Lecture 6 Circular motion Instantaneous velocity and speed For an object travelling in the uniform circular motion, its instantaneous velocity is not constant because the direction of the object is continuously changing, however, its instantaneous speed is constant. constant speed = average speed = distance travelled t = circumference of the circle period = 2 π r T where r is the radius of the circle and T is the time taken for the object to complete a revolution. If the object travelling in the uniform circular motion completes a full revolution, the average velocity over the entire revolution should be zero because the displacement is zero. A toy car completes a single lap of a circular track in 15 s with an average speed of 1.3 m/s. Assume that the speed of the toy car is constant. a. hat is the radius of the track? b. hat is the magnitude of the toy car's instantaneous velocity halfway through the lap? c. hat is the average velocity of the toy car after half of the lap has been completed? d. hat is the average velocity of the toy car over the entire lap? As all objects with changing velocities are experiencing an acceleration, this means that all objects that are moving in a circle are accelerating. According to ewton's Second Law of Motion, an acceleration can be caused only by a non-zero net force. Physics 1
Centripetal acceleration is always directed towards the centre of the circular path and is given by 2 2 v 4π r a = = 2 r T where a = centripetal acceleration (ms -2 ) v = speed (ms -1 ) r = radius of circle (m) T = period of motion (s) This formula provides a way of calculating the centripetal acceleration of a mass moving with uniform circular motion having speed v and radius r. The net force involved in circular motion is called centripetal force which is always directed towards the centre of the circle. F net = ma = = r 2 2 mv m 4π r T 2 where F net = net force or centripetal force on the object () m = mass (kg) v = speed (ms -1 ) r = radius of circle (m) T = period of motion (s) A motorcyclist is riding around a circle of radius of 100 m. The surface is flat and horizontal. The motorcyclist is travelling at a constant speed of 30.0 ms -1. The motorcycle with rider has a mass of 250 kg. hat is the magnitude of the net force on the motorcycle with rider? Draw an arrow to show the direction of the acceleration on the motorcycle. Physics 2
s of centripetal force As long as an object is in uniform circular motion, the net force (centripetal force) acting on the object must be towards the centre of the circle. The following are some typical examples. 1. Tension hen travelling around a roundabout, ick notices that the dice suspended from the rear-vision mirror swing out. If ick is travelling at 10 m/s and the roundabout has a radius of 5.0 m, what angle will the string connected to the dice (mass 100 g) make with the vertical? θ 2. Friction hen a car travels around a corner, the sideways frictional forces contribute to the centripetal force. If the sideways frictional forces are not sufficient, the net force on the car will be less than the centripetal force required to keep the car moving in a circle. Fr A car of mass 1200 kg travels around a bend with a radius of 10 m. The total sideways friction on the wheels is 14400. The road is not banked. Calculate the maximum constant speed at which the car can be driven around the bend without skidding off the road. Physics 3
Cyclists and motorcyclists also rely on sideways frictional forces to enable them move around corners. To increase the size of the sideways frictional forces, they often lean into the corner. The reaction force exerted by the ground will have a component towards the centre of the circular motion. Rsin(θ ) reaction force (R) normal reaction force θ Rcos(θ ) sideways friction A cyclist rounds a bend. The surface of the bend is horizontal. The cyclist is forced to lean at an angle of 20º to the vertical to just take the bend successfully. The total sideways frictional force on the tyres is 360. The cycle had a mass of 20 kg. hat is the mass of the cyclist? Going around the circular banked track Fnet θ Fr F = Fr cos( θ ) + sin( θ ) net In designing a bicycle track at a racing track, the designer wants to bank the track on a particular corner so that the bicycles will go around the corner with no sideways frictional force required between the tyres and the track at 10 m/s. Figures below show the banked track and the bicycle. On Figure B draw two arrows to show the two forces acting on the bicycle and rider (treated as a single object). Figure A Figure B Physics 4
The circular path of the bicycle has a constant radius of 100 m, and the bicycle will be travelling at a constant 10 m/s. hat should be the value of the angle of the bank, θ, so that the bicycle travels around the corner with no sideways frictional force between the tyres and the track? A demonstration at a show involves a bike being ridden around a circular banked track. The horizontal path the bike takes is a circle with radius of 20 m, and the bike travels at a constant speed of 15 m/s. The bike and rider have a total mass of 300 kg. Ignore retarding friction. Figure A Figure B hat is the magnitude of the net force on the bike and rider? Figure B shows the only two forces on the bike and rider. On Figure B draw an arrow to show the direction of the net force on the bike and rider. Explain how the forces shown cause this net force. hat is the correct angle of the banked track, θ, to achieve this net force? Physics 5
3. ormal reaction force For a person in the Gravitron it is the normal reaction force from the wall provides the centripetal force which produces a circular motion. Fr all Sarah, who weighs 50 kg, is at a fair ground and takes a ride on the Rotor. The Rotor is a large cylinder which, when spinning, pin riders against the internal wall. Figure A is a sketch of the situation. Initially Sarah is standing on the floor, and the Rotor speeds up. She feels the wall pushing harder and harder against her back. hen the friction between the wall and her back is large enough, the floor is lowered. Sarah is held against the wall, and does not slip down. The diameter of the Rotor is 16 m. Figure A Figure B Figure B is a force diagram showing all the forces acting on Sarah: force of wall on Sarah (C), weight force () and friction force (F). hen the Rotor is moving just fast enough to stop Sarah slipping, what is the magnitude of force F? The Rotor continues to speed up so that its final speed at the wall is 11 m/s. hat is the magnitude of the force (C) of the wall on Sarah? Physics 6
on-uniform circular motion (vertical circular track) 1. Energy transformations Let's review the conservation of mechanical energy. If the gravitational force and the restoring force are the only forces doing work on an object, the kinetic energy and the potential energy of the object interconvert while the total mechanical energy of the system (comprising the object and the earth or spring) remains constant. A skateboarder with mass 50 kg enters the half-pipe at point A, as shown in Figure below. Assume the frictional forces are negligible. A 4.0 m B hat is the skateboarder's speed at point B? hat is the centripetal force on the skateboarder at point B? hat is the apparent weight of the skateboarder at point B? 2. ewton's laws of motion By performing the force analysis, we could find the the net force which provides the centripetal force for the circular motion at the top or bottom of the circular track. Physics 7
A ride at a fun fair involves a car travelling around a vertical circle of radius 15 m. The arrow indicates the direction of travel as shown in Figure below. hat is the minimum speed that the car can have at position A if it is not to leave the rails and start to fall? Sarah, who has a mass of 60 kg, is a passenger in the car. hat will be Sarah's apparent weight at position A when the car is travelling at this minimum speed? Explain your answer. As the car passes point B at the bottom of the circle, it has a speed of approximately 5.5 m/s. hat is Sarah's apparent weight at this point? Show your working. Physics 8