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MM203 Mechanics of Machines: Part 1 1 Module • Lectures • Tutorials • Labs • Why study dynamics? • Problem solving 2 Vectors PQ R R R Q Q PQ QP P P R P R Q R Q 5 2 7 3 1 4 5 5 3 3 −Q P P Q R P Q R 3 Unit vectors - components 4 1 0 3 40 31 v vx i v y j vz k v v vx v y vz 2 2 2 4 3 ? 4 4 3 3 0 4 Direction cosines • l, m, and n – direction cosines between v and x-, y-, and z-axes vy vx vz l , m , n v v v v vli mj nk , l m n 1 2 – Calculate 3 direction cosines for 2 2 3 4 6 8 5 Dot (or scalar) product P Q Px Qx Py Q y Pz Qz P Q PQ cos Q P • Component of Q in P direction P Q QP Q cos P P Q QP 6 Angle between vectors P Q PxQx Py Qy Pz Qz cos PQ PQ lP lQ mP mQ nP nQ i i ? ij ? PP P 2 7 Dot product • Commutative and distributive P Q Q P P Q R P Q P R 8 Particle kinematics • What is kinematics? • What is a particle? • • • • Rectilinear motion - review Plane curvilinear motion - review Relative motion Space curvilinear motion 9 Rectilinear motion ds v dt dv a dt • Combining gives v dv a ds • What do dv and ds represent? 10 Example • The acceleration of a particle is a = 4t − 30 (where a is in m/s2 and t is in seconds). Determine the velocity and displacement in terms of time. (Problem 2/5, M&K) 11 Vector calculus d uP uP uP dt • Vectors can vary both in length and in direction P Px i Py j Pz k 12 Plane curvilinear motion • Choice of coordinate system (axes) – Depends on problem – how information is given and/or what simplifies solution – Practice 13 Plane curvilinear motion • Rectangular coordinates • Position vector - r y j r xi yj v r xi yj vx i v y j i x a v r xi yj ax i a y j • e.g. projectile motion • ENSURE CONSISTENCY IN DIRECTIONS 14 Plane curvilinear motion • Normal and tangential coordinates • Instantaneous radius of curvature – r ds r db ds db v v r dt dt y Note that dr/dt can be ignored in this case – see M&K. • What is direction of v? r db ds x 15 Plane curvilinear motion v vet rbet d vet a v ve t vet dt e t be n v 2 an vb rb 2 r en et r at v s 16 Example • A test car starts from rest on a horizontal circular track of 80 m radius and increases its speed at a uniform rate to reach 100 km/h in 10 seconds. Determine the magnitude of the acceleration of the car 8 seconds after the start. (Answer: a = 6.77 m/s2). (Problem 2/97, M&K) 17 Example • To simulate a condition of “weightlessness” in its cabin, an aircraft travelling at 800 km/h moves an a sustained curve as shown. At what rate in degrees per second should the pilot drop his longitudinal line of sight to effect the desired condition? Use g = 9.79 m/s2. (Answer: db/dt = 2.52 deg/s). (Problem 2/111, M&K) b 18 Example • A ball is thrown horizontally at 15 m/s from the top of a cliff as shown and lands at point C. The ball has a horizontal acceleration in the negative x-direction due to wind. Determine the radius of curvature of the path at B where its trajectory makes an angle of 45° with the horizontal. Neglect air resistance in the vertical direction. (Answer: r = 41.8 m). (Problem 2/125, M&K) A B 50 m C 40 m x 19 Plane curvilinear motion • Polar coordinates r re r e r e , e e r v r re r re r v re re r y e er r x 20 Plane curvilinear motion a ar e r a e where 2 a r r r 1 d 2 a r 2r r r dt 21 Example • An aircraft flies over an observer with a constant speed in a straight line as shown. Determine the signs (i.e. +ve, -ve, or 0) for r , r , r, , , and y • for positions • A, B, and C. • (Problem 2/134, M&K) C B v A r x 22 Example • At the bottom of a loop at point P as shown, an aircraft has a horizontal velocity of 600 km/h and no horizontal acceleration. The radius of curvature of the loop is 1200 m. For the radar tracking station shown, determine the recorded values of d2r/dt2 and d2/dt2 for this instant. (Answer: d2r/dt2 = 12.5 m/s2, d2/dt2 = 0.0365 rad/s2). (Problem 2/141, M&K) P r 400 m 600 km/h 1000 m 23 Relative motion • Absolute (fixed axes) • Relative (translating axes) • Used when measurements are taken from a moving observation point, or where use of moving axes simplifies solution of problem. • Motion of moving coordinate system may be specified w.r.t. fixed system. 