outcomes Candidates should be able to: (a) Define displacement, speed, velocity and acceleration. (b) Use graphical methods to represent distance travelled, displacement, speed, velocity and acceleration. (c) Find displacement from the area under a velocity-time graph. (d) Use the slope of a displacement-time graph to find the velocity. (e) Use the slope of a velocity-time graph to find the acceleration. (f) Derive, from the definitions of velocity and acceleration, equations which represent uniformly accelerated motion in a straight line. (g) Solve problems using the equations which represent uniformly accelerated motion in a straight line, including the motion of bodies falling in a uniform gravitational field without air resistance. (h) Describe qualitatively the motion of bodies falling in a uniform gravitational field with air resistance. (i) Describe and explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction. By Stella Seah Page 1 of 14

11 2. KINEMATICS Instantaneous velocity at t 1 = ds dt t=t 1 or gradient of tangent at t = t 1 Note: Velocity-time graph can be obtained by differentiating displacement-time graph. 1.3 Velocity-time graphs Average acceleration = Total Change in Velocity = v t Total Time Taken Instantaneous acceleration at t 1 = dv dt t=t 1 or gradient of tangent at t = t 1 Note: Acceleration-time graph can be obtained by differentiating velocity-time graph. From v = ds dt t, s = 2 v dt t 1 Change in displacement s from t = t 1 to t = t 2 can be found from area under velocity-time graph from t = t 1 to t = t 2. Note: Final displacement = Initial displacement + Change in displacement. We usually define initial point as origin, i.e. initial displacement = 0. Hence, final displacement at t = t 1 is simply the area under v-t curve from 0 to t = t 1. General shape of displacement-time graph can be obtained from integration of velocity-time graph. By Stella Seah Page 3 of 14

12 2. KINEMATICS Worked Example 5 (N96/1/4) A car is travelling along a straight road. The graph shows the variation with time of its acceleration during part of the journey. Indicate on the graph the point at which the car is travelling at its greatest velocity. 2. Equations of Motion Consider an object which moves in a straight line with a uniform acceleration a. If its velocity increases from u to v in a time t, from the definition of acceleration, a = dv a = v-u t dt v = u + at (1) From definition of velocity, v = ds s = dt v dt = Area under velocity-time graph = Area of trapezium s = + (u + v)t, (2) Substituting Eqn (1) into (2): s = 1 2 u + (u+at) t = 1 2 2u + at t s =ut at2 (3) By Stella Seah Page 6 of 14

13 2. KINEMATICS 3 The graph shows the variation with time t of the velocity v of a bouncing ball, released from rest. Downward velocities are taken as positive. At which time does the ball reach its maximum height after bouncing? 4 The brake in a car moving at a speed of 30 ms -1 is applied to stop the car in 20 s. The graph below shows the variation with time t of the acceleration a of the car moving along a straight line. 5 Sketch the corresponding graph showing the variation with time t of the velocity v of the car. By Stella Seah Page 3 of 5

14 2. KINEMATICS (c) the horizontal component of the initial velocity of the ball [13.3 m s -1 ] (d) the magnitude and direction of the initial velocity of the ball. [14.0 m s -1, 18.4 o above horizontal] 10 (a) Distinguish between deceleration and negative acceleration. The ball leaves the right hand at an angle of 80.0 to the horizontal and reaches a maximum height of 1.75 m above the level of the juggler s hands. (b) Show that the ball s initial speed of projection is 5.95 m s -1. (c) Calculate how far the juggler must position his hands apart so that the ball lands on his left hand. [1.23 m] (d) For a fixed speed of projection, suggest two advantages for the juggler to throw the balls at such a large angle to the horizontal. (e) Explain whether the speed of projection should be greater or smaller in reality, for a fixed angle of projection. By Stella Seah Page 5 of 5

15 3. DYNAMICS DYNAMICS Newton s Laws of Motion Linear Momentum and its conservation Learning Outcomes Candidates should be able to: (a) State each of Newton's laws of motion. (b) Show an understanding that mass is the property of a body which resists change in motion. (c) Describe and use the concept of weight as the effect of a gravitational field on a mass. (d) Define linear momentum and impulse. (e) Define force as rate of change of momentum. (f) Recall and solve problems using the relationship F = ma, appreciating that force and acceleration are always in the same direction. (g) State the principle of conservation of momentum. (h) Apply the principle of conservation of momentum to solve simple problems including elastic and inelastic interactions between two bodies in one dimension (Knowledge of the concept of coefficient of restitution is not required.). (i) Recognise that, for a perfectly elastic collision between two bodies, the relative speed of approach is equal to the relative speed of separation. (j) Show an understanding that, whilst the momentum of a system is always conserved in interactions between bodies, some change in kinetic energy usually takes place. By Stella Seah Page 1 of 13

