Circular Motion – Part II

Goal of the class To understand the forces involved in circular motion Question of the day: How does the tension in a string vary in circular motion?

Motion in a Horizontal Circle e.g. on a frictionless table A=v2/r from last time Mg = N as no vertical acceleration Fnet = ma = mv2/r towards the centre The force is just the tension T = mv2/r

Conical Pendulum Speed v = constant Angle = theta radius r, Length L Only 2 forces, mg and T Resolve T into x and y The pendulum acted on by 3 forces, mg down, Tcos theta up and T sin theta towards centre As a_vert = 0 then F_vert = 0 as F=ma This implies mg = Tcos theta and cancel out (1) Fnet = Tsin theta = mv2/r (2) Divide to eliminate T Tan theta = v2/rg or v2 = rg tan theta

Going around the bend 1) Frictionless 2) Friction Only forces are mg and N But for circular motion a =v2/r This needs a force towards the centre that doesn’t exist Therefore no circular motion on frictionless road For friction fs =mus N So fs = mv2/r or v2 = r. fs /m = r mus N / m = r mus g

Problem A car rounds a banked round. The radius of curvature of the road is R, the banking angle is θ, and the coefficient of static friction is μ . Find the range of speeds the car may have without slipping up or down the bank Don’t move x and y as you need v2/r in x direction Resolve and factor out N Sub 1 into 2 and find v2

Problem 2 A large vertical cylinder spins about its axis fast enough that any person inside is held against the wall when the floor drops away. The radius of the cylinder is R and the coefficient of static friction is μs. Find the maximum period of revolution required to stop the person from falling. Minimum speed N=mv2/r Mu_s mv2/r >= mg V2 >= Rg/mu_s V = root that T = 2piR/V Solve for T

Science Fair I have a sign up sheet here for people who can sign up to the science fair if they wish Friday 28th November

Extra Problem A pail of water is spun in a vertical circle of radius 0.7m. What is the minimum speed at the top so that no water spills out? For no water to come out it means N must be bigger than 0 Mv2/r > mg