Uniform Circular Motion 2.4 Knowledge and Skills Checklist Do I know that uniform circular motion means that a body is moving in a circular path with constant speed? Do I know that, although the speed of the body is constant, the velocity is constantly changing (in Q a e direction)? Do I know that the velocity of the body at any point of its path is in the direction of the tangent to the circle at that point? Do I know that, and can I explain why, the body has an acceleration at all points of its path? Do I know how to find the average acceleration of a body in some time interval At in two dimensions? Do I know that instantaneous acceleration at any time can be approximated by finding the average acceleration over a very small time interval At, centred on that time? Can I show, using a vector argument, that the change of velocity, and hence the acceleration, of the body over a very small time interval is directed toward the centre of the circle? Do I know that this acceleration, directed toward the centre of the circle, is called centripetal acceleration? v DO I know that centripetal acceleration can be calculated by the formula a, = -? r Do I know that the magnitude of the centripetal acceleration is constant for a given speed and radius? Do I know, and can I use, the fact that centripetal acceleration is directly proportional to the square of the speed of the body? Do I know, and can I use, the fact that centripetal acceleration is inversely proportional to the radius of the circle? Do I know what is meant by the period of the circular motion? Can I derive the formula that describes the relationship between the period of the motion and the speed 27cr of the body v = -? T v 27cr Can I solve problems using ac = - and v = -? r T Given the rate of rotation in revolutions per second or revolutions per minute (rpm), can I find the period of the motion? Do I know that there must be a force acting to cause circular motion? Can I explain why there must be a force acting using Newton's first law of motion? Can I explain why there must be a force acting using Newton's second law of motion? Do I know that, and can I explain why, the force causing centripetal acceleration is directed toward the centre of the circle? Do I know that, in order to produce uniform circular motion, a force must act on a body moving with constant velocity? Do I know that this force must be perpendicular to the velocity of the body and must remain perpendicular to the velocity as the direction of motion of the body changes? Can I describe situations in which the centripetal acceleration is caused by a tension force? Can I describe situations in which the centripetal acceleration is caused by a frictional force? Can I describe situations in which the centripetal acceleration is caused by a gravitational force? Can I describe situations in which the centripetal acceleration is caused by a normal force? Do I know the two vertical forces on a vehicle moving with constant velocity on a flat horizontal road? Do I know the three horizontal forces on a vehicle moving with constant velocity on a flat horizontal road? Do I know that, and do I appreciate why, all forces in any one direction on the above vehicle must cancel each other out? Can I explain why banking a bend reduces the reliance on fi-iction, by utilising the horizontal component of the normal force? Can I derive the formula for the optimal banking angle for a given speed and radius? Can I solve problems using this formula?
Physics Essentials Stage 2 Physics 2.5 Exercises 1 The diagram below shows a body moving with constant speed v ms-' in a circle of radius r metres. (1) In the space to the right of this diagram, construct a vector diagram to show the change in the velocity of the body Av, as it rotates from A to B. (2) If the time interval of rotation fiom A to B is very small, what is the direction of the vector Av? (3) What is the direction of the acceleration of the body? Explain your answer. v (4) Prove that the magnitude of the acceleration is given by a = -. r (5) A car rounds a bend of radius 400m at a speed of 20111s-l. What centripetal acceleration does it experience?
Uniform Circular Motion 2 An electron in an energy level of an atom moves in a circular path............ around the nucleus. The radius of the path is 1-20x 10-I 'm and the speed of the electron is 1-03 x 107ms-'. 9. (1) Find the centripetal acceleration of the electron. (2) Draw an arrow on the diagram to indicate the direction of the forces on the nucleus and on the electron. (3) Which experiences the greater force - the electron or the nucleus? (4) Find the magnitude of the force causing the centripetal acceleration. (5) What is the nature of the force that causes the centripetal acceleration. (6) Determine the number of protons in the nucleus of the atom. 3 Two cars, labelled A and B, with masses r n = ~ loookg and r n = ~ 2000kg, move around a bend at the same speed. Find the following ratios. Explain your answers. (1) The centripetal acceleration of A : the centripetal acceleration of B. (2) The force on A : the force on B.
Physics Essentials Stage 2 Physics 4 A car moves around a circular bend at fixed speed. State and explain ~ hhappens a ~ to the force causing the centripetal acceleration on the car if the following occur. (1) The speed of the car doubles. (2) The speed of the car is reduced to % of the original speed, and the radius of the bend is halved. A locomotive of a model railway has a mass of 0.60kg and is moving around a circular track of radius 2.4m with a constant speed of 0-60ms-'. (1) What is the net acceleration of the locomotive. (2) What is the net force on the locomotive causing it to move in a circle. A car is driven at a constant speed of 60kph around a circular track of radius 100m. (1) What is the magnitude of the acceleration of the car? (2) If the mass of the car and its contents is 2000kg, what is the magnitude of the force on the car? (3) What is the direction of the force on the car?
