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Contents 1. Introduction ...................................................................................................................................... 2 2. Theory ............................................................................................................................................... 3 3. Equipment/Apparatus....................................................................................................................... 6 3.1 TQ Centrifugal Force Apparatus.................................................................................................... 6 3.2 The Electronic tachometer............................................................................................................ 7 3.3 The Speed Control Unit ................................................................................................................. 7 4. Procedure .......................................................................................................................................... 8 5. Precautions taken ............................................................................................................................. 9 6. Results ............................................................................................................................................. 10 7. Conclusion ....................................................................................................................................... 15 8. References ...................................................................................................................................... 16 1. Introduction A centripetal force acts towards the centre of a circular path and keeps the body moving in this path; otherwise the body would observe a straight line motion. In line with Newton's second law of motion, the centripetal force also produces an acceleration having the same direction it has. By definition, there exists a direct relationship between the centripetal force of a body and the square of its speed and an inverse relationship between the force and radial distance from the center of motion. Centripetal forces are responsible for the orbiting of the sun by the earth and other satellites. Other examples include motion of a car round a curved track, the frictional force between the tire and the ground serves the purpose. The study of centripetal force is of vital importance due to its impact in many engineering designs and applications. For modern machinery, rotational speed is of vital importance and also, centrifugal force resulting from slightly displaced parts can result in failure of vital components. The study of centripetal forces also helps in the design of curved roads, which are banked so that for a particular range of velocity deemed safe, cars can drive through the tracks without skidding off. 2. Theory A body moving along a curved path experiences changes to its acceleration. This means at each instantaneous point along this path, the particles has a component of acceleration perpendicular to the path, even if its speed is constant. Consider a body moving in circle with uniform speed about a fixed point O, Fig 1.0 Circular motion If the body moves from A to B through an angle, (measured in radians), its angular speed, ω, about O is defined as the change in angle per second. If t is time taken by the object to move from A to B, Also, it should be noted that the time taken for a complete revolution in a circular path, T Assuming, s is the length of circular distance between A and B, Dividing both sides of the equation by the time taken to move from A to B, t This implies that the tangential velocity (v), is Fig 2 Acceleration in a circle Considering the scenario above ,assume the body has moved a small radial distance ,the velocities at X and Y are different even though their magnitude are the same and equal to tangential velocity, v due to the fact that they have different directions . Therefore, acceleration occurs as there is a change in velocity. When t is small, small and the velocity change acts towards the centre. Therefore, Taking limits as A Body moving in a circular path of radius r with constant speed v has an acceleration directed towards the centre and by extension experiences a force which is also directed towards the centre. This force is called the centripetal force. From Newton's 2nd law of motion, Since for circular motion, Centripetal Force, 3. Equipment/Apparatus 3.1 TQ Centrifugal Force Apparatus The TQ Centrifugal Force Apparatus is designed to demonstrate the relationship between centrifugal force, mass of a body, its distance from the axis and its angular velocity. It is made up of two pivoted counterbalanced bell-cranks housed in slideable blocks as shown above. Different combinations of masses which can be increased by 25g are pivoted while the slideable blocks are held in place by locating pins. Four equally spaced holes allow each block fit in five different radial positions in each end of the horizontal rotating member. A transparent safety dome covers the rotating assembly. Removal of the dome disconnects the motor from power supply. 