KINETICS of PARTICLES Newton

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KINETICS of PARTICLESNewtons 2nd Law The
Equation of Motion
Lecture VI
Or
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Newtons 2nd Law The Equation of Motion

• Kinetics is the study of the relations between
  the unbalanced forces and the changes in motion
  that they produce.
• Newtons 2nd law states that the particle will
  accelerate when it is subjected to unbalanced
  forces. The acceleration of the particle is
  always in the direction of the applied forces.
• Newtons 2nd law is also known as the equation of
  motion.
• To solve the equation of motion, the choice of an
  appropriate coordinate systems depends on the
  type of motion involved.
• Two types of problems are encountered when
  applying this equation
• The acceleration of the particle is either
  specified or can be determined directly from
  known kinematic conditions. Then, the
  corresponding forces, which are acting on the
  particle, will be determined by direct
  substitution.
• The forces acting on the particle are specified,
  then the resulting motion will be determined.
  Note that, if the forces are constant, the
  acceleration is also constant and is easily found
  from the equation of motion. However, if the
  forces are functions of time, position, or
  velocity, the equation of motion becomes a
  differential equation which must be integrated to
  determine the velocity and displacement.
• In general, there are three general approaches to
  solve the equation of motion the direct
  application of Newtons 2nd law, the use of the
  work energy principles, and the impulse and
  momentum method.

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Newtons 2nd Law The Equation of Motion (Cont.)
F2
ma
FR ?F

F1
P
P
Kinetic Diagram
Free-body Diagram
Note The equation of motion has to be applied in
such way that the measurements of acceleration
are made from a Newtonian or inertial frame of
reference. This coordinate does not rotate and is
either fixed or translates in a given direction
with a constant velocity (zero acceleration).
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Newtons 2nd Law The Equation of Motion (Cont.)
Curvilinear Motion
Rectilinear Motion
Rectangular Coordinates
n-t Coordinates
Polar Coordinates
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Newtons 2nd Law The Equation of Motion
Exercises
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Exercise 1

• 3/1 The 50-kg crate is projected along the
  floor with an initial speed of 7 m/s at x 0.
  The coefficient of kinetic friction is 0.40.
  Calculate the time required for the crate to come
  to rest and the corresponding distance x traveled.

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Exercise 2

• 3/2 The 50-kg crate of Prob. 3/1 is now
  projected down an incline as shown with an
  initial speed of 7 m/s. Investigate the time t
  required for the crate to come to rest and the
  corresponding distance x traveled if (a) q 15
  (b) q 30.

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Exercise 3
3/4 During a brake test, the rear-engine car is
stopped from an initial speed of 100 km/h in a
distance of 50 m. If it is known that all four
wheels contribute equally to the braking force,
determine the braking force F at each wheel.
Assume a constant deceleration for the 1500-kg
car.
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Exercise 4
3/6 The 300-Mg jet airliner has three engines,
each of which produces a nearly constant thrust
of 240 kN during the takeoff roll. Determine the
length s of runway required if the takeoff speed
is 220 km/h. Compute s first for an uphill
takeoff direction from A to B and second for a
downhill takeoff from B to A on the slightly
inclined runway. Neglect air and rolling
resistance.
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Exercise 5

• 3/17 The coefficient of static friction between
  the flat bed of the truck and the crate it
  carries is 0.30. Determine the minimum stopping
  distance s which the truck can have from a speed
  of 70 km/h with constant deceleration if the
  crate is not to slip forward.

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Exercise 6
3/28 The system is released from rest with the
cable taut. For the friction coefficients ms
0.25 and mk 0.20, calculate the acceleration of
each body and the tension T in the cable. Neglect
the small mass and friction of the pulleys.
20 kg
B
Problem 3/28
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Exercise 7
3/43 The sliders A and B are connected by a
light rigid bar of length l 0.5 m and move with
negligible friction in the horizontal slots
shown. For the position where xA 0.4 m, the
velocity of A is vA 0.9 m/s to the right.
Determine the acceleration of each slider and the
force in the bar at this instant.
P 40 N
Problem 3/43
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Exercise 8
3/51 If the 80-kg ski-jumper attains a speed of
25 m/s as he approaches the takeoff position,
calculate the magnitude N of the normal force
exerted by the snow on his skis just before he
reaches A.
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Exercise 9
3/54 The hollow tube is pivoted about a
horizontal axis through point O and is made to
rotate in the vertical plane with a constant
counterclockwise angular velocity q. 3 rad/s.
If a 0.l-kg particle is sliding in the tube
toward O with a velocity of 1.2 m/s relative to
the tube when the position q 30 is passed,
calculate the magnitude N of the normal force
exerted by the wall of the tube on the particle
at this instant.
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Exercise 10
3/71 A small object A is held against the
vertical side of the rotating cylindrical
container of radius r by centrifugal action. If
the coefficient of static friction between the
object and the container is ms, determine the
expression for the minimum rotational rate q. w
of the container which will keep the object from
slipping down the vertical side.