Partnership for Integration of Computation into Undergraduate Physics

1) Consider the case of a stiff rope (i.e no elasticity). Assume the climber, mass $m$, is initially at rest a horizontal distance $r_\text{0}$ from their highest anchor prior to falling and there is no tension or slack in the belay rope. Also assume the mass of the rope can be neglected. - Obtain expressions for $v_\text{b}$, the climber speed and $F_\text{max,b}$ the tension in the rope at the bottom of the pendulum swing. - What is the force exerted by the rope on the highest carabiner when the climber is swinging through the lowest point of their trajectory? - Find values of $v_\text{b}$ and $F_\text{rope,b}$ for the case $r_\text{0}=3.0$ m and a climber mass $m=80$ kg. 2) A more realistic model of the pendulum fall includes the elasticity of the belay rope and we model the falling climber and rope as a spring pendulum. We again assume the climber is initially at rest a horizontal distance $r_\text{0}$ from their highest anchor prior to falling, and there is no tension or slack in the rope. Assuming a linear response of the rope, i.e. $F_\text{rope} =k_\text{0} s$, the equations of motion for this spring-pendulum model may be written: $$\ddot{z}=(r_\text{0}+z)\dot{\phi}^2+g \cos\phi -\frac{1}{m} k_\text{0}s\hspace{2cm} $$ $$\ddot{\phi}=-\frac{2\dot{z}\dot{\phi}}{(r_\text{0}+z)}-\frac{g\sin\phi}{(r_\text{0}+z)}\hspace{4cm}$$ with $s=z/l_0$. Numerically solve equations (9) and (10) using an Excel spreadsheet utilizing the E-C method to find the climber trajectory. Initially (at $t=0$) $\ddot{z}=0$, $\dot{z}=0$, $z=0$, $\ddot{\phi}=-\frac{g}{r_\text{0}}$, $\dot{\phi}=0$, $\phi=1.5708$ ($90^0$). Increment the time by an amount $\Delta t$ and re-calculate each of these quantities. You will need to use equations 9 and 10 to recalculate the accelerations. Repeat this to generate a numerical solution of $\phi$ and $z$ as a function of time. Select a sufficiently small $\Delta t$ - a number in the $10^{-4}$ second range should be sufficient. See the pseudocode table for help with coding the excel cells. With the above solution, you can extract the trajectory in the form $(y, x)$, the tension $F_{rope}$ in the belay rope as a function of $x$, and the climber speed $v=\sqrt((r_\text{0}+z)^2\dot{\theta}^2+\dot{z}^2)$ as a function of $x$. a) As a first check that your excel spreadsheet is working as expected, select a sufficiently large value of $k_\text{0}$ so that your rope is rigid, i.e. there is no significant elasticity. A value of $k_0\geq 1\times 10^7$ N will suffice. Consider the case $r_\text{0}=3.0$ m, $l_\text{0}=60$ m and a climber mass $m=80$ kg. Start by generating plots for: - the trajectory $(y, x)$ of the climber during the fall during the first half-swing ($\phi=\pi/2 \rightarrow -\pi /2$) - the tension in the belay rope $F_{rope}$ versus position $x$. - the speed $v$ of the climber as a function of position $x$ Now do the following: - check that your answer for the speed at the bottom of the swing and the tension in the belay rope as determined from your excel spreadsheet agree with those calculated in exercise 1. b) Numerical errors accumulate over time. This accumulated error can be minimized by selecting a sufficiently small time increment. - Devise a check that your time increment in Exercise 2a is sufficiently small and use it to check your results. Since there is no friction in the model, the total mechanical energy (kinetic + potential) should be constant. - To check this, plot the total mechanical energy of the climber as a function of time. - Comment on what this plot implies about the dependability of your numerical result. c) Look at your numerical results for trajectory as a function of time. - How long does it take for the climber to reach the bottom of the swing? - Using the parameters of part 2a), do a direct calculation of the period of the swing and compare to the period found from your spreadsheet - do they agree? Hint: For your direct calculation of the period, the climber speed $v$ during the pendulum fall is given by $\sqrt(2gr_\text{0}\cos\phi )$ for this stiff rope. Integrate $dt=ds/v$ ($ds$ is an arc length) to find the period - you will need to do the integral numerically or look it up. d) Now we'll examine the climber pendulum swing for a more realistic belay rope with some elasticity. Select $r_\text{0}=3.0$ m, $l_\text{0}=60$ m, $k_0=25.1$ kN, $c=0$ corresponding to the bluewater rope assumed to be linear. Make plots (as you did in Exercise 2a)) of the trajectory, climber speed, and belay rope tension. Compare the speed of the climber at the bottom of their swing to that for the stiff rope of Exercise 2a). - Which is smaller? - Comment on any differences in the climber trajectory or belay rope tension compared to the stiff rope. e) For your numerical calculations of part 2d), devise checks you can do to assure your numerical calculation is dependable. 3) Repeat 2d) modeling the rope more accurately by including the non-linear part of the rope response. Now you will be solving the following set of ordinary differential equations: $$\ddot{z}=(r_\text{0}+z)\dot{\phi}^2+g \cos\phi -\frac{1}{m} (k_\text{0} s+c s^2)\hspace{2cm} $$ $$\ddot{\phi}=-\frac{2\dot{z}\dot{\phi}}{(r_\text{0}+z)}-\frac{g\sin\phi}{(r_\text{0}+z)}.\hspace{4cm}$$ Make plots (as you did in part 2a)) of the trajectory, and climber speed versus $x$. - Is there any change in the climber trajectory compared to the linear case of Exercise 2d? - What do you find the climber speed at the bottom of the pendulum to be for this more realistic case? How does it compare to the stiff rope of 2a) and the elastic rope of 2d)? - Devise checks you can do to assure your numerical calculation is dependable.as you did in Exercises 2b and 2e. 4) Next we'll look at longer time behavior as the climber swings back and fourth. There are several sources of friction including air resistance and friction between the belay rope and top carabiner as the rope stretches and contracts during the swinging. Over longer time this friction will remove energy from the climber as they swing back and forth. We'll look at a special case where the belayer is located at the highest carabiner so that $r_o = l_o$. Rope-carabiner interactions as a source of friction are eliminated leaving air resistance as the only significant friction. You'll be solving the two differential equations describing the generalized pendulum at the end of the theory section: $$\ddot{z}=(r_\text{0}+z)\dot{\phi}^2+g \cos\phi -\frac{1}{m} (k_\text{0} s+c s^2)-sign(\dot{z})\frac{1}{m}F_\text{air}\cos\psi\hspace{2cm} $$ $$\ddot{\phi}=-\frac{2\dot{z}\dot{\phi}}{(r_\text{0}+z)}-\frac{g\sin\phi}{(r_\text{0}+z)}-sign(\dot{\phi})\frac{F_\text{air}\sin\psi}{m(r_\text{0}+z)}.$$ Start with the parameters $r_o = l_o = 10$ m, and include air resistance by setting $D=1.0$ m. We have chosen a larger value of $r_o$ so that the climber speed takes on larger values and the air resistance force is significant - this value of $r_o$ represents a somewhat extreme pendulum fall. - Plot the trajectory in space ($x, y$), and in time ($\phi, t$). You should see a reduction in amplitude in this latter plot. - Plot the reduced mechanical energy $E/mgr_\text{0}$ as a function of time $t$ for at least two swings (back and fourth) of the climber. If you are using the excel spreadsheet supplied, you will need to copy/paste more lines below line 22,500. - Can you describe the physical origin of any pattern you see in this latter plot? Change the value of $r_o$ (you select a new value) and repeat the reduced mechanical energy plot. - What happens to the rate of decrease of energy? - Does this behavior make sense from a physical point of view? 5) A consistency check may be done on the generalized pendulum-fall spreadsheet. If air resistance has removed all the kinetic energy from the problem, the final height (below the anchor) and the final mechanical energy may be calculated directly. In your excel spreadsheet choose a large value of $D (> 1 \space m) $so that the air resistance force is large. - Plot the climber trajectory and climber mechanical energy as a function of time (you'll need several swings). - Are the 'long-time' values of these quantities what you expect?