Unit Dynamics Module The Dynamics of Circular Motion page 1 of 2 Forces and Uniform Circular Motion In uniform circular motion acceleration a v2 is always toward the center of r the circle In uniform circular motion a large velocity and a small radius imply large force In uniform circular motion the contact force normal force is always toward the center of the circle In a conical pendulum the velocity that will produce uniform circular motion is v gR tan Recall the kinematics of uniform circular motion The position vector has a magnitude of r the radius of the circle The velocity vector is always tangent to the curve and it has a constant magnitude The direction of the acceleration vector is toward the v2 center and its magnitude is a r An object is twirled on a string parallel to the ground Use Newton s second law to study the forces involved If you draw a force diagram you find that there is only one force in the x direction the tension T in the string pulls the object toward the center of the circle Set the tension equal to the product of the mass of the object and the acceleration T mv 2 r Notice that as you twirl the object faster the tension increases and if you decrease the length of the string the tension increases Notice that the tension is in the same direction as the acceleration In uniform circular motion the force is toward the center of the circle or centripetal www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1923 doc rev 03 27 2001 Unit Dynamics Module The Dynamics of Circular Motion page 2 of 2 Forces and Uniform Circular Motion A conical pendulum is a pendulum that is swung in a circle instead of back and forth For a given angle of the string away from the vertical there is a certain speed that will produce uniform circular motion You would like to know that speed as a function of angle Use Newton s second law Consider the forces on the pendulum bob in the ydirection The object experiences the force of gravity weight in the downward direction and the ycomponent of the tension in the upward direction There is no acceleration in the vertical direction so the right side of Newton s second law is zero You can solve for the value of the tension In the x direction there is one force the xcomponent of the tension towards the center of the circle The acceleration is v 2 r You can solve for v by substituting mg cos for T and recalling that tan is the same as sin cos The result shows that if you increase the angle at which you spin the conical pendulum you will increase the speed required for uniform circular motion www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1923 doc rev 03 27 2001
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