16 3. DYNAMICS Impulse of a force is defined as the change in momentum. Impulse = Change in momentum Δp = F dt (Area under force-time graph) Note: Final momentum = Initial momentum + Change in momentum (impulse). Worked Example 5 In a collision, when a truck of mass kg runs into the back of a car of mass 1200 kg, a constant force of N acts for 0.25 s. Calculate the change in velocity of (a) the car (b) the truck Worked Example 6 The figure below shows a variable force acting on a 200 kg object travelling in a straight line with an initial velocity, in the positive direction along the same line as the line of action of the force, of 15 m s -1 at t = 0 s. Calculate the magnitude of the final velocity of the object at t = 30.0 s. By Stella Seah Page 5 of 13

17 3. DYNAMICS Worked Example 9 Ball A of mass 1 kg moves at 10 m s -1. If it collides with Ball B of mass 100 kg at rest and rebounds at 9 m s -1, (a) find the velocity with which the Ball B moves off. (b) calculate the fraction of initial kinetic energy carried off by Ball B. 2.1 Elastic Collision An elastic interaction is one where the total kinetic energy of the system remains constant before and after the interaction. Before Collision After Collision By conservation of linear momentum, m A u A + m B u B = m A v A + m B v B àm A (u A v A ) = m B (v B u B ) (1) By conservation of kinetic energy, ½ m A u A 2 + ½ m B u B 2 = ½ m A v A 2 + ½ m B v B 2 m A u A 2 m A v A 2 = m B v B 2 m B u B 2 m A (u A v A ) (u A + v A ) = m B (v B u B ) (u B + v B ) (2) Substitute (1) into (2) m B (v B u B ) (u A + v A ) = m B (v B u B ) (u B + v B ) u A + v A = u B + v B u A u B = - (v A v B ) à Relative speed of approach = Relative speed of separation By Stella Seah Page 9 of 13

18 3. DYNAMICS Newton s Laws of Motion 1 A resultant force of 1 N acts on a stationary mass of 1 kg for a duration of 1 s. How far will the mass move during that 1 s. [0.5 m] 2 A force of 30 N halves a body s velocity in 900 m. If the mass of the body is 5 kg, calculate the original velocity and the duration for which the force acts on it. [120 m s -1, 10 s] 3 The variation of the force F with time t acting on a body of mass 200 g is shown in the graph below Given that the velocity of the body at t = 75 ms is 15 m s -1, what is the velocity of the body at t = 150 ms? [39 m s -1 ] 4 A helicopter of mass 1000 kg hovers at the same height by imparting a downward velocity v m s -1 to the air displaced by its rotors. The area swept out by the rotor blades is 80 m 2. The density of air is 1.3 kg m -3. Considering the air displaced downwards per second, (a) what is the volume of air displaced? (b) what is the mass of air displaced? (c) what is the momentum of the air displaced? (d) what is the force exerted on the helicopter by the air? (e) what is the value of v? [80v, 104v, 104v 2, 104v 2, 9.71 m s -1 ] 5 Two blocks A and B are connected using an inextensible string on a horizontal frictionless floor. Another inextensible string is tied to block A and pulled by a force P so that both the blocks have an acceleration of 2.0 m s -2. Given that the masses of blocks A and B are 50 kg and 10 kg respectively, what are the values of the force P and the tension T in the string connecting the two blocks? [120 N, 20 N] 6 A steady 50 m s -1 wind hits a rigid wall which is in a plane perpendicular to the wind direction and is stopped completely by the wall. Estimate the pressure exerted by the wind on the wall. The density of air is 1.25 kg m -3. [3130 Pa] Hint: Use the same approach as Question 4. Conservation of Momentum 7 A 2 kg rifle fires a bullet of mass 20 g with a velocity of 350 m s -1. (a) Calculate the recoil velocity of the rifle. (b) If the mass of the man holding the rifle is 70 kg, what will be the combined recoil velocity of the man and the rifle? [3.5 m s -1 ; m s -1 ] By Stella Seah Page 2 of 4

