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Motion in a Vertical Circle Consider a ball moving in a circular path as shown. What happens to the speed of the ball? What happens to the tension in the string? Vertical circle Or here V min Fw F s1 Because of gravity, the speed of the ball is not uniform. The ball accelerates on the downward path and decelerates on the upward path. Speed is at a minimum at the top and a maximum at the bottom. F s2 Fw V max At the top: Fc 1 Fs 1 Fw 2 Fc 1 mv min r Fs 1 Fw Since vmax > vmin then At the bottom: Fs2 > Fs1 Fc 2 Fs 2 Fw i.e. the string tension is greater at the bottom! 2 Fc 2 mv max r Fs 2 Fw Recall This implies that there is some velocity below which the ball will not move in a circular path. The string will slacken at the ball’s highest point. This could also happen if a car did not have enough speed to make a loop de loop. The minimum value is when Fs1 = 0. At this time , the ball just makes it around the circle. At the top: Fc 1 Fs 1 Fw 2 mv min 0 mg r 2 v min g r v rg This value of vmin is called the critical velocity. It depends only on the radius of the circle and the acceleration due to gravity. It does NOT depend on the mass of the object in motion. A satellite in orbit around a planet moves with the critical velocity for it’s radius of orbit (altitude) See animation Note that the radius of orbit for a satellite is equal to the radius of the planet (Earth) plus it’s altitude. alt Re R o rbit Sample problem #3