24 Relative motion • Set of translating axes (x-y) attached to Y particle B (arbitrarily). The position of A relative to the frame x-y (i.e. relative to B) is O y rA A rA/B B x rB X rA / B xi yj 25 Relative motion • Absolute positions of points A and B (w.r.t. fixed axes X-Y) are related by rA rB rA / B or rB rA rB / A where rB / A rA / B 26 Relative motion • Differentiating w.r.t. time gives rA rB rA / B v A v B v A/ B a A a B a A/ B v A/ B v B / A a A / B a B / A • Coordinate systems may be rectangular, tangential and normal, polar, etc. 27 Inertial systems • A translating reference system with no acceleration is known as an inertial system. If aB = 0 then a A a A/ B • Replacing a fixed reference system with an inertial system does not affect calculations (or measurements) of accelerations (or forces). 28 Example • A yacht moving in the direction shown is tacking windward against a north wind. The log registers a hull speed of 6.5 knots. A “telltale” (a string tied to the rigging) indicates that the direction of the apparent wind is 35° from the centerline of the boat. What is the true wind velocity? (Answer: vw = 14.40 knots). (Problem 2/191, M&K) vw 50° 35° 29 Example • To increase his speed, the water skier A cuts across the wake of the boat B which has a velocity of 60 km/h as shown. At the instant when = 30°, the actual path of the skier makes an angle b = 50° with the tow rope. For this position, determine the velocity vA of the skier and the value of d/dt. (Answer: vA = 80.8 km/h, d/dt = 0.887 rad/s). (Problem 2/193, M&K) A b B m 0 1 vB 30 Example • Car A is travelling at a constant speed of 60 km/h as it rounds a circular curve of 300 m radius. At the instant shown it is at = 45°. Car B is passing the centre of the circle at the same instant. Car A is located relative to B using polar coordinates with the pole moving with B. For this instant, determine vA/B and the values fo d/dt and dr/dt as measured by an observer in car B. (Answer: vA/B = 36.0 m/s, d/dt = 0.1079 rad/s, dr/dt = −15.71 m/s). (Problem 2/201, M&K) A r B 31 Space curvilinear motion • • • • Rectangular coordinates (x, y, z) Cylindrical coordinates (r, , z) Spherical coordinates (R, , f) Coordinate transformations – not covered • Tangential and normal system not used due to complexity involved. 32 Space curvilinear motion • Rectangular coordinates (x, y, z) – similar to 2D R xi yj zk v xi y j zk a xi yj zk 33 Space curvilinear motion • Cylindrical coordinates (r, , z) R rer zk z R y r v re r re zk z x 2 a r r e r r 2r e zk 34 Space curvilinear motion • Spherical coordinates (R, , f) v R e R R cos fe Rfef z a aR e R a e af ef where Rf 2 R 2 cos 2 f aR R cos f d 2 a R 2 Rf sin f R dt 1 d 2 af R f R 2 sin f cos f R dt R y f R x 35 Example • A section of a roller-coaster is a horizontal cylindrical helix. The velocity of the cars as they pass point A is 15 m/s. The effective radius of the cylindrical helix is 5 m and the helix angle is 40°. The tangential acceleration at A is gcosg. Compute the magnitude of the acceleration of the passengers as they pass A. (Answer: a = 27.5 m/s2). (Problem 2/171, M&K) 5m A A g = 40° 36 Example • The robot shown rotates about a fixed vertical axis while its arm extends and elevates. At a given instant, f = 30°, df/dt = 10 deg/s = constant, l = 0.5 m, dl/dt = 0.2 m/s, d2l/dt2 = −0.3 m/s2, and W = 20 deg/s = constant. Determine the magnitudes of the velocity and acceleration of the gripped part P. (Answer: v = 0.480 m/s, a = 0.474 m/s2). (Problem 2/177, M&K) y 5 0.7 O l W P m f x 37 Particle kinetics • Newton’s laws F ma • Applied and reactive forces must be considered – free body diagrams • Forces required to produce motion • Motion due to forces 38 Particle kinetics • Constrained and unconstrained motion • Degrees of freedom • Rectilinear motion – covered • Curvilinear motion 39 Rectilinear motion - example F ma • The 10 Mg truck hauls a 20 Mg trailer. If the unit starts from rest on a level