19 4. FORCES FORCES Types of forces Equilibrium of forces Centre of gravity Turning effects of forces Learning Outcomes Candidates should be able to: (a) Recall and apply Hooke s law to new situations or to solve related problems (b) Deduce the elastic potential energy in a deformed material from the area under the forceextension graph (c) Describe the forces on mass, charge and current in gravitational, electric and magnetic fields, as appropriate. (d) Solve problems using the equation p = ρgh (e) Show an understanding of the origin of the upthrust acting on a body in a fluid (f) State that an upthrust is provided by the fluid displaced by a submerged or floating object (g) Calculate the upthrust in terms of the weight of the displaced fluid (h) Recall and apply the principle that, for an object floating in equilibrium, the upthrust is equal to the weight of the object to new situations or to solve related problems (i) Show a qualitative understanding of frictional forces and viscous forces including air resistance (No treatment of the coefficients of friction and viscosity is required) (j) Use a vector triangle to represent forces in equilibrium (k) Show an understanding that the weight of a body may be taken as acting at a single point known as its centre of gravity (l) Show an understanding that a couple is a pair of forces which tends to produce rotation only (m) Define and apply the moment of a force and the torque of a couple (n) Show an understanding that, when there is no resultant force and no resultant torque, a system is in equilibrium (o) Apply the principle of moments to new situations or to solve related problems By Stella Seah Page 1 of 17

20 4. FORCES 1. Types of Forces 1.1 Free Body Diagram Free-body diagrams are diagrams used to show the magnitude and direction of all forces acting upon an object in a given situation. The direction of the arrow reveals the direction which the force is acting. The relative length of the force indicates the magnitude of the force. The free-body diagrams should depict ONLY forces which act on that object in the given situation. Force Remarks Diagram Weight Directed downwards towards Earth or along gravitational field Normal Contact Force Force starts from centre of gravity Perpendicular (i.e. normal) to the surface. Tension In direction of the rope or string (away from the object). Spring force Opposite to the direction of displacement x from equilibrium position Static friction Kinetic friction Parallel to the surface, drawn from surface of object Direction depends on what is needed to prevent object from moving. Parallel to the surface, drawn from surface of object Opposite to the direction of the velocity Buoyancy force Upwards By Stella Seah Page 2 of 17

21 4. FORCES 1.2 Spring Force Hooke s Law states that the force F required to stretch (or compress) an object (e.g. spring, wire) is directly proportional to the amount of extension (or compression) x, if the limit of proportionality is not exceeded. Mathematically, Hooke s Law can be expressed as where: F = kx F is the external force required to extend (or compress) the wire or spring, (SI unit: N) k is a positive constant known as the force constant or spring constant, (SI unit: N m -1 ) x is the displacement of the spring from equilibrium position, i.e. extension (or compression) of the spring (unit: m) The spring force (force exerted by the spring) is given by: F = -kx The negative sign indicates that the spring force is always in the opposite direction of the displacement of the object, i.e. the spring force is always directed in the equilibrium position. An external force must be applied in the same direction as displacement of spring for compression (or extension) of spring to take place. Hence, direction of external force is the same as the object s displacement. The spring force is the force exerted by the spring. Consider an already compressed or extended spring, a spring force would act on the object such that it is directed back to its equilibrium unstretched position. The direction of spring force is opposite from the object s displacement. By Stella Seah Page 3 of 17

22 4. FORCES For an object partially immersed, the volume of fluid displaced is the volume of object submerged in fluid Worked Example 7 Calculate the upthrust experienced by an object (0.1 m x 0.1 m x 0.1 m) made of iron and of mass 5.0 kg when entirely immersed in seawater. (Take g = 9.81 m s -2, density of iron = 7.86 x 10 3 kg m -3 and density of seawater = 1.03 x 10 3 kg m -3 ) Worked Example 8 The area of the horizontal cross-section of a ship is A. The ship has a mass of 5.0 x 10 6 kg and sinks to a depth of 2.00 m in fresh water when unloaded. Find the extra depth that the ship will sink in fresh water when loaded with 750 x 10 3 kg of cargo. Worked Example 9 Consider a cup filled to the brim with water with a piece of large ice floating on top. Explain if the water will overflow after the ice completely melted. By Stella Seah Page 12 of 17

