Lecture 13 AS - Circular Motion PDF

PHYS 0100 Preliminary Physics I Ms. Amanda Sankaran [email protected] Circular Motion Learning Outcomes After this lecture, students should be able to: Understand circular motion. Apply the equations for centripetal acceleration. Use equations for centripetal force. ...

PHYS 0100 Preliminary Physics I Ms. Amanda Sankaran [email protected] Circular Motion Learning Outcomes After this lecture, students should be able to: Understand circular motion. Apply the equations for centripetal acceleration. Use equations for centripetal force.  Uniform Circular Motion Uniform circular motion: motion of a circle of constant radius at constant speed. Instantaneous velocity is always tangent to the circle. Why study Circular Motion? To understand ✓ Motion of planets. ✓ Motion of electrons around the nucleus. ✓ Motion of a giant wheel. ✓ Motion of space stations. ✓ Motion of moon and satellites.  As the wheel turns the carts exhibits circular motion because they revolve around a single axis of rotation. This axis runs through the axis of the wheel perpendicular to the plane that contains it. The carts move around with a constant radius, thus tracing a circle. In describing the speed of the cart, we can discuss the tangential speed (instantaneous speed along the tangent line). This is because the inertial of an objects causes it to try and maintain a straight line path. Tangential speed is constant – Uniform Circular Motion Tangential speed will depend on the distance from the axis of rotation. The uniform circular motion of the cart – Centripetal Acceleration. At any given moment, the cart has a tangential velocity to the wheel. ✓ Magnitude is constant ✓ Direction changes ✓ Velocity changes there will be acceleration ✓ Centripetal acceleration always points towards the center. Speed – in uniform circular motion, distance = the circumference of the circle. Tangential speed – the distance travelled in one rotation is the circumference. 𝑠 = 2𝜋𝑟 Period (T) is the time (or number of seconds) to make one revolution. 𝑠 2𝜋𝑟 𝑣= = 𝑡 𝑇 Frequency (f) is the number of revolutions completed by an object in a given time. It is measured in Hertz. 1 1 𝑓= → 𝑇= 𝑇 𝑓 Speed expressed in period and frequency is: 𝑣 = 2𝜋𝑟𝑓 2𝜋𝑟 𝑣= 𝑇 Common terms to remember when dealing with circular motion. Angular displacement = angle swept out 𝜃. Angular displacement is measured in radians rather than in degrees. Distance moved along a circle: 𝑠 = 𝑟𝜃. s –arc length, r is the radius of the circle. 2𝜋 radians = 360 degrees ▪ An object moving in a circle at a constant radius and constant speed still experiences a changing velocity. ▪ The velocity is changing because the direction of the velocity vector is constantly changing. ▪ Therefore the object is accelerating and the acceleration is referred to as a centripetal acceleration as it is directed towards the centre of the circle. Velocity The velocity of an object in uniform circular motion has a constant magnitude and a constant change in direction. The object does not have constant velocity since its direction changes. The velocity is always tangent to the path of the object. Centripetal Acceleration The acceleration has constant magnitude but changing direction and is radially directed inwards. ac Vector Rate of change of tangential velocity. Always perpendicular to the path of motion. Points towards the center of the circle.  Centripetal Acceleration Magnitude calculated by: Note: If you double the tangential speed, you quadruple the centripetal acceleration If you double the radius, you halve the centripetal acceleration. Angular Speed Centripetal Force (Latin for “center seeking”) It is the inward net force which keeps an object moving with a uniform velocity along a circular path. This force is directed along the radius towards the center. Properties of Centripetal Force It is a real force. Is necessary for maintaining uniform circular motion. Acts along the radius of the circle. Directed towards the center of the circle. Does not 𝟐 do any work. 𝒎𝒗 𝑭= 𝒓 Units – Newton (kg m/s2) Examples of Centripetal force A body tied to a string and whirled in a horizontal circle. The centripetal force is provided by the tension in the string. For a car travelling around a circular road with uniform speed, the centripetal force is provided by the static friction between tires of the car and the road. In case of electrons revolving around the nucleus, the centripetal force is provided by the electrostatic force of attraction between the nucleus and the electrons. In the case of the motion of the moon around the earth, the centripetal force is provided by the _______ between the Earth and the moon. = 𝑚𝑣𝜔 = 𝑚𝜔2 𝑟 Question - Centripetal force  Question - Centripetal force