road with a tractive force of 20 kN between the driving wheels and the road, compute the tension T in the horizontal drawbar and the acceleration a of the rig. (Answer: T = 13.33 kN, a = 0.667 m/s2). (Problem 3/5, M&K) 20 Mg 10 Mg 40 Example • The motorized drum turns at a constant speed causing the vertical cable to have a constant downwards velocity v. Determine the tension in the cable in terms of y. Neglect the diameter and mass of the small pulleys. (Problem 3/48, M&K) 2b y v • Answer: 2 2 m b v 2 2 T b y g 3 2y 4y m 41 Curvilinear motion • Rectangular coordinates F x max , F y may • Normal and tangential coordinates F n man , • Polar coordinates F r mar , F ma t t F ma 42 00 10 • A pilot flies an airplane at a constant speed of 600 km/h in a vertical circle of radius 1000 m. Calculate the force exerted by the seat on the 90 kg pilot at point A and at point A. (Answer: RA = 3380 N, RB = 1617 N). (Problem 3/63, m Example M&K) 600 km/h 43 Example • The 30 Mg aircraft is climbing at an angle of 15° under a jet thrust T of 180 kN. At the instant shown, its speed is 300 km/h and is increasing at a rate of 1.96 m/s2. Also is decreasing as the aircraft begins to level off. If the radius of curvature at this instant is 20 km, compute the lift L and the drag D. (Lift and drag are the aerodynamic forces normal to and opposite to the flight direction, respectively). (Answer: D = 45.0 kN, L = 274 kN). (Problem 3/69, M&K) T 44 Example • A child's slide has a quarter circle shape as shown. Assuming that friction is negligible, determine the velocity of the child at the end of the slide ( = 90°) in terms of the radius of curvature r and the initial angle 0. • Answer v 2 gr 1 sin 0 45 Slide • Does it matter what profile slide has? • What if friction added? 46 Example • A flat circular discs rotates about a vertical axis through the centre point at a slowly increasing angular velocity w. With w = 0, the position of the two 0.5 kg sliders is x = 25 mm. Each spring has a stiffness of 400 x N/m. Determine the value of x for w = 240 rev/min and the normal force exerted by the side of the slot on the block. Neglect any friction and the mass of the springs. (Answer: x = 118.8 mm, N = 25.3 N). (Problem 3/83, w x 80 mm 80 mm M&K) 47 Work and energy • Work/energy analysis – don’t need to calculate accelerations A′ • Work done by force F F dr A dU F dr F ds cos a a where ds dr R+dr r O • Integration of F = ma w.r.t. displacement gives equations for work and energy 48 Work and energy • Active forces and reactive forces (constraint forces that do no work) • Total work done by force U F dr Fx dx Fy dy Fz dz or U Ft ds • where Ft = tangential force component 49 Work and energy • If displacement is in same direction as force then work is +ve (otherwise –ve) • Ignore reactive forces • Kinetic energy T mv 1 2 2 • Gravitational potential energy Vg mgh 50 Example • A small vehicle enters the top of a circular path with a horizontal velocity v0 and gathers speed as it moves down the path. Determine the angle b (in terms of v0) at which it leaves the path and becomes a projectile. Neglect friction and treat the vehicle as a particle. (Problem 3/87, M&K) • Answer: v0 R b 2 v0 1 2 b cos 3 3 gR 51 Example • The small slider of mass m is released from point A and slides without friction to point D. From point D onwards the coefficient of kinetic friction between the slider and the slide is mk. Determine the distance s travelled by the slider up the incline beyond D. m A 2R s B R D 30° (Problem 3/125, M&K) • Answer: s 4R C 1 mk 3 52 Example • A rope of length pr/2 and mass per unit length r is released with = 0 in a smooth vertical channel and falls through a hole in the supporting surface. Determine the velocity v of the chain as the last part of it leaves the slot. (Problem 3/173, M&K) • Answer: r p 4 v gr 2 p 53 Linear impulse and momentum • Integration of F = ma w.r.t. time gives equations of impulse and momentum. • Useful where time over which force acts is very short (e.g. impact) or where force acts over specified length