MM203 Mechanics of Machines: Part 1

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MM203Mechanics of Machines Part 1
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Module

• Lectures
• Tutorials
• Labs
• Why study dynamics?
• Problem solving

3
Vectors
4
Unit vectors - components
5
Direction cosines

• l, m, and n direction cosines between v and x-,
  y-, and z-axes

• Calculate 3 direction cosines for

6
Dot (or scalar) product

• Component of Q in P direction

7
Angle between vectors
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Dot product

• Commutative and distributive

9
Particle kinematics

• What is kinematics?
• What is a particle?
• Rectilinear motion - review
• Plane curvilinear motion - review
• Relative motion
• Space curvilinear motion

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Rectilinear motion

• Combining gives
• What do dv and ds represent?

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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, MK)

12
Vector calculus

• Vectors can vary both in length and in direction

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Plane curvilinear motion

• Choice of coordinate system (axes)
• Depends on problem how information is given
  and/or what simplifies solution
• Practice

14
Plane curvilinear motion

• Rectangular coordinates
• Position vector - r

• e.g. projectile motion
• ENSURE CONSISTENCY IN DIRECTIONS

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Plane curvilinear motion

• Normal and tangential coordinates
• Instantaneous radius of curvature r
• What is direction of v?

Note that dr/dt can be ignored in this case see
MK.
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Plane curvilinear motion
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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,
  MK)

18
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, MK)

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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, MK)

20
Plane curvilinear motion

• Polar coordinates

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Plane curvilinear motion
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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
• for positions
• A, B, and C.
• (Problem 2/134, MK)

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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 d2q/dt2 for this instant.
  (Answer d2r/dt2 12.5 m/s2, d2q/dt2 0.0365
  rad/s2). (Problem 2/141, MK)

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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.

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Relative motion

• Set of translating axes (x-y) attached to
  particle B (arbitrarily). The position of A
  relative to the frame x-y (i.e. relative to B) is

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Relative motion

• Absolute positions of points A and B (w.r.t.
  fixed axes X-Y) are related by

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Relative motion

• Differentiating w.r.t. time gives
• Coordinate systems may be rectangular, tangential
  and normal, polar, etc.

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Inertial systems

• A translating reference system with no
  acceleration is known as an inertial system. If
  aB 0 then
• Replacing a fixed reference system with an
  inertial system does not affect calculations (or
  measurements) of accelerations (or forces).

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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, MK)

30
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
  q 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 dq/dt. (Answer vA 80.8 km/h,
  dq/dt 0.887 rad/s). (Problem 2/193, MK)

31
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 q 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 dq/dt and dr/dt as measured by an
  observer in car B. (Answer vA/B 36.0 m/s,
  dq/dt 0.1079 rad/s, dr/dt -15.71 m/s).
  (Problem 2/201, MK)

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Space curvilinear motion

• Rectangular coordinates (x, y, z)
• Cylindrical coordinates (r, q, z)
• Spherical coordinates (R, q, f)
• Coordinate transformations not covered
• Tangential and normal system not used due to
  complexity involved.

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Space curvilinear motion

• Rectangular coordinates (x, y, z) similar to 2D

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Space curvilinear motion

• Cylindrical coordinates (r, q, z)

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Space curvilinear motion

• Spherical coordinates (R, q, f)

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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, MK)

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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, MK)

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Particle kinetics

• Newtons laws
• Applied and reactive forces must be considered
  free body diagrams
• Forces required to produce motion
• Motion due to forces

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Particle kinetics

• Constrained and unconstrained motion
• Degrees of freedom
• Rectilinear motion covered
• Curvilinear motion

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Rectilinear motion - example

• 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, MK)

41
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, MK)

• Answer

42
Curvilinear motion

• Rectangular coordinates
• Normal and tangential coordinates
• Polar coordinates

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Example

• 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, MK)

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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 q 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, MK)

45
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 (q 90) in terms of the radius of
  curvature r and the initial angle q0.
• Answer

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Slide

• Does it matter what profile slide has?
• What if friction added?

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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 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, MK)

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Work and energy

• Work/energy analysis dont need to calculate
  accelerations
• Work done by force F
• Integration of F ma w.r.t. displacement gives
  equations for work and energy

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Work and energy

• Active forces and reactive forces (constraint
  forces that do no work)
• Total work done by force
• where Ft tangential force component

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Work and energy

• If displacement is in same direction as force
  then work is ve (otherwise ve)
• Ignore reactive forces
• Kinetic energy
• Gravitational potential energy

51
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, MK)
• Answer

52
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. (Problem 3/125, MK)
• Answer

53
Example

• A rope of length pr/2 and mass per unit length r
  is released with q 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, MK)
• Answer

54
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.

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Linear impulse and momentum

• If mass m is constant then sum of forces time
  rate of change of linear momentum
• Linear momentum of particle
• Units kgm/s or Ns
• Scalar form

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Linear impulse and momentum

• Integrate over time
• Product of force and time is called linear
  impulse
• Scalar form

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Linear impulse and momentum

• Note that all forces must be included (i.e. both
  active and reactive)

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Linear impulse and momentum

• 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)

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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 coefficient of
  restitution

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Impact

• For particle 1, (v0)n being the intermediate
  normal velocity component (of both particles) and
  (v1)'n being normal velocity component after
  collision
• Similarly for particle 2

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Impact

• Combining gives
• e 0 for plastic impact, e 1 for elastic
  impact
• Note that tangential velocities are not affected
  by impact

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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,
  MK)

63
Example

• 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, MK)

64
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) (Problem 3/199,
  MK)

65
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, MK)

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Cross (or vector) product

• Magnitude of cross-product
• Direction of cross-product governed by right-hand
  rule

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Right-hand rule

• 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.

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Cross (or vector) product

• Distributive

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Cross (or vector) product

• Derivative

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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 kgm/sm or Nms

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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.

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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 z-axis.

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Angular impulse and momentum

• Is angular momentum of P about O positive or
  negative? governed by right-hand rule

74
Right-hand rule

• Curl fingers in. Rotation indicated by fingers is
  in direction of thumb. Is this positive or
  negative in this case?

75
Angular impulse and momentum

• Direction of component about z-axis is in
  z-direction

76
Angular impulse and momentum
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Angular impulse and momentum

• Note

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Angular impulse and momentum

• The resultant moment of all forces about O is
• From Newtons 2nd law
• Differentiate w.r.t. time
• Now
• so

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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.
• 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.

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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?

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Angular impulse and momentum

• i.e. change in angular momentum is equal to total
  angular impulse

82
Angular impulse and momentum

• Example ice skater

83
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 Nms) (Problem 3/221, MK)

84
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, MK)

85
Example

• 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, MK)

86
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, MK)

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Relative motion

• Fixed reference frame X-Y
• Moving reference frame x-y

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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.

89
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 dq/dt 3 rad/s
  when q 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, MK)

90
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,
  MK)
• Answer