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University of Toronto St. George
Erich Poppitz

PHY254 Lecture 2 ▯ Text Readings: Morin: 2.1-2.2, 3.1-3.2. Start going through compwiki website. ▯ In this section of the course, we review some of the techniques in me- chanics from ▯rst year, and introduce some new ideas. The approach is really problem based since the theory isn’t so new. ▯ In most of the course, we’ll deal with everyday forces: { Conservative body forces: gravity, electrostatic force { Contact forces: tension, normal force, friction / normal force. Setting up F = ma problems: Statics ▯ Statics is the study of mechanical systems at rest. In static systems, forces and torques are in balance; contact forces like static friction stabilize the system. ▯ The key is to draw a diagram, draw a free body diagram where appro- priate, break down the force equations into components, and use the torque equation as an extra constraint, if applicable. ▯ A nice force balance example is on p.26 of Morin: \Rope wrapped around a pole". 1 Suppose you hold the rope with tension T . What is maximum load the 0 rope can bear without slipping? (Another way of putting it: Imagine the boat had a motor and was trying to get away, what is the maximum force that the rope can bear without slipping relative to the pole, given your ▯xed T ?0 ▯ Answer (in book): We need to draw an FBD for a small segment of the rope wrapped around the pole. There will be a couple of math tricks we use because we are dealing with a small segment. I’ll point them out when we get to them.... x-balance: N ▯ T(▯ + d▯)sin(d▯=2) ▯ T(▯)sin(d▯=2) = 0: y-balance: T(▯ + d▯)cos(d▯=2) ▯ T(▯)cos(d▯=2) ▯ F = 0 s Since we want the maximum load the rope can hold, we will set the static friction to its maximum value: F s;max= ▯ s Main math trick: d▯ is a SMALL angle so we will use some approxima- tions. Basically, we can use Taylor expansions of the trig functions and T(▯+d▯) and only keep the biggest terms. Here are the approximations: for small angle d▯; sin(d▯) ▯ d▯; cos(d▯) ▯ 1 2 for the tension: dT T(▯ + d▯) = T(▯) + d▯ + h.o.t. d▯ So the approximate force balances are: Approx x-balance: ! N ▯ d▯ T(▯) + dT d▯ + T(▯) = 0 ) 2 d▯ 1 dT N ▯ T(▯)d▯ ▯ d▯ = 0 2 d▯ Now for small angle d▯, d▯ << d▯ so we can ignore the last term on the left hand side and we are left with N = T(▯)d▯ Approx y-balance: ! dT (1) T(▯) + d▯ ▯ T(▯) ▯ F =s0 ) d▯ dT d▯ = ▯sN d▯ Using the x-equation’s solution for N in this equation we get: dT d▯ = ▯ s(▯)d▯ ) d▯ dT = ▯ T(▯) (1) d▯ s This is a di▯erential equation we can solve by separation of variables: dT = ▯ s▯ ) (2) T 3 Z Z dT = ▯ d▯ ) (3) T s lnT = ▯ ▯s+ c ) (4) Taking the exponential of both sides and using the fact that e a+b = e e b we get: ▯s▯ T = c 1 ) (5) c (where c 1 e is just another constant). We can solve for the constant using an initial condition. The rope begins to turn around the pole at ▯ = 0. At that point, we are providing a tension force of T on0that end of the rope. So the initial condition is that at ▯ = 0; T = T .0Plugging this in we can solve for c 1 0 T(0) = T =0c e 1 c = T 1 0 and we end up with: T = T e0 ▯s▯ (6) Since we applied the maximum static friction force in the analysis, this means that the rope will not slip as long as the tension in it is LESS than the amount above. So the rope does not slip provided T ▯ T e 0 ▯s▯. Lets try and understand what this means physically. { ▯ increases as the rope wraps around the pole. { The maximum T
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