of time. 54 Linear impulse and momentum d F m v m v G dt • If mass m is constant then sum of forces = time rate of change of linear momentum • Linear momentum of particle G mv • Units – kg·m/s or N·s • Scalar form: F G , F G , F G x x y y z z 55 Linear impulse and momentum • Integrate over time F dt G t2 t1 2 G1 G or G1 t2 t1 F dt G 2 • Product of force and time is called linear impulse t • Scalar form t Fx dt mvx 2 mvx 1 , etc. 2 1 56 Linear impulse and momentum • Note that all forces must be included (i.e. both active and reactive) 57 Linear impulse and momentum ? m 2 v2 F m 1 v1 −F ? • If there are no unbalanced forces acting on a system then the total linear momentum of the system will remain constant (principle of conservation of linear momentum) 58 Impact • How to determine velocities after impact? • Forces normal to contact surface. Fd is force during deformation period while Fr is force during recovery period. • The ratio of the restoration impulse to the deformation impulse is called the coefficiente of restitution (V2)n m2 (V1)n m1 t t0 t0 0 Fr dt Fd dt 59 Impact • For particle 1, (v0)n being the intermediate normal velocity component (of both particles) and (v1)′n being normal velocity component after collision e t t0 t0 0 m1 v1 n m1 v0 n Fd dt m1 v0 n m1 v1 n Fr dt • Similarly for particle 2 m2 v2 n m2 v0 n e m2 v0 n m2 v2 n 60 Impact • Combining gives v2 n v1 n e v1 n v2 n • e = 0 for plastic impact, e = 1 for elastic impact • Note that tangential velocities are not affected by impact 61 Example • A 75 g projectile traveling at 600 m/ strikes and becomes embedded in the 50 kg block which is initially stationary. Compute the energy lost during the impact. Express your answer as an absolute value and as a percentage of the original energy of the system. (Problem 3/180, M&K) 75 g 600 m/s 50 kg 62 Example d/2 d/2 • The pool ball shown must be hit so as to travel into the side pocket as shown. Specify the location x of the cushion impact if e = 0.8. (Answer: x = 0.268d) (Problem 3/251, M&K) x d 63 Example • The vertical motion of the 3 kg load is controlled by the forces P applied to the end rollers of the framework shown. If the upward velocity of the cylinder is increased from 2 m/s to 4 m/s in 2 seconds, calculate the average force Rav under each of the two rollers during the 2 s interval. Neglect the small mass of the frame. (Answer: Rav = 16.22 N) v P 3 kg P (Problem 3/199, M&K) 64 Example • A 1000 kg spacecraft is traveling in deep space with a speed vs = 2000 m/s when struck at its mass centre by a 10 kg meteor with velocity vm of magnitude 5000 m/s. The meteor becomes embedded in the satellite. Determine the final velocity of the spacecraft. (Answer: v = 36.9i + 1951j – 14.76k m/s) (Problem 3/201, M&K) z vm 4 x 2 5 vs y 65 Cross (or vector) product • Magnitude of cross-product P Q PQ sin Q P PQ sin PQ sin Q P • Direction of cross-product governed by righthand rule 66 Right-hand rule PQ R • Middle finger in direction of R if thumb in direction of P and index finger in direction of Q. • Use right-handed reference frame for x,y, and z. 67 Cross (or vector) product i j k , etc. i i 0 , etc. P×Q Q P Q×P=−P×Q • Distributive P Q R P Q P R 68 Cross (or vector) product P Q y z i j k P Q Px Py Pz Qx Qy Qz Pz Qy i Px Qz Pz Qx j Px Qy Py Qx k • Derivative d P Q P Q PQ dt 69 Angular impulse and momentum • The angular momentum of a particle about any point is the moment of the linear momentum about that point. • Units are kg·m/s·m or N·m·s 70 Angular impulse and momentum • Planar motion • There are 3 components of the angular momentum of P about arbitrary point O: i.e. about x-,y-, and z-axes. y mv P r O x 71 Angular impulse and momentum • Since P is coplanar with x- and y-axes, it has no moment about these axes. It only has a moment about the zaxis. y mv P r O x 72 Angular impulse and momentum • Is angular momentum of P about O positive or negative? – governed by right-hand rule y mv P r O x 73 Right-hand rule • Curl fingers in. Rotation indicated by fingers is in direction of thumb. Is