Uniform Circular Motion and Pendulum

Uniform Circular Motion Uniform circular motion is the motion of an object that travels a circular path with constant velocity. The characteristic quantities of this motion are: Period T is the time it takes for the object to complete a turn (measured in seconds s) The frequency f is the number of turns per second and is measured in Hertz Hz; 1 Hz = 1 turn per second The angular velocity is the ratio between the difference of angular position and the time interval spent Centripetal acceleration is the acceleration that at any moment is directed toward the center of the circumference. Centripetal acceleration as a function of angular velocity: Deepening Uniform circular motion is a type of motion in which a body moves in rotation along a circular trajectory with a constant velocity magnitude Uniform circular motion is the motion of an object that travels a circular trajectory by moving at a constant speed. Tangential velocity To find the value of constant speed with which a point moves describing a circular trajectory in uniform circular motion, we must think about the definition of speed as the ratio between the distance travelled and the time spent. In the case of circular motion, if we think about the complete 360° turn and so the distance is equal to the length of the circumference. The time needed to make a complete turn is always the same at each turn and is called period. In uniform circular motion the speed is constant in magnitude but not in direction. In fact, it is called tangential velocity because it is a vector which, point by point, is always tangent to the circular trajectory and perpendicular to the radius. The direction of the velocity vector is given by the direction of the object (clockwise or counterclockwise) Angular velocity The angular velocity is defined as the ratio between the angle traveled by the point and the time interval spent and which is indicated by the Greek letter w Since the value of the speed is constant in the uniform circular motion, to calculate the angular speed it is sufficient to consider a complete turn: the angle traveled is 360° - in radians and the time spent is the period Frequency The frequency is a quantity that determines how many revolutions are made every second. It is defined as the reciprocal of the period and its unit of measurement is the Hertz Centripetal acceleration Speed is a vector and to vary a vector or change its magnitude or direction or both logically. Therefore, although uniform circular motion is a motion at constant velocity, we are still in the presence of acceleration. In the uniform circular motion the magnitude of speed remains constant but its direction changes constantly. On the velocity vector acts an acceleration, called centripetal acceleration because it always directed towards the center of the circumference. Differences between tangential and angular velocity The fundamental difference between the two speeds is that the tangential speed depends on the radius. For example, in a race between two athletes, one is in the first lane and one in the last. To facilitate the example, let's imagine that they start from the same point and finish, after an entire lap of the field, even, and that neither of them has left their lane. Logically, the athlete in the outermost lane so the one whose trajectory has the greatest radius - will have a greater tangential speed, while the athlete in the first lane, therefore closer to the centre, will turn more slowly Pendulum A simple pendulum is a weigh suspended from a pivot. It must have a mass m, hanging by an inextensible wire of length I and without mass. From the moment the pendulum is moved from its vertical, it will begin an oscillatory motion that will continue identical until a new interaction with the environment. Let's draw the pendulum to understand which forces act on it: As the mass point m oscillates, it moves away from the origin of a distance x representing the amplitude of the oscillation motion. O is defined as the angle that the rope creates with the vertical. Only two forces act on the point: the weight P directed downwards and the tension T of the rope. The force will be broken down into its two components, the parallel and the perpendicular. If we wanted to find the acceleration acting on the mass point m, we would have to apply the Newton's second Law. Only the parallel component of the weight appears on the horizontal axis: Logically g is the acceleration of gravity. Considering that a rectangular triangle is formed with hypotenuse the rope length L and for horizontal cathetus the distance x we can write the relation: Then we can rewrite the parallel component of the weight as: Since the P// component points to the negative side of the x-axis, then we write the Newton's second Law as: Simplify it all and with a reverse formula we'll get it