Uniform Circular Motion 7 A merry-go-round has a diameter of 9-Om. A girl of mass 50kg rides on a horse 3-Om from the centre. The merry-go-round takes 20s to complete one revolution. (1) What is the speed with which the girl is moving. (2) What is the net force on the girl. (3) What is the speed of a point on the outer edge of the merry-go-round? 8 A rock of mass 4-0kg is attached to a wire and is whirled in a horizontal circle of radius 040m with constant speed. The tension in the wire is 80N. (1) Find the acceleration of the rock. (2) Find the speed of the rock. (3) Find the period of the motion of the rock. (4) Find the number of revolutions that the rock makes in one minute.
32 Physics Essenfials Stage 2 Physics -- 9 (1) Explain why the force that causes an object to move in a circle does not change the speed of the object, but does change its velocity. (2) Explain why the kinetic energy of an object moving with uniform circular motion does not change, even though it has a force continually acting on it. 10 Describe situations where the force causing centripetal acceleration is provided by (1) an electrostatic force (2) a gravitational force (3) a fi-ictional force (4) a tension force (5) a magnetic force 11 The Moon takes 27-32 days to orbit the Earth. radius of the Moon's orbit is 3-844x10*m. (1) Find the linear speed of the moon in its orbit. (2) Find the centripetal acceleration of the moon.
Uniform Circular Motion 12 A racing car of mass 800kg takes 12.0s to travel at a constant speed around a flat circular racing track of radius 100m. (1) Find the speed of the car as it moves around the track. (2) Find the average force that each of the four tyres must exert against the track to maintain its circular motion. Remember to give both magnitude and direction. (3) What is the average force that each tyre exerts against the road due to the weight of the car? (4) What is the total average force that each tyre exerts against the road? 13 (1) A car moves around a flat corner at constant speed. On the diagram at right draw labelled vectors to show each of the following. (a) (b) (c) The normal force on the car FN. The gravitational force on the car FG. The force causing the centripetal acceleration of the car Fc. (2) The car now moves around a banked bend at constant speed. On the diagram at right draw labelled vectors to show each of the following. (a) (b) (c) (d) (e) The normal force on the car FN. The gravitational force on the car FG. The. force causing the centripetal acceleration of the car Fc. The vertical component of the normal force Fv. The horizontal component of the normal force FH. ' k-
14 (1) Explain why roads are often banked on curves. (2) A car is to travel around a bend of radius r metres at a speed of v ms-'. Show that the car can go around the bend with no reliance on frictional force between the tyres and the track if the track is v banked at an angle 8, where tan 8 = -. rg -- -.- (3) A car of mass 1 lookg goes around a banked track of radius 600m at a constant speed of 60ms-'. (a) What is the banking angle if the car negotiates the bend with no reliance on friction? (b) What is the magnitude of the centripetal acceleration of the car as it goes around the bend?
Uniform Circular Motion 15 A car moves around a banked track with a banking angle of 14". This angle is optimal for a car travelling with a speed of 50ms-l. The normal force on the car is 20,000N. (1) Find the horizontal and vertical components of the normal force. OOOON (2) What is the gravitational force on the car? " " (3) What is the mass of the car? (4) What is the magnitude of the force which causes the centripetal acceleration of the car? (5) What is the centripetal acceleration of the car?. " " " -.--- (6) What is the radius of the bend? 16 At one end of a racing track the radius of the bend is 300m and the banking angle is 25". Find the speed v at which are car must travel around this bend so that there is no reliance on frictional force to provide the centripetal acceleration.
Physics Essentials Stage 2 Physics 17 On 11/6/1995 in Stuttgart, Gemany, Russian athlete Yuriy Georgiyevich Sedykh set a new world record of 88.74m for the hammer throw. In this event a steel ball (the hammer) of mass 7.62kg attached to a metal chain 1-2m long is whirled at high speed in a circle and then released. In this throw the hammer was 1.8m above ground level at the moment of release and the launch angle was 43". Assume that when the hammer was at 1-8m above ground level, just before hitting the ground, it had travelled a horizontal distance of 86.0m. u2 sin 28 (1) The range R of a projectile launched at ground level is given by R = -, where u is its g initial speed and 8 is the launch angle. Use this to- determine the speed of the hammer at the moment of its release. (2) Assuming that, at the moment of release, the hammer was rotating in a circle of radius 2.23m, find its centripetal acceleration just before release. (3) What force was Yuriy Sedykh applying to the handle of the chain just before releasing the hammer? (4) At what rate (in revolutions per minute) was Yuriy Sedykh spinning at the moment that he released the hammer? (5) What was the kinetic energy of the hammer at the moment of release? (6) From the moment that Yuriy Sedykh began his spin, to the moment of releases of the hammer, 3.4 seconds elapsed. What power did Yuriy Sedykh generate in his throwing action?