23 4. FORCES Forces and Equilibrium 1 [PJC/2010/H2/P2/Q2] A solid iron sphere of density 8000 kg m 3 and volume m 3 is completely submerged in a liquid of density 800 kg m 3. The iron sphere is resting on a spring of spring constant k as shown below. The spring is compressed by 10.2 cm. (a) Find the upthrust acting on the iron sphere. (b) Calculate the spring constant k. A string of breaking strength 32.0 N is used to lift the iron sphere vertically upwards. The iron sphere is then lifted partially out of the liquid. (c) Explain why the spring breaks. (d) Calculate the volume of the fluid displaced at the instant when the string breaks. [3.53 N, 312 N m -1, 4.23 x 10-4 m 3 ] 2 A mass hanging from a spring balance in air gives a reading of 60N. When completely immersed in water, the reading becomes 35N. When completely immersed in another liquid, the reading becomes 40N. Find the density of the liquid. Take density of water to be 1000 kg m -3. [800 kg m 3 ] By Stella Seah Page 2 of 6

24 5. WORK, ENERGY AND POWER WORK, ENERGY AND POWER Work Energy conversion and conservation Potential energy and kinetic energy Power Learning Outcomes Candidates should be able to: (a) show an understanding of the concept of work in terms of the product of a force and displacement in the direction of the force (b) calculate the work done in a number of situations including the work done by a gas which is expanding against a constant external pressure: W = p V (c) give examples of energy in different forms, its conversion and conservation, and apply the principle of energy conservation to simple examples (d) derive, from the equations of motion, the formula E k = ½ mv 2 (e) recall and apply the formula E k = ½ mv 2 (f) distinguish between gravitational potential energy, electric potential energy and elastic potential energy (g) show an understanding of and use the relationship between force and potential energy in a uniform field to solve problems (h) derive, from the defining equation W = Fs, the formula E p = mgh for potential energy changes near the Earth s surface (i) recall and use the formula E p = mgh for potential energy changes near the Earth's surface (j) show an appreciation for the implications of energy losses in practical devices and use the concept of efficiency to solve problems (k) define power as work done per unit time and derive power as the product of force and velocity. By Stella Seah Page 1 of 15

25 5. WORK, ENERGY AND POWER 3. Work done can be positive or negative. If work done is positive, it means kinetic energy of object increases. If work done is negative, it means kinetic energy of object decreases. Scenario Work done by force Example s = 0 zero Man pushing on a wall that does not budge i.e. applied force does not move the object kinetic energy remains the same Force in direction of motion F = 0 zero i.e. Force is perpendicular to object s motion Force has a resolved component in the same direction as displacement Force has a resolved component in the opposite direction as displacement kinetic energy remains the same positive kinetic energy increases negative kinetic energy decreases An object moving on a horizontal surface: no work done by weight An object dropping on the ground: weight and displacement in same direction à object accelerates as it drops An object moving on a rough surface: friction in opposite direction as object s motion à object slows down. Worked Example 1 A man pushes a box of mass 1.0 kg with a horizontal force of 2.0 N along a smooth horizontal surface. Calculate the work done on the object due to (a) the horizontal force (b) the normal contact force (c) the weight of the box. Worked Example 2 A 0.50 kg ball slides down from the top of a rough ramp of length 4.0 m which is inclined at an angle 30 o above the horizontal. The frictional force acting on the ball is 5.0 N. Calculate the work done on the ball by (a) the weight of the box (b) the normal contact force (c) the frictional force. By Stella Seah Page 3 of 15

26 5. WORK, ENERGY AND POWER Worked Example 6 [2010/SAJC/H2/P2/Q2] A system of two bodies A and B initially at rest are connected by an inextensible cord over a frictionless pulley and are resting on inclined planes as shown. Body A of mass 2.00 kg and body B of mass 5.00 kg move, in the directions as indicated, a distance of m and each experiences a frictional force of 3.00 N. (a) Calculate the total change in gravitational potential energy of the system. (b) Find the speeds of the bodies after travelling m. Worked Example 7 A bullet of mass kg is travelling horizontally with a speed of 150 m s -1 when it strikes a wooden block of mass 2.0 kg which is hanging at rest from vertical strings. If the bullet embeds itself in the block, calculate the vertical height risen by the block and bullet. By Stella Seah Page 8 of 15

