OCR A Physics A-level Circular Motion Notes PDF

OCR A Physics A-level Topic 5.3: Circular motion Notes www.pmt.education  Kinematics of circular motion The radian Radians can be used instead of degrees to measure angles. A radian is the angle subtended by a circular arc with a length equal to the radi...

OCR A Physics A-level Topic 5.3: Circular motion Notes www.pmt.education  Kinematics of circular motion The radian Radians can be used instead of degrees to measure angles. A radian is the angle subtended by a circular arc with a length equal to the radius of the circle. There are 2π radians in a full circle. To convert an angle from degrees to radians, divide it by 360 and then multiply by 2π. Period and frequency The period, T, is the time, in seconds, taken for an object to travel a full circle. The frequency, f, measured in Hertz (Hz) is the number of full circles completed in one second. The two are related by the formula f = 1/T. Angular velocity The angular velocity, ω, of an object moving in a circle is the rate of change of angle. It is 𝜃𝜃 given by the formula 𝜔𝜔 = 𝑡𝑡 , where θ is the angle the object has travelled through in the time t. It is measured in rads-1, so the angle must be measured in radians and the time in seconds. In a time of one period, T, the object will complete a full circle, with an angle equal to 2π radians. The formula for angular velocity can therefore be written as 2𝜋𝜋 𝜔𝜔 = = 2𝜋𝜋𝜋𝜋 𝑇𝑇 Centripetal force Centripetal force When an object travels in a circle, its direction is constantly changing. As velocity is given with respect to direction, this means the velocity of the object is also constantly changing, even though the speed maybe constant. In order for the velocity to change, the object must be accelerating, and this acceleration is provided by the centripetal force. Centripetal force is the net force which acts perpendicular to the direction of the velocity, towards the centre of the circle. The speed v of a object moving in a circle of radius r is given by the formula 2𝜋𝜋𝜋𝜋 𝑣𝑣 = = 𝜔𝜔𝜔𝜔 𝑡𝑡 The acceleration of the object is equal to the rate of change of velocity. The centripetal acceleration for an object moving in a circle is given by the formula 𝑣𝑣 2 𝑎𝑎 = = 𝜔𝜔2 𝑟𝑟 𝑟𝑟 www.pmt.education Using the formula F=ma, the centripetal force can be determined by substituting in the expression for centripetal acceleration, to give 𝑚𝑚𝑣𝑣 2 𝐹𝐹 = = 𝑚𝑚𝜔𝜔2 𝑟𝑟 𝑟𝑟 The centripetal force can come from many places, e.g. gravitational fields, friction, tension in a rope. It is important to remember that centripetal force is the net force acting towards the centre of the circle, so when an object experiences multiple forces (such as normal contact force acting in and gravitational force acting out) the resultant force must be used. Techniques to investigate circular motion Circular motion can be investigated experimentally by tying a bung, with mass m, to a piece of string, and threading it through a glass tube. The other end of the string has a weight, with mass M, suspended from it. This provides the centripetal force, F = Mg, as the tension throughout the string is constant. The string is whirled in a circle, and the time taken for a complete rotation is recorded. The mass of the weight is altered and the experiment repeated. 𝑚𝑚𝑣𝑣 2 As the centripetal force is given by the formula , and provided by the weight on the end of 𝑟𝑟 the string, Mg, we can equate the two to get 𝑚𝑚𝑣𝑣 2 𝑀𝑀𝑀𝑀 = 𝑟𝑟 By measuring the radius of the circle and using the time for one complete oscillation, the velocity can be determined. When v2 is plotted against M, a straight line graph which passes through the origin should be produced. www.pmt.education