2. 1© Manhattan Press (H.K.) Ltd. Cyclists rounding bends on a level road Vehicles rounding bends on a level road Vehicles rounding bends on a level road 5.3 Vehicles rounding bends Vehicles rounding banked bends Vehicles rounding banked bends

3. 2 © Manhattan Press (H.K.) Ltd. Cyclists rounding bends on a level road 5.3 Vehicles rounding bends (SB p. 178) Centripetal force is provided by static friction on wheel

4. 3 © Manhattan Press (H.K.) Ltd. Vehicles rounding bends on a level road 5.3 Vehicles rounding bends (SB p. 179) 1. No skidding sideway friction (f) between tyres and ground provides centripetal force f = F c

5. 4 © Manhattan Press (H.K.) Ltd. Vehicles rounding bends on a level road 5.3 Vehicles rounding bends (SB p. 179) 2. Skidding (f not sufficient to provide centripetal force) Vehicle fails to follow curved track and tyres slide sideways. The car skids

6. 5 © Manhattan Press (H.K.) Ltd. Vehicles rounding bends on a level road 5.3 Vehicles rounding bends (SB p. 179) Notes: 1. Since the required centripetal force (F c = ) is provided solely by the friction between the tyres and the ground, the speed of the vehicle must not be too high, otherwise, it will skid. 2. The limiting speed of the vehicle without skidding is independent of its mass. 3. When the road is wet, the friction reduces and thus the maximum speed of the vehicle is lower. Go to Example 5 Example 5

7. 6 © Manhattan Press (H.K.) Ltd. Vehicles rounding banked bends 5.3 Vehicles rounding bends (SB p. 181) Centripetal force is provided by - horizontal component of normal reaction towards centre Go to Common Error

8. 7 © Manhattan Press (H.K.) Ltd. Vehicles rounding banked bends 5.3 Vehicles rounding bends (SB p. 181) Note: Ideal banking angle θ is independent of m, each value of θ is ideal for one speed only. In actual situation, there is a maximum allowed speed for the track with radius r. Therefore, θ can be designed to prevent the vehicles from skidding. Go to More to Know 2 More to Know 2

9. 8 © Manhattan Press (H.K.) Ltd. End

10. 9 © Manhattan Press (H.K.) Ltd. Q: Q:A Land Rover of mass 1 600 kg travelling at a constant speed of 30 m s –1 turns along a circular path of radius 50 m to the left. If the driver exceeds this speed, the inside wheels just lose contact with the ground. (a) Calculate the acceleration of the Land Rover towards the centre of the circle. (b) Calculate the frictional force (F) causing this acceleration. 5.3 Vehicles rounding bends (SB p. 180)

11. 10 © Manhattan Press (H.K.) Ltd. Q: Q: (c) There is a maximum height h of the centre of gravity above the ground for which the Land Rover does not lose contact with the ground at the inside wheels. Calculate h by taking moments about the centre of gravity. R is the normal reaction acted on the Land Rover by the ground. Solution 5.3 Vehicles rounding bends (SB p. 180)

12. 11 © Manhattan Press (H.K.) Ltd. Solution : Return to Text (b) Frictional force provides the centripetal force for the Land Rover is: F = ma = 1 600 ×18 = 28 800 N (c) If the Land Rover does not turn over, the net moment on it must be zero. Fh = 0.8R 5.3 Vehicles rounding bends (SB p. 180)

13. 12 © Manhattan Press (H.K.) Ltd. Centripetal force is NOT a separate new force, it is only the resultant of forces acting on the vehicle. Return to Text 5.3 Vehicles rounding bends (SB p. 181)

14. 13 © Manhattan Press (H.K.) Ltd. As seen in the equation tanθ = v 2 /rg, for a given radius of banked bend, the value of θ is ideal for one speed. However, friction (f) between the tyres of the vehicle and the ground reduces the danger of skidding when it turns. At lower speed, friction points up along the ground so as to prevent the vehicle from skidding inwards. At higher speed, friction points down along the ground so as to prevent the vehicle from skidding outwards. For higher speed, we have: Horizontal direction: R sinθ + f cosθ= mv 2 /r Vertical direction: R cosθ – f sinθ= mg 5.3 Vehicles rounding bends (SB p. 182) Return to Text