27 5. WORK, ENERGY AND POWER Consider an increase of energy as distance from source, x, increases (i.e. du > 0). Since F = du < 0, it points in the direction towards the source, which is a point of lower potential dx energy. Similarly, when potential energy decreases as distance from source, x, increases, i.e. du dx < 0, F >0, meaning it points in the direction away of source, which is a point of lower potential energy. Hence, the negative sign indicates that force always points in the direction of decreasing potential. An object always moves to a point of lowest potential. Since we know that for an object to be at equilibrium, force acting on it should be 0. Therefore, du = 0, which corresponds to the following three situations: dx dx Stable equilibrium Minimum point on U-x graph Unstable equilibrium Maximum point on U-x graph Neutral equilibrium Constant potential energy U-x graph is a horizontal line Worked Example 8 The potential energy of a body when it is at a point P a distance x from a reference point O is given by V = kx 2, where k is a constant. What is the force acting on the body when it is at P? A B C D E 2 kx in the direction OP kx in the direction OP zero kx in the direction PO 2 kx in the direction PO By Stella Seah Page 11 of 15

28 5. WORK, ENERGY, POWER Work and Energy Conversion 1 On braking, 500 kj of heat was produced when a vehicle of total mass 1600 kg was brought to rest on a level road. What was the speed of the vehicle just before the brakes were applied? [25 m s -1 ] 2 Two bodies A and B have identical light springs attached as shown. The spring constant is N m -1. Body A is of mass 3.0 kg and is moving with a velocity of 5.0 m s -1 over the frictionless plane towards body B which has a mass of 2.0 kg and is at rest. At a particular time t 0 during the collision, the springs undergo maximum compression and the two bodies move at a common velocity v 0. Determine the compression of each spring at time t 0. [0.016 m] 3 A metal spring of natural length 20.0 cm fixed to the ceiling such that the bottom end is at a height of 30.0 cm from the ground as shown. (a) When the box of mass 5.00 kg hangs in equilibrium, the bottom end of the spring is at a distance of 25.0 cm from the ground. Determine the spring constant of the spring. The box was then brought to a higher point, such that the bottom of the spring was 40.0 cm above the ground. The box was subsequently released from rest. (b) Calculate the speed of the box when the bottom of the spring is 30.0 cm above the ground. (c) Determine the nearest distance of the bottom of the spring from the ground. (d) Explain why in practice, we expect the answer obtain in (c) to be larger. [981 N m -1, 1.98 m s -1, m] By Stella Seah Page 2 of 5

29 6. CIRCULAR MOTION CIRCULAR MOTION Kinematics of uniform circular motion Centripetal acceleration Centripetal force Learning Outcomes Candidates should be able to: (a) express angular displacement in radians (b) understand and use the concept of angular velocity to solve problems (c) recall and use v = rω to solve problems (d) describe qualitatively motion in a curved path due to a perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle (e) recall and use centripetal acceleration a = rω 2, a = v 2 /r to solve problems (f) recall and use centripetal force F = mrω 2, F = mv 2 /r to solve problems. By Stella Seah Page 1 of 8

30 6. CIRCULAR MOTION 1. Angular Displacement and Angular Velocity 1.1 Angular Displacement When a particle is constrained to move in a circular path, it is convenient to use the polar coordinates (r, θ) instead of Cartesian coordinates (x, y) to describe the position of the particle. The angle θ, expressed in radians, is a ratio of arc length s to radius r. θ = s r where: θ is the angle measured in radian (SI unit: rad) s is the curved arc length (SI unit: m) r is the radius of the circle (SI unit: m) Note: For one complete revolution, θ = circumference = 2πr = 2π radius r Since one revolution is 360 o, 360 o 2π rad. Worked Example 1 The laser on a CD player is 5.0 cm from the central hole of the disc. What length of the disc is scanned by the laser when the disc turns through an angle of 0.45 radians? 1.2 Angular Velocity For an object moving in circular motion, it is hard to describe its linear velocity at every point since the direction of velocity changes. Instead, angular velocity ω is used. Angular velocity, ω, is the rate of change of angular displacement. (SI unit: rad s -1 ) The period T is the time taken for an object to make one complete revolution. (SI unit: s) By Stella Seah Page 2 of 8

