Title: Curvilinear Motion
1Curvilinear Motion Equations of Motion
2Lecture Goals
- Normal and Tangential Coordinates
- Rectilinear Equations of Motion
3Equations of motion
From Newtons Second Law of motion where the
force and acceleration are vector quantities.
4Equations of motion
The vectors can be broken into components similar
to equilibrium
5Equations of motion
From equilibrium equations
6Equations of motion
The difference between the equilibrium equations
and the equations of motion is that the right
hand side is no longer zero.
7Procedure 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
8Example 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.
9Example Problem Equation of Motion
Draw the free-body diagram of the skydiver.
10Example Problem Equation of Motion
Rewrite the equation The acceleration is going
to be a constant
11Example Problem Equation of Motion
The acceleration is defined as
12Example Problem Equation of Motion
Plug in the acceleration equation
13Class 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.
14Example 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.
15Example 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.
16Example Problem Equation of Motion
Find the equations of motion for the block.
17Example Problem Equation of Motion
If there is no friction resistance then aAy
aBx
18Example Problem Equation of Motion
The force in the cable is T 245 N
19Class 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.
20Example 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.
21Example Problem Rectilinear motion
Draw the free body diagrams of the bodies
22Example Problem Rectilinear motion
The equations are
23Example Problem Rectilinear motion
Use the constraints of the two length of the
cable The direction so the block are moving
the upward
24Example Problem Rectilinear motion
The frictional force is Rearrange the
equations in terms of T
25Example Problem Rectilinear motion
The equations are
26Example Problem Rectilinear motion
The equations are
27Example Problem Rectilinear motion
The velocity can be found by using integration of
the acceleration
28Equations of motion
The difference between the linear and curvilinear
equations of motion is the defined system.
29Example 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. .
30Example Problem Curvilinear motion
Draw the free body diagram of the mass .
31Example Problem Curvilinear motion
Solve for the acceleration .
32Example Problem Curvilinear motion
Compute the tangential velocity .
33Example 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.
34Example Problem Curvilinear motion
Determine the equation of motion for the car
with a radius r 400 ft. banked through an
angle, q 18o.
35Example Problem Curvilinear motion
Solve for R and find an.
36Example Problem Curvilinear motion
Compute the tangential velocity .
37Class 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.
38Homework (Due 4/23/03)
Problems
15-10, 15-21,15-23, 15-57, 15-60, 15-61, 15-62,
15-63, 15-64
39Bonus 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.
40Bonus Problem Curvilinear motion
Draw the free body diagram of the block B.
41Bonus Problem Curvilinear motion
Determine the velocity of the r component
Substitute into the equation for r double dot
42Bonus Problem Curvilinear motion
Integrate
43Bonus Problem Curvilinear motion
Insert the
to find the force.