Curvilinear Motion

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Curvilinear Motion Equations of Motion

• ENGR 221
• April 16, 2003

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Lecture Goals

• Normal and Tangential Coordinates
• Rectilinear Equations of Motion

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Equations of motion
From Newtons Second Law of motion where the
force and acceleration are vector quantities.
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Equations of motion
The vectors can be broken into components similar
to equilibrium
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Equations of motion
From equilibrium equations
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Equations of motion
The difference between the equilibrium equations
and the equations of motion is that the right
hand side is no longer zero.
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Procedure for solving the Equations of Motion

• Draw the Free-Body Diagram.
• Write the equations of motion
• Solve for the velocities and accelerations need
  or obtain the equations for the components.
• Solve for the forces

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Example Problem Equation of Motion
A 80-kg skydiver falls at 85 m/s when she opens
her parachute. If her speed is reduced to 5 m/s
during the next 60 m of fall, determine the
average force exerted on her body by the
parachute during this interval.
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Example Problem Equation of Motion
Draw the free-body diagram of the skydiver.
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Example Problem Equation of Motion
Rewrite the equation The acceleration is going
to be a constant
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Example Problem Equation of Motion
The acceleration is defined as
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Example Problem Equation of Motion
Plug in the acceleration equation
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Class Problem Rectilinear Motion
The 2000-lb elevator cage is brought to rest from
an initial speed of 25 ft/s in a distance of 50
ft. Determine the uniform deceleration and the
tension in the elevator cable while the cage is
coming to rest.
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Example Problem Rectilinear Motion
Two bodies A and B with masses of 25 kg and 30
kg, respectively. During motion of the bodies,

• Determine the acceleration of body A and tension
  in the cable connecting the bodies.
• Determine the acceleration of body B if body A is
  replaced with a constant force of 245 N.

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Example Problem Rectilinear Motion
Draw the free body diagram of the boxes A and B.
The normal weight, NB and R is resistance due to
friction and the tension T.
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Example Problem Equation of Motion
Find the equations of motion for the block.
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Example Problem Equation of Motion
If there is no friction resistance then aAy
aBx
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Example Problem Equation of Motion
The force in the cable is T 245 N
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Class Problem Rectilinear Motion
Two bodies A and B with masses of 100 kg and 300
kg, respectively. During motion of the bodies,

• Determine the acceleration of body A and tension
  in the cable connecting the bodies if the fiction
  is neglected.

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Example Problem Rectilinear motion
Twp bodies A (40 kg) and B (30 kg) are connected
by a flexible cable. The kinetic coefficient of
friction between body A and the inclined surface
is mk 0.25 and the system is released from
rest. During the motion of the body determine.

• The acceleration of A
• The tension in the cable
• The velocity of body B after 5 sec.

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Example Problem Rectilinear motion
Draw the free body diagrams of the bodies
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Example Problem Rectilinear motion
The equations are
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Example Problem Rectilinear motion
Use the constraints of the two length of the
cable The direction so the block are moving
the upward
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Example Problem Rectilinear motion
The frictional force is Rearrange the
equations in terms of T
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Example Problem Rectilinear motion
The equations are
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Example Problem Rectilinear motion
The equations are
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Example Problem Rectilinear motion
The velocity can be found by using integration of
the acceleration
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Equations of motion
The difference between the linear and curvilinear
equations of motion is the defined system.
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Example Problem Curvilinear motion
The bob of a 2 m pendulum describes an arc of
circle in a vertical plane. If the tension in
the cord is 2.5 times the weight of the bob for
the position show, find the velocity and
acceleration of the bob at the position. .
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Example Problem Curvilinear motion
Draw the free body diagram of the mass .


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Example Problem Curvilinear motion
Solve for the acceleration .
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Example Problem Curvilinear motion
Compute the tangential velocity .
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Example Problem Curvilinear motion
Determine the rated speed of a highway curve of
radius r 400 ft. banked through an angle, q
18o. The rated speed of the banked curve road is
the speed at which a car should travel if no
lateral friction force is to be exerted on its
wheel.
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Example Problem Curvilinear motion
Determine the equation of motion for the car
with a radius r 400 ft. banked through an
angle, q 18o.
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Example Problem Curvilinear motion
Solve for R and find an.
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Example Problem Curvilinear motion
Compute the tangential velocity .
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Class Problem Curvilinear motion
An airplane is descending at an angle, q 20o
with respect to the horizontal when it drops a
bomb. If the altitude at the time of release is
5000 m and speed of the plane 750 km/hr,
determine the range (horizontal distance
traveled) of the bomb and the elapsed time before
it strikes the ground. Neglect air friction.
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Homework (Due 4/23/03)
Problems
15-10, 15-21,15-23, 15-57, 15-60, 15-61, 15-62,
15-63, 15-64
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Bonus Problem Curvilinear motion
A block B of mass m may slide freely on a
frictionless arm OA, which rotates in a
horizontal plane at a constant rate ,
Knowing that B is released at a distance r0 from
O and express as a function of r (a) the
components vr of the velocity of B along AO, (b)
the magnitude of the horizontal force F exerted
on B by the arm OA.
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Bonus Problem Curvilinear motion
Draw the free body diagram of the block B.


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Bonus Problem Curvilinear motion
Determine the velocity of the r component
Substitute into the equation for r double dot
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Bonus Problem Curvilinear motion
Integrate
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Bonus Problem Curvilinear motion
Insert the
to find the force.