this positive or negative in this case? 74 Angular impulse and momentum p H O mv r cos mv r sin 2 y • Direction of component about zaxis is in z-direction HO mv r sin k mv P r O x 75 Angular impulse and momentum H O x mvy y mvx k mvy mv y P mvx r O x x 76 Angular impulse and momentum H O r mv • Note r mv mv r H x m y vz z v y , etc. 77 Angular impulse and momentum • The resultant moment of all forces about O is MO r F • From Newton’s 2nd law MO r mv • Differentiate w.r.t. time H O r mv v mv r mv H • Now • so v mv 0 O M H O O 78 Angular impulse and momentum • The moment of all forces on the particle about a fixed point O equals the time rate of change of the angular momentum about that point. M H O x O x , etc. • If moment about O is zero then angular momentum is constant (principle of conservation of angular momentum). • If moment about any axis is zero then component of angular momentum about that axis is constant. 79 Angular impulse and momentum • Particle following circular path at constant angular velocity. Is angular momentum about O varying with time? • Is angular momentum about O′ varying with time? • Is component about z-axis varying with time? z O′ O y x w m 80 Angular impulse and momentum M H O O M t2 t1 O dt H O 2 H O1 • i.e. change in angular momentum is equal to total angular impulse M t2 t1 Ox dt H O x 2 H O x 1 , etc. 81 Angular impulse and momentum • Example – ice skater w1 w2 82 Example • Calculate HO, the angular momentum of the particle shown about O (a) using the vector definition and (b) using a geometrical approach. The centre of the particle lies in the x-y plane. (Answer: HO = 128.7k N·m·s) (Problem 3/221, M&K) y 2 kg 7 m/s 30° 8m O 6m x 83 Example • A particle of mass m moves with negligible friction across a horizontal surface and is connected by a light spring fastened at point O. The velocity at A is as shown. Determine the velocity at B. (Problem 3/226, M&K) A 350 mm vA = 4 m/s 54° O 230 mm B 65° vB 84 Example r D 2r w0 • Each of 4 spheres of mass m is treated as a particle. Spheres A and B are mounted on a light rod and are rotating initially with an angular velocity w0 about a vertical axis through O. The other two spheres are similarly (but independently) mounted and have no initial velocity. When assembly AB reaches the position indicated it latches with CD and the two move with a common angular velocity w. Neglect friction. Determine w and n the percentage loss of kinetic energy. (Answer w = w0/5, n = 80%). (Problem 3/227, r O B A 2r M&K) C 85 Example • The particle of mass m is launched from point O with a horizontal velocity u at time t = 0. Determine its angular momentum about O as a function of t. (Answer H0 = −½mgut2k). (Problem 3/233, M&K) m u O y x 86 Relative motion • Fixed reference frame X-Y • Moving reference frame x-y a A a B a rel y A F ma A ma B a rel Y F m a rel rA/B = rrel rA x B rB O X 87 Relative motion • Special case – inertial system or “Newtonian frame of reference” with zero acceleration • Note that work-energy and impulse momentum equations are equally valid in inertial system – but relative momentum/relative energy etc. will, in general, be different to those measured relative to fixed frame of reference. 88 Example • The ball A of mass 10 kg is attached to the light rod of length l = 0.8 m. The rod is attached to a carriage of mass 250 kg which moves on rails with an acceleration aO as shown. The rod is free to rotate horizontally about O. If d/dt = 3 rad/s when = 90°, find the kinetic energy T of the system if the carriage has a velocity of 0.8 m/s. Treat the ball as a particle. (Answer: T = 112 J). (Problem 3/311, M&K) A O l aO 89 Example • The small slider A moves with negligible friction down the tapered block, which moves to the right with constant speed v = v0. Use the principle of work-energy to determine the magnitude vA of the absolute velocity of the slider as it passes point C if it is released at point B with no velocity relative to the block. (Problem 3/316, M&K) • Answer: 2 v A v0 2 gl sin 2v0 cos 2 gl sin B v A l C 90