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LF Mechanics 9. Circular Motion LF Mechanics – 9 N Drury [email protected] Learning Outcomes

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LF Mechanics 9. Circular Motion LF Mechanics – 9 N Drury N.Drury@Bham.Ac.Uk Learning Outcomes LF Mechanics 9. Circular motion LF Mechanics – 9 N Drury [email protected] Learning Outcomes Knowledge and application of: • Circular motion and key terms • Angular velocity • Centripetal force • Centripetal acceleration • Frames of reference and centrifugal force • Simulating gravity Circular motion • The path is a circle and the object is moving at constant speed 풗 along the path. • Period of rotation (푻): the time taken for one full rotation, measured in seconds. • Frequency (풇): the number of rotations per second, measured in Hertz (Hz), which is equivalent to s-1. Linear and angular velocity Linear velocity 풗 in Angular velocity 흎 in ms-1 rad s-1 • Its vector is always • Its vector is tangent to the path. perpendicular to the • Defined as change in plane of rotation. displacement per unit • Defined as angle time. covered per unit time. Angular speed • If an object rotates with a time period T, the angle it turns through per second is: w = 2p / T as f = 1 / T Then w = 2pf Linear speed 푑 • Linear speed, 푣 = 푡 • The circumference of the circle is 2pr • So the linear speed 2휋푟 is given by 푣 = 푇 • This can be written as 푣 = 휔 푟 Useful equations 1 1 • 푓 = and 푇 = 푡 푓 • 푣 = 휔푟 where r is the path radius 푑푠 2휋푟 • 푣 = = 푑푡 푇 푑휃 2휋 • 휔 = = where θ is measured in radians 푑푡 푇 What provides the force on the hammer to make it go round in a circle? What provides the force on the satellite to make it go round in a circle? What provides the force on the car to make it go round in a circle? What provides the force on the bike to make it go round in a circle? Circular motion For an object to go round in a circle you need an inward force that changes the direction of motion. The force must always act towards the centre of the circle and is said to be a Centripetal force. Centripetal (radial) Acceleration 푣2 Its magnitude: 푎 = 푟 Its direction: towards the centre of the circle Centripetal force 푚푣2 Its magnitude: 퐹 = 푐 푟 Its direction: towards the centre of the circle The centripetal force is not a force that appears in addition to the existing forces. Instead, from the forces already present, one (or a component of one) is responsible for the circular motion and it is therefore the centripetal force. Why don’t you fall out of the car when you go up side down? As the bucket spins around the weight provides the centripetal force on the water and non of its weight is used to • Is the centripetal force constant during make it fall a circle? out. • At which point does the bucket ‘feel’ heavier and lighter? • What are the forces at A, B, C and D? Question 1 A pulley wheel rotates at 300 rev min-1. Calculate (a)its angular velocity in rad s-1 (b)the linear speed of a point on the rim if the pulley has a radius of 150mm (c)the time for one revolution. Frames of reference • A frame of reference wherein a free body exhibits no acceleration is called inertial frame of reference. Newton’s laws are seen to hold exactly in an inertial frame (e.g. empty space). • A rotating frame of reference, in contrast, is an accelerating frame of reference. Newton’s laws are not valid in an accelerating frame of reference (e.g. Earth). Centrifugal force in a rotating frame of reference • An occupant inside a rotating system seems to experience an outward force. This apparent outward force is called centrifugal (meaning centre fleeing). • Centrifugal force: a fictitious force, not a real force which can simulate gravity in a rotating frame of reference. Simulating gravity • Inertial frame: gravitational force is an interaction between one mass and another. • Accelerating frame: for centrifugal force no such agent exists –there is no counterpart. Nothing produces it; it is a result of rotation. This is why it is called “apparent” or “fictitious” or “inertial” force and not a real force. Misconceptions When the string breaks the bucket moves in a straight line tangent to –not outward from –the centre of its circular path In a tumble dryer the clothes are forced into a circular path but the water is not. WRONG: bucket/water flies outwards due to the centrifugal force CORRECT: bucket/water flies outwards due to the absence of a centripetal force that will keep It in a circular path Question 2 A heavy iron ball is attached by a spring to the rotating platform. Two observers, one in the rotating frame and one on the ground at rest, observe its motion. Which observer sees the ball being pulled outward, stretching the spring? Which sees the spring pulling the ball into circular motion? Question 3 If Earth were to spin faster about its axis, you would feel lighter. If you were in a rotating space habitat that increased its spin rate you would feel heavier. Explain why greater spin rates produce opposite effects in these cases. Examples: 1. Over the hill R mg r Centre of circle mg – R = mv2 / r Examples: 2. Roundabout velocity Friction r Centre of circle Friction, F = mv2 / r Examples: 3. Pendulum Cable, length L Tension, T T – mg = mv2 / L velocity mg Question 4 An object of mass 0.30kg is attached to the end of a string and is supported on a smooth horizontal surface. The object moves in a horizontal circle of radius 0.50m with a constant speed of 2.0ms-1. Calculate (a) the centripetal acceleration (b) the tension in the string. Question 5 An object of mass 4.0kg is whirled round a vertical circle of radius 2.0m with a speed of 5.0ms-1. Calculate the maximum and minimum tension in the string connecting the object to the centre of the circle. Assume g=10ms-2. Question 6 A car travels over a humpback bridge of radius of curvature 45m. Calculate the maximum speed of the car if its road wheels are to stay in contact with the bridge (g=10ms-2). Question 7 A conical pendulum consists of a small bob of mass 0.20kg attached to an inextensible string of length 0.80m. The bob rotates in a horizontal circle of radius 0.40m, of which the centre is vertically below the point of suspension. Calculate (a) the linear speed of the bob, (b) the period of rotation of the bob (c) the tension in the string (g=10ms-2). Question 8 An aircraft flies with its wings tilted as shown in the figure in order to fly in a horizontal circle of radius r. The aircraft has mass 4.0 x 104kg and has a constant speed of 250ms-1. With the aircraft flying in this way, two forces acting on the aircraft in the vertical plane are the force P acting at an angle of 35° to the vertical and the weight W. (a) calculate P (b) determine the acceleration of the aircraft (c) calculate the radius of the path. Summary Knowledge and application of: • Circular motion and key terms • Angular velocity • Centripetal force • Centripetal acceleration • Frames of reference and centrifugal force • Simulating gravity Further reading Review • OpenStax College Physics: Chapter 6: Uniform circular motion and gravitation p 201 - 230 Read ahead • OpenStax College Physics: Chapter 7: Work, energy and energy resources p 241 - 271.
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