Advanced Higher Mathematics of Mechanics Course Outline (2016-2017) Block 1: Change of timetable to summer holiday Assessment Standard Assessment 1 Applying skills to motion in a straight line (Linear & Parabolic Motion 1.1) Applying skills to vector associated with motion 2 (Linear & Parabolic Motion 1.2) Block 2: August to October holiday Assessment Standard Assessment 3 Complete work on relative motion 4 Applying skills to projectiles moving in a vertical plane (Linear & Parabolic Motion 1.3) 5 Applying skills to forces associated with dynamics and equilibrium (Linear & Parabolic Motion 1.4) Block 3: October to Christmas holiday Assessment Standard Assessment 6 Applying skills to principles of momentum, impulse, work, power and energy (Force, Energy and Periodic Motion 1.1) 7 Applying skills to Simple Harmonic Motion (Force, Energy and Periodic Motion 1.3) 8 Applying skills to motion in a horizontal circle with uniform angular velocity (Force, Energy and Periodic Motion 1.2) Unit test on Linear and Parabolic motion Extension test on Linear & Parabolic Motion
Block 4: January to Easter holiday Assessment Standard Assessment 9 Applying calculus skills through techniques of integration (Mathematical techniques for Mechanics 1.3) 10 Applying skills to Centre of Mass (Force, Energy and Periodic Motion 1.4) 11 Revision of work on Mathematical techniques for Mechanics Unit test on Force, Energy and Periodic Motion Prelim Unit test on Mathematical techniques for Mechanics
Block 1 Applying skills to motion in a straight line (Linear & Parabolic Motion 1.1) Miss Mackay Skills include: Working with rates of change with respect to time in one dimension and know the dot notation for this, i.e. Velocity, v = ds dx = s (or = = x ) dt dt Acceleration, a = dv = d2 s = s (or = d2 x = x ) dt dt 2 dt 2 Use calculus to determine corresponding expressions connecting displacement, velocity and acceleration Differentiate Differentiate Displacement, s Velocity, v Acceleration, a Integrate Integrate At this stage will require simple integration only. Separation of variables and a = v dv not required until ds differential equations are covered towards the end of the course (Mathematical Techniques for Mechanics) Working with time dependent graphs including velocity/time graphs for both constant and variable acceleration. Calculating displacement as the area under a velocity/time graph for constant (simple area calculation) and variable acceleration (area under curve by integration). The latter will be covered in revision of Mathematical techniques for Mechanics at the end of the course. Using calculus to derive the equations of motion (stuva) Applying the equations of motion (stuva) and/or calculus to problems Identify modelling assumptions made in particular contexts. For example air friction (resistance) in vertical motion, constant velocity and interpretation of negative velocity, variation in the value of g and the need to model all bodies as particles should be discussed. Sketch and annotate, interpret and use displacement/time, velocity/time and acceleration/time graphs Determine the distance travelled using the area under a velocity/time graph Use calculus to determine corresponding expressions connecting displacement, velocity and acceleration Using calculus derive the equations of motion Applying equations of motion or calculus to more complex problems Applying skills to vectors associated with motion (Linear & Parabolic Motion 1.2) Mrs Hall Skills include: Finding resultant velocity by adding column vectors, and by using sine rule in vector diagrams. Giving velocity in terms of magnitude (speed) and direction (bearing). Using the cosine rule to find speed in vector diagrams. Conditions for setting a course across a river to cross directly; to reach a point upstream and for the shortest time. Finding relative velocity v A B = v A v B using column vector method and vector diagram method. Finding true velocity using column vector method and vector diagram method. Finding intercept/collision course. Finding closest distance apart if two bodies do not collide.