31 6. CIRCULAR MOTION Worked Example 8 Three identical masses are tied together using strings and made to rotate around a pin on a smooth horizontal table as shown in the figure. The three masses remain in a straight line as they rotate. Determine the ratio of the tension in string 1: string 2: string Non-Uniform Circular Motion The speeds of objects in a vertical circular motion are generally not constant (and difficult to maintain at constant speed), hence is not uniform circular motion. This implies the centripetal acceleration cannot be the same magnitude throughout the motion. Consider an object of mass m tied to a string, describing a circular path in a vertical plane: At the bottom At the side At the top Net force = T mg = mv2 T = mg + mv2 r At the bottom, tension is the maximum as it has to provide the centripetal force necessary for circular motion. r Centripetal force = T = mv2 T = mv2 r Weight is parallel to direction of motion so there is work done on the object. Hence, speed of object is NOT a constant. r Net force = T + mg = mv2 T = mv2 r mg In order to barely make the top of circle, tension at the top is zero. Hence, mv min 2 = mg r v min = gr r By Stella Seah Page 7 of 8

32 6. CIRCULAR MOTION 3 In an amusement park ride in which a person of mass 63 kg rotates in a uniform vertical circle at constant speed of radius 6.6 m. The time taken for one revolution is 3.2 s. Find the force which is exerted by the seat on the rider when the rider is (a) at the bottom of the circle A, (b) at the top of the circle B, (c) half way up C. [2200 N, 990 N, 1700 N] Long Questions 4 A 2.5 kg pendulum bob is attached to a 15.0 cm inextensible string and fixed to point B of a L shape structure. The structure is mounted on a rotating disc A. The spinning causes the pendulum bob to swing at an angle of θ to the vertical. (a) Assuming that the structure rotates with a constant angular velocity such that point C on the circumference of the disk has speed of 13.6 cm s -1, find the rate of rotation of the disc A in terms of rev s -1 if the diameter of the disc is 6.00 cm. (b) As the structure rotates, the pendulum bob swings and maintains a constant angle, θ with the vertical, draw a free body diagram for the pendulum bob. (c) Given that θ = 30 o, determine the tension in the string and hence calculate the length d. (d) Explain qualitatively whether angle θ will increase or decrease if length d increases. (e) Explain how the angle θ will vary if the mass of the pendulum bob doubles. [0.721 rev s -1 ; 0.20 m] By Stella Seah Page 2 of 4

33 7. GRAVITATIONAL FIELD GRAVITATIONAL FIELD Gravitational field Force between point masses Field of a point mass Field near to the surface of the Earth Gravitational potential Learning Outcomes Candidates should be able to: (a) show an understanding of the concept of a gravitational field as an example of field of force and define gravitational field strength as force per unit mass (b) recall and use Newton's law of gravitation in the form F = Gm 1m 2 r 2 (c) derive, from Newton's law of gravitation and the definition of gravitational field strength, the equation g = GM for the gravitational field strength of a point mass r2 (d) recall and apply the equation g = GM for the gravitational field strength of a point mass to r2 new situations or to solve related problems (e) show an appreciation that on the surface of the Earth g is approximately constant and equal to the acceleration of free fall (f) define potential at a point as the work done in bringing unit mass from infinity to the point (g) solve problems using the equation φ = GM r for the potential in the field of a point mass (h) recognise the analogy between certain qualitative and quantitative aspects of gravitational and electric fields (i) analyse circular orbits in inverse square law fields by relating the gravitational force to the centripetal acceleration it causes (j) show an understanding of geostationary orbits and their application. By Stella Seah Page 1 of 12

34 7. GRAVITATIONAL FIELD 2. Gravitational Field and Gravitational Field Strength 2.1 Gravitational Field A gravitational field is a region of space where a mass will experience a gravitational force when placed in that field. A diagram of field lines is used to describe the gravitational field. The direction of a field at a point in space is along a tangent to the field line at that point. The strength of a field is given by the proximity of the lines. For a uniform field, the field lines are represented by equally spaced parallel lines. Gravitational field strength due to a point mass is drawn radially inwards. Near the surface of a point mass, the gravitational field is almost constant. E.g. gravitational field strength near the surface of Earth is 9.81 m s -2. By Stella Seah Page 3 of 12

35 7. GRAVITATIONAL FIELD For a uniform spherical mass of radius R, the variation of gravitational field strength with distance from centre of mass is shown below, where g s is the gravitational field strength at the surface of the sphere. For r > R, gravitational field strength is inversely proportional to the square of the distance away from the centre, as seen from Newton s Law of Gravitation. For r < R, gravitational field strength is directly proportional to the distance away from the centre. Consider the gravitational field strength at a distance of away from the centre of a uniform sphere of density ρ, The gravitational force a point mass experiences within the mass is due to the mass of the inner core. GM Since g = the object is: and the mass of the inner core M = 4.'/ 0 ' & / g = 4 3 Gπρr, the gravitational field strength within By Stella Seah Page 5 of 12

