b) When the rope starts to brake the fall at t = to the rope is stretched resulting in a force F. Let y = y(t) be the length of the rope stretched at time t. Formulate an initial value problem that describes the fall for t > to = 0.
b) When the rope starts to brake the fall at t = to the rope is stretched resulting in a force F. Let y = y(t) be the length of the rope stretched at time t. Formulate an initial value problem that describes the fall for t > to = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Question b)
Transcribed Image Text:We are going to create a model for the forces acting when a
climber with mass m falls on a rope. To simplify the problem, we
will I assume that there is no external and internal friction
associated with the rope, and that there is no slack in the rope
when the climber falls a height h/2 above the last belay, see
Figure 1. The rope has length L (in meters m) when the fall
occurs. All motion in the model is vertical (along the y-axis), and
we will assume that the positive direction is downward.
Figure 1: Climber with mass m falling freely for a length h until
the elastic rope starts to brake the fall at y = 0 and t = to = 0. At
this time, the climber has speed vo. The rope has length L, is
fixed at point S, and is without slack when the climber falls.
σ = Eε
Here, tension o is defined as force per area:
a) The climber is in free fall until the rope starts to catch the fall. Set up the model that applies
to free fall and derive the expression for the speed vo the climber has when the rope starts to
brake the fall. In this subquestion and in subquestion d) you can assume that the speed of the
climber is only a function of y such that:
dy
v(y).
d²y dv
dt² dy
= V
dt
b) When the rope starts to brake the fall at t = to the rope is stretched resulting in a force F.
Let y = y(t) be the length of the rope stretched at time t. Formulate an initial value problem that
describes the fall for t > to = 0.
F
A
and & is the deformation (strain) of the rope given by the relative stretch
c) In the problem you have found in b) the force from the rope that brakes the fall is included. We will
assume that this follows Hook's law. Using the terminology from elasticity theory, we can express
Hook's law by the tension (stress) o on the rope:
y(t)
-
F₂ (y)
9
W²
0 =
ε =
(1 - cos(wt)) +
Y
L
EA
L
L
Elastic modulus E is a material constant and A is the cross section of the rope. We will assume
that EA is constant. Show that the tensile forceF, in the model with these assumptions
becomes
Y
(1)
and that the solution of the initial value problem from b) given the force (4) becomes:
√2gh
W
-sin(wt)
(2)
(3)
h
y = 0
(4)
(5)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