Finding course for closest approach. Give the displacement, velocity and acceleration of a particle as a vector and understand that speed is the magnitude of the velocity vector. Resolve position, velocity and acceleration vectors into 2 and 3 dimensions and use these to consider resultant or relative motion. Apply position, velocity and acceleration vectors to practical problems including navigation, the effects of winds or currents and other relevant contexts. Solve a simple problem involving collision eg. Given position and velocity vectors for two bodies, prove that they will collide. Finding the course set to cross a river directly to the other side Finding the true velocity of the wind Course for collision/interception Conditions for closest approach
Block 2 Applying skills to projectiles moving in a vertical plane (Linear & Parabolic Motion 1.3) Miss Mackay Skills include: Be able to work with horizontal projection starting with a = 0i + ( g)j to derive equations for x, y, x, y, x and y. Know how to split initial velocity into components (u = u cos α i + u sin α j) and be able to model particles falling under gravity. Know how to find key points on a particle s projectile either using stuva or: Maximum height when y = 0 or T H = 1 2 T F Time of flight (usually) when y = 0 or T F = 2T H Range when t = T F Be able to derive formulae associated with projectile motion: Time of flight, T = 2u sin α g, Maximum height, H = u2 sin 2 α 2g, Horizontal Range R = u2 sin 2α g Know and be able to demonstrate that maximum horizontal range is achieved when α = 45 or π 4 radians. Derive the equation of a trajectory (realise that this represents a quadratic equation/parabola): y = x tan α gx2 2u 2 cos 2 α Use the equation of the trajectory to find points or to work backwards and find the angle of projection. Projection from an inclined plane is not required but will offer good extension. Derive formula associated with projectile motion Use these formula to calculate time of flight, maximum height, and range Be able to calculate maximum range and the angle of projection used to achieve this. Solve problems in two-dimensional motion involving projectiles under a constant gravitational force (height of projection may be different from height on landing) Derive and use the equation of a trajectory Intersection of trajectories (link to relative motion)
Applying skills to forces associated with dynamics and equilibrium (Linear & Parabolic Motion 1.4) Miss Mackay/Mrs Hall Skills include: Combine forces to find a resultant/net force Using Newton s first (every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force) and third (for every action force in there is an equal and opposite reaction force) laws of motion to understand equilibrium Know that in equilibrium resultant/net force is zero. Know how to split forces and resolve in perpendicular directions Understand link between resolving and vectors and be able to use both approaches For simple problems involving 3 forces be familiar with alternative approaches such as triangle of forces and Lamii s theorem Consider equilibrium between connected particles (Tension) and know to model each particle separately Understand the concept of static, dynamic and limiting friction Know that in limiting equilibrium and dynamic Friction = F max = μr and also that for static/stationary particles F μr Be able to demonstrate that coefficient of friction, μ = tan θ Use the relationships F max = μr and μ = tan θ to solve problems involving particles on a rough inclined plane and be able to state modelling assumptions in such problems Be able to model problems where an external force is required to keep a body in equilibrium (minimum and maximum value) and consider limiting equilibrium for movement along the line of greatest slope. Resolve forces in two dimensions to find their components, e.g. a body sitting at rest on a rough inclined plane or a particle hanging in equilibrium suspended by two strings Combine forces to find the resultant force, e.g. by splitting into horizontal and vertical components or by expressing in vector form. Equilibrium of connected particles, e.g 2015 Applied Maths Mechanics exam question 9 Static friction F μr
Block 3 Applying skills to principles of momentum, impulse, work, power and energy (Force, Energy and Periodic Motion 1.1) Miss Mackay Skills include: Know that momentum is a vector quantity, p = mv Know that Force is rate of change of momentum, F = dp Work with impulse I = F dt Know that for constant force applies Newton s second law (F = ma) gives I = Ft = mv mu Use impulse appropriately in simple situations using the above equations Use the concept of conservation of linear momentum m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 or m 1 u 1 + m 2 u 2 = (m 1 + m 2 )v for bodies that coalesce. Ensure candidates are familiar with resolving/splitting momentum or expressing in component vector form for such problems, e.g. 2015 Applied Maths Mechanics question 1. Know that work done, W = F i dx Understand that work can be done by (+) or against (-) a force Be able to calculate work done by a constant force in one dimension W = Fs Be able to calculate work done by a constant force in two dimensions W = F s (i.e. use scalar product) Be able to calculate work done by a variable force acting along a straight line using integration: W = F idx Be able to calculate work done during rectilinear motion by a variable force using integration: W = F vdt where v = i dx dt dt Use impulse to solve a simple problem (bouncing balls, collision of particles etc.) making use of the equations I = Ft = mv mu or I = F dt Use the concept on conservation of linear momentum to solve a simple problem Evaluate work done by a constant force for a problem involving motion in one dimension (W = Fs) Use conservation of linear momentum to solve problems e.g. linear motion in lifts, recoil of a gun, pile drivers etc. Evaluate work done by a constant force for a problem involving motion in two dimensions (W = F s) Determine work done by a variable force using integration (W = F idx or W = F vdt)