36 7. GRAVITATIONAL FIELD 4.1 True Weight and Apparent Weight True weight of an object can be calculated using the following relation: W = mg Where: W is the true weight of the object, m is the mass of the object g is the gravitational field strength at that particular point True weightlessness is experienced when the gravitational field strength at the point is zero. Recall that apparent weightlessness is experienced when the normal contact force acting on an object is zero. At the equator, the gravitational force has to provide the centripetal force necessary for the person to rotate along the Earth s axis. Hence, F g N = mω 2 r N = mg mω 2 R Where ω = angular velocity of Earth s rotation about its own axis and R = radius of Earth. At the poles, since the person is not rotating (radius of rotation r = 0), F g = N In general, for a person at an angle θ above the Earth s equator, Where r = R cos θ F g N = mω 2 r Hence, the apparent weight N is given by N = mg mω 2 R cos θ By Stella Seah Page 10 of 12

37 7. GRAVITATIONAL FIELD 6 The figure below shows the variation of gravitational potential between the surface of Moon and the surface of Earth along the line joining the centres. Mass of Earth = 5.98 x kg, Mass of Moon = 7.35 x kg Distance between Earth and Moon = 3.84 x 10 5 km. (a) State how the resultant gravitational field strength can be deduced from the figure. (b) State the gravitational field strength at Point P. (c) Determine the distance X. (d) Find the gravitational potential at Point P. A rocket of mass 2.7 x 10 6 kg on a mission to the Moon is to be launched from Earth. In order to reach the surface of the Moon, the rocket must be launched with a minimum speed. (e) Using the figure and the value calculated in (d), determine this minimum speed. Explain your working clearly. (f) With this minimum speed, calculate the speed at which the rocket will land on the Moon s surface. [0 N kg -1, 3.46 x 10 8 m, 1.29 x 10 6 J kg -1, 1.10 x 10 4 m s -1, 2280 m s -1 ] By Stella Seah Page 3 of 5

38 8. OSCILLATIONS OSCILLATIONS Simple harmonic motion Energy in simple harmonic motion Damped and forced oscillations: resonance Learning Outcomes Candidates should be able to: (a) describe simple examples of free oscillations (b) investigate the motion of an oscillator using experimental and graphical methods (c) understand and use the terms amplitude, period, frequency, angular frequency and phase difference and express the period in terms of both frequency and angular frequency (d) recognise and use the equation a = ω 2 x as the defining equation of simple harmonic motion (e) recall and use x = x 0 sin ωt as a solution to the equation a = ω 2 x (f) recognize and use v= v 0 cos ωt and v = ±ω (x 0 2 x 2 ) (g) describe, with graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion (h) describe the interchange between kinetic and potential energy during simple harmonic motion (i) describe practical examples of damped oscillations with particular reference to the effects of the degree of damping and the importance of critical damping in cases such as a car suspension system (j) describe practical examples of forced oscillations and resonance (k) describe graphically how the amplitude of a forced oscillation changes with frequency near to the natural frequency of the system, and understand qualitatively the factors which determine the frequency response and sharpness of the resonance (l) show an appreciation that there are some circumstances in which resonance is useful and other circumstances in which resonance should be avoided. By Stella Seah Page 1 of 11

39 8. OSCILLATIONS 1. Oscillation and its Terminologies An object undergoing free oscillation vibrates at its natural frequency. There is no periodic external forces acting on the oscillator. Examples include, assuming the absence of air resistance/friction, (1) a simple vertical spring-mass system, (2) a simple horizontal spring-mass system, (3) a simple (swinging) pendulum An oscillatory system without energy losses can be graphically represented by a sinusoidal function: x = x 0 sin ωt x = x 0 cos ωt Oscillation starts from equilibrium position Oscillation starts from furthest point Physical Quantity Definition SI Unit Type and Symbol Amplitude, x 0 Magnitude of maximum displacement metre (m) scalar from equilibrium position Displacement, x Distance from equilibrium position at a metre (m) vector specified time in a certain direction Period, T Time taken for one complete oscillation second (s) scalar Frequency, f Number of complete oscillations per unit hertz (Hz) scalar time f = 1 T Angular frequency, ω scalar Phase angle, ωt Number of complete cycles (in radian) per unit time. One cycle = 2π rad ω = 2πf = 2π T Stage of oscillation that has been completed at a specific reference time in terms of angle, e.g. ¼ oscillation = π/2 rad ½ oscillation = π rad ωt = 2π rad radians per second (rad s -1 ) radian (rad) scalar By Stella Seah Page 2 of 11

