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« B         Main » Optional Unit V - Kinematics & Dynamics » Uniform Circular Motion   ### Uniform Circular Motion

 Learning Outcomes Define the following terms: centripetal acceleration, centripetal force. Explain why an object travelling in a circular path at a constant speed undergoes a change in velocity. Illustrate the direction of the velocity vector, the centripetal acceleration vector, and the centripetal force vector for a moving object at a specific position on a circular path. Use a vector diagram to illustrate a change in velocity when the magnitude of the velocity vector remains constant but the direction changes. Recognize that if an object were suddenly released from its circular path, it would tend to continue to move in the direction of the velocity vector, unless it was acted upon by some external force. Explain that centripetal acceleration acts in the same direction as the change in velocity. Explain that centripetal force acts in the same direction as centripetal acceleration. Use mathematical relationships for centripetal acceleration and centripetal force to solve problems involving circular motion. Recognize that to place a satellite into orbit, it must be travelling such that the force of gravity acting on it (i.e., its weight) provides a force equivalent to the centripetal force needed to maintain its motion. Explain that the orbital velocity of a satellite does not depend on the mass of the satellite. Describe some useful applications of satellites.  Key Concepts If an object is travelling in a circular path at constant speed its velocity is changing because, even though the magnitude of the velocity remains constant, the direction of the velocity is changing continuously. The velocity vector points along a tangent to the circle, in the direction that the object would tend to move if it were suddenly released. The acceleration acts in the same direction as the change in velocity. The acceleration is called centripetal acceleration. It always acts inward, toward the centre of the circle, in the same direction as the change in velocity, perpendicular to the velocity vector. The instantaneous acceleration ( ) at any point on the circular path is: The magnitude of the centripetal acceleration is given by: where R is the radius of the circle, T is the period of revolution, f is the frequency of revolution, and ac is the magnitude of the centripetal acceleration. The subscript c serves as a reminder of the vector nature of the acceleration. Its direction is constantly changing at every position along the circular path. From Newton's Second Law: The force acts in the same direction as the acceleration. The force, directed towards the centre of the circle, is called centripetal force. (Other relationships can be obtained by substituting the equations for centripetal acceleration into the equation for Newton's Second Law, e.g. ) (It is important that students clear up any misconceptions they might have about centrifugal force, which is a fictitious force, that appears to act in an accelerated frame of reference.) The minimum velocity needed at the top of the loop for an object to perform a loop-the-loop is: To place a satellite into orbit, it must be travelling such that the force of gravity acting on it (i.e., its weight) provides a force equivalent to the centripetal force needed to maintain its motion. The orbital velocity does not depend on the mass of the satellite.        