Applying skills to motion in a horizontal circle with uniform angular velocity (Force, Energy and Periodic Motion 1.2) Miss Mackay Skills include: Applying equations to motion in a horizontal circle with uniform angular velocity. Understand how to derive equations associated with circular motion under uniform angular velocity starting with r = r cos θ i + r sin θ j where r is constant and θ is variable before considering the special case where θ = ωt leading to the tangential and radial components for velocity (v t = rω and v r = 0) and acceleration (a t = 0 and a r = rω 2 ). This is covered in old unit 2 support notes. Understanding of how to derive these equations using a non-vector approach is also useful. Also covered in old unit 2 support notes. Know and make use of the equations: θ = ωt, a = ω 2 r, v = rω = rθ, T = 2π a = rω 2 = rθ 2 = v2 ω r, Be able to identify all forces acting on a particle and by resolving radially and vertically be able to identify the net force providing the acceleration towards the centre. Be familiar with common applications such as a particle sitting on a rotating disc, a particle rotating around a ring, a particle on a rod/string rotating around a point, a particle on the inner surface of a rotating cylinder. Use equations of motion in a horizontal circle alongside Newton s Inverse Square Law of Gravitation: a 1 or a = k where d is the distance between the centre of 2 particles, one d 2 d2 orbiting the other. Be familiar with Newton s Universal Law of Gravitation F = GMm. Note the universal constant, G, r 2 will not necessarily be given so students will need to be familiar with ways to get around this, e.g. F = ma = mrω 2 = GMm simplifying to constant GM = r 2 r3 ω 2. Escape velocity could be considered at this point or when doing differential equations at the end of the course. Solve simple problems involving motion in a horizontal circle making use of the equations: θ = ωt, v = rω = rθ, a = rω 2 = rθ 2 = v2 r, a = ω 2 r, T = 2π ω Solve a simple problem using Newton s Inverse square law of gravitation Solve more complex problems involving skidding, banking etc. Use Newton s inverse square law to solve more complex problems involving satellites, moons etc.
Applying skills to Simple Harmonic Motion (Force, Energy and Periodic Motion 1.3)
Block 4 Applying skills to Centre of Mass (Force, Energy and Periodic Motion 1.4) Moments about a point A = force x perpendicular distance from A to the line of action of the force. Idea that the point A is really an axis about which we take moments. Units for moments are the Newtonmetre. Algebraic sum of moments. Convention that clockwise moments are positive, though answers should be given in the form xnm clockwise or ynm anticlockwise. Resolving parallel and perpendicular to a force to find translator effect. Idea of a couple having a turning effect but no translatory effect. Like and unlike forces. Proving a system resolves to a couple. Finding the couple required for equilibrium. Replacing parallel forces with a resultant force. Position of the resultant. Parallel forces in equilibrium. Non-parallel forces in equilibrium. Replacing a system of forces with a single resultant force. Centre of gravity of a system of particles. Centre of gravity of a rigid body: lamina, rod, circular lamina, triangular lamina, semi-circular lamina. Centre of gravity of a composite lamina. Centre of gravity of a suspended lamina. Use of calculus to find the centre of gravity of a uniform composite lamina of area A bounded by a given curve y = f(x) and the lines x = a and x = b b a Ax = xydx b a Ay = 1 2 y2 dx Evaluating the turning effect of a single force or a set of forces acting on a body, considering clockwise and anticlockwise rotation. Equating the moments of several masses acting along a line to that of a mass acting at a point on the line m i x i = x m i where (x, 0) is the centre of mass of the system. Finding the positions of centres of mass of standard uniform plane laminas, including rectangle, triangle, circle and semicircle. For a circle, the centre of mass will be 2 along the 3 median from the vertex. For a semicircle, the centre of mass will be 4r along the axis of symmetry from the diameter. 3π Extending the idea of moments of several masses to two perpendicular directions to find the centre of mass of a set of particles arranged in a plane m i x i = x m i and m i y i = y m i where (x, y ) is the centre of mass of the system. Applying integration to find the centre of gravity of a uniform composite lamina of area A bounded by a given curve y = f(x) and the lines x = a and x = b b a Ax = xydx b a Ay = 1 2 y2 dx Applying calculus skills through techniques of integration (Mathematical techniques for Mechanics 1.3) Miss Mackay
Revision of work on Mathematical techniques for Mechanics
Topic: Further Calculus (Chain Rule and applications) Relationships and Calculus unit, Assessment standard 1.3 Applying calculus skills of differentiation (Chain rule only) Relationships and Calculus unit, Assessment standard 1.4 Applying calculus skills of integration (Reverse of chain rule only) Applications unit, Assessment standard 1.4 Applying calculus skills to optimisation and area
Block 4 Topic: Vectors Expressions and Functions unit, Assessment standard 1.4 Applying geometric skills to vectors