40 8. OSCILLATIONS 4. Forced Oscillations When an external periodic force is applied to a light-damped oscillatory system, it sets the system into forced oscillation. The system then oscillates at the frequency of the external periodic force. Consider a man pushing a swing: If the swing is displaced slightly and released, it will start to swing freely. The frequency at which the swing oscillates is the natural frequency of the swing. Ideally, without any damping effects due to energy loss, the swing will continue to swing at a certain amplitude infinitely. The swing is now pushed at fixed time interval. The frequency at which the force is applied is called the driver frequency. If the periodic force is applied in the opposite direction as the motion of the swing, the swing s velocity will decrease (left). If the periodic force is applied in the same direction as the motion of the swing, the swing will increase in its velocity. When the driver frequency matches the natural frequency, i.e. the swing is pushed every time it reaches the man, the pusher will add more energy to the swing and the amplitude is going to increase every time the periodic force is increased. This causes the swing to oscillate with maximum amplitude. This effect is known as resonance. Resonance is a phenomenon in which a forced oscillating system oscillates with maximum amplitude when the external driving frequency is equal to the natural frequency of the system. By Stella Seah Page 10 of 11

41 8. OSCILLATIONS (a) Calculate the angular velocity of the truck. (b) Calculate the shortest time taken t for the truck to oscillate from its lowest point to a point m below its equilibrium position. (c) If the truck travels at a certain speed over the series of speed bumps, the vertical oscillations can be very large. Explain why this is so. [8.7 rad s -1, s] Damped Oscillations 5 A student sets up the apparatus illustrated in figure below in order to investigate oscillations. The variation with time t of the displacement y of the end of the ruler is shown in figure below. (a) The student claims that the curve in figure above may be represented by the equation y = y o sin ωt. Identify two features which show that the student s claim is incorrect. (b) Calculate the angular frequency of the oscillation. (c) Calculate the acceleration of the ruler at time t = 1.5 s. (d) The card on the ruler is replaced with one having a larger surface area and the same mass. The experiment is then repeated. On the same figure above, draw another graph to show the effect of this change on the variation with t of the displacement of the ruler. By Stella Seah Page 3 of 6

42 9. THERMAL PHYSICS THERMAL PHYSICS Internal energy Temperature scales Specific heat capacity Specific latent heat First law of thermodynamics The ideal gas equation Kinetic energy of a molecule Learning Outcomes Candidates should be able to: (a) show an understanding that internal energy is determined by the state of the system and that it can be expressed as the sum of a random distribution of kinetic and potential energies associated with the molecules of a system (b) relate a rise in temperature of a body to an increase in its internal energy (c) show an understanding that regions of equal temperature are in thermal equilibrium (d) show an understanding that there is an absolute scale of temperature which does not depend on the property of any particular substance, i.e. the thermodynamic scale (e) apply the concept that, on the thermodynamic (Kelvin) scale, absolute zero is the temperature at which all substances have a minimum internal energy (f) convert temperatures measured in Kelvin to degrees Celsius: T / K = T / C (g) define and use the concept of specific heat capacity, and identify the main principles of its determination by electrical methods (h) define and use the concept of specific latent heat, and identify the main principles of its determination by electrical methods (i) explain using a simple kinetic model for matter why: (i) melting and boiling take place without a change in temperature (ii) the specific latent heat of vaporisation is higher than specific latent heat of fusion for the same substance (iii) cooling effect accompanies evaporation (j) recall and use the first law of thermodynamics expressed in terms of the change in internal energy, the heating of the system and the work done on the system (k) recall and use the ideal gas equation pv = nrt, where n is the amount of gas in moles (l) show an understanding of the significance of the Avogadro constant as the number of atoms in kg of carbon-12 (m) use molar quantities where one mole of any substance is the amount containing a number of particles equal to the Avogadro constant (n) recall and apply the relationship that the mean kinetic energy of a molecule of an ideal gas is proportional to the thermodynamic temperature to new situations or to solve related problems. By Stella Seah Page 1 of 23