CHAPTER 7 CIRCULAR MOTION 1 Describing Circular Motion Picture of Uniform Circular Motion Is the object accelerating? Linear speed vs Angular Speed Period of Rotation Page 1
2 Centripetal Acceleration and centripetal force Vector addition and subtraction: Centripetal acceleration by tip to tail method Page 2
Centripetal Force and Circular Motion [HKALE] A small object of mass 0.05 kg is released from rest at the rim of a heavy, smooth semi-spherical bowl of radius 10 cm. Find the force acting on the object by the bowl when it passes the bottom of the bowl. Page 3
[HKALE] Assuming the Earth to be a perfect sphere, what would its angular velocity of rotation have to be for an object at the equator to be weightless (i.e. to give a spring balance reading of zero)? (Radius of the Earth = 6.4 x 10 6 m.) 3 Examples of Uniform Circular Motion 1. Car Turning on Horizontal Road Coefficient of Friction Page 4
2. Car Turning on Banked Road The ideal banking angle: What if θ is fixed, but v is too small or too large? Page 5
[HKALE] An aircraft flies with a constant speed in a horizontal circle of radius 10 km. If its wings slant at an angle of 21.8 to the horizontal, find the speed of the aircraft. [HKALE] A car of mass m is moving with speed v on a banked road along a circular path of horizontal radius r. The angle of inclination of the road is θ. If the centripetal force arises entirely from a component of the normal reaction N from the road, which of the following relations is correct? A. N cos D. v 2 θ= mg B. N= mg cosθ C. gr = E. tanθ N sin θ = mv r 2 v 2 = gr sinθ Page 6
3. Bicycle around a track [On Horizontal Rough Road] [On Frictionless Banked Road] Page 7
4. Car on Level Road: Skidding and Overturning [Analysis] [Condition for Overturning] [Condition for Skidding] Page 8
[HKALE] [Circular Motion] (a) The centripetal acceleration, a, of a body undergoing a uniform circular motion of radius r can be 2 v expressed as either (1) a = or (2) a = ω 2 r, where v and ω are linear and angular speeds of the r body respectively. For a uniform circular motion of constant period, student A thinks that a decreases with r according to equation (1), while student B argues that a increases with r according to equation (2). Comment on their arguments. (2 marks) (b) To study a circular motion, a small rubber bung of mass m is attached to one end of a piece of string passing through a thin glass tube, which has a weight W hanging at its other end. The rubber bung is set into a horizontal circular motion by a student holding the glass tube. L A glass tube paper marker rubber bung W (i) Draw a diagram to show the forces acting on the rubber bung and explain why the string is not horizontal but dips at a small angle θ. Page 9
(ii) Show that the weight W equals mω 2 L in theory, where ω is the angular speed and L is the length of the string beyond the upper opening of the glass tube. Give TWO reasons to explain why there are discrepancies between the experimental and theoretical results. (iii) Suggest TWO ways to increase the rate of revolution of the rubber bung. (iv) When the rubber bung is at position A, the string suddenly breaks. Describe and explain its subsequent motion. (7 marks) Page 10
[HKALE] [Circular Motion] (a) Figure 1 shows a man of mass 70 kg standing against the wall of a cylindrical compartment called a rotor. The level of the rotor's floor can be adjusted. The diameter of the rotor is 5.0 m. The rotor is spun at a certain speed about its central vertical axis so that, at this angular speed, the man remains pinned against the wall even if the floor of the rotor is pulled downwards. (i) Name the forces F A and F B acting on the man. (2 marks) (ii) It is known that the maximum value of F A equals 0.4 F B. Find the minimum angular speed, in rad s -1, of the rotor needed to keep the man pinned against the wall. (3 marks) (iii) If the mass of the man is greater than 70 kg, would the result in (a)(ii) increase, decrease or remain unchanged? Explain briefly. (2 marks) Page 11
(b) Figure 2 shows a donut-shaped space station that is far from any planetary objects. It is designed such that the astronauts live at the periphery 1.0 km from the centre. Describe how an artificial gravity of 10 N kg -1 can be created at the periphery. (3 marks) FINAL REMARKS Circular motion, even with uniform speed, is a motion under acceleration and hence the object is being acted by a constant centripetal force. However, the effect of such force is to change only the direction of the motion of the object but not its speed. Hence the centripetal force always does no work on the object. Students should always remember that centripetal force is the reason for producing circular motion, but not circular motion produces centrifugal force ( 離心力 ) which is a common misconception in daily speaking. There are many different situations for questions to be set on circular motion. Students should NOT remember the results (for example 2 v tanθ= ) but you should be able to analyze the situation using basic principles. A lot rg of practice is required for mastering this chapter. Page 12