3.2 The Electronic tachometer The tachometer measures rotational speed using optical sensor mounted on the experimental apparatus. 3.3 The Speed Control Unit The E67 Speed Control Unit produces a variable DC output voltage which can be used to drive the horizontal rotating member of the centrifugal force apparatus motors. 4. Procedure 1. The locating pins on the sliding blocks were raised and the blocks positioned so that they were both on the same distance from the centre. The pins were then pushed down to locate the blocks firmly on the horizontal member. The distance from the axis to the pivots of the bell-cranks was measured. 2. A mass of 25g was screwed unto each vertical arm of the two bell cranks then, a mass of 175g was then screwed on the horizontal arm of the two bell-cranks. It was made sure the magnitude of the masses on the respective arms of the bell cranks was equal. 3. The safety dome was replaced and the motor started and observed using the speed control unit. The speed was slowly increased until the bell-cranks were flung outwards with an audible click. The approximate speed at which it occurred was noted. Also; the movement of the bell-cranks was observed from a position level with the plane of rotation. 4. The speed the bell-cranks were decreased until they returned to their original positions, then the speed was increased very slowly and the reading repeated. The speed indicated on the Tachometer at the instant when the upper arm of the bell-cranks moved outwards was recorded. If the bell-cranks did not move simultaneously, the speed when the first one moved was recorded. 5. The magnitude on each horizontal arm of the bell cranks was reduced by 25g while the body of mass on the vertical arm constant was kept constant .Steps three and four was repeated and more results down to the 50g mass were obtained. 6. The value of the horizontal masses on the bell cranks was then kept at 175g while the value of the vertical mass was increased by 25g to 50g .Steps three and four was repeated. 5. Precautions taken 1. Placing the dome on the centrifugal force apparatus before putting it on. 2. Fixing the masses securely on the axis. 3. Accurate taking of readings on the tachometer. 6. Results Table 1: Results for varying speed and mass for a constant radius of 8cm r = 8cm ma = 25 ma = 50 ma = 75 Centrifugal mb (g) Force mbg (N) Rev/min ω2 (rad/s)2 Rev/min ω2 (rad/s)2 Rev/min ω2 (rad/s)2 175 1.715 230 581 185 376 162 288 150 1.47 215 507 170 317 147 237 125 1.225 195 417 158 274 137 206 100 0.98 175 336 140 215 122 163 75 0.735 152 254 120 158 107 126 Table 2: Results for varying speed and radius for a constant mass of 50 g ma = 50 r = 8 cm r = 9.5 cm r = 11cm Centrifugal mb (g) Force mbg (N) Rev/min ω2 (rad/s)2 Rev/min ω2 (rad/s)2 Rev/min ω2 (rad/s)2 175 1.715 179 352 165 299 150 247 150 125 100 75 1.47 1.225 0.98 0.735 162 150 135 124 288 247 200 169 150 140 130 110 247 215 185 133 140 130 120 100 215 185 158 110 Graphs of radial force (F = Mbg) against ω2 as shown in figure 3. At constant upper mass ma, a straight line profile passing through origin was obtained demonstrating the linear relationship between radial force and angular velocity. The deviation of the linear theoretical profile is attributable to experimental errors, measurement errors and wears on the apparatus due to friction. Fig. 3: Variation F with speed and mass at constant radius (r= 8cm) The radial force also varies linearly with the upper mass ma as shown in figure 4. This was determined by reading values of F from each line on figure 1 for a particular value of angular speed ω2. The values are shown in table 3. Table 3: Values of F at constant speed ω2 = (300rad/s) F ma 0.84 25 1.28 50 1.8 75 Fig 4: Variation of mass at constant speed (r = 8cm, ω2 =300 (rad/s)2) Varying the radius and maintaining the upper mass ma, constant at 50g, the force was plotted against angular speed as shown in figure 5. The result shows a linear profile demonstrating that radial force is directly proportional to angular speed. Fig. 5: Variation of F with speed and radius at constant mass (ma = 50g) At a constant speed (ω2 = 200 (rad/s)2), the corresponding values of F on each of the lines in figure 3 was determined as shown in table 4. Table 4: Values of F at constant speed ω2 = (200rad/s) F r (cm) 1 8 1.16 9.5 1.32 11 A graph of F against radius r was plotted. The obtained profile, figure 6 shows that radial force is directly proportional to radius. Fig. 6: Variation of radius at constant speed (Ma = 50g, ω2 = 200 (rad/s2) 7. Conclusion The complete series of results affirm the theoretical relationship between radial force and each of the independent variables therefore validating the equation of radial force, F = mω2r. 8. References Nelkon, Parker. (1958). Advanced Level Physics. Oxford: Heinemann Educational Publishers. School of Mechanical and Manufacturing Engineering:centrifugal force apparatus. (n.d.). Retrieved November 2, 2012, from School of Mechanical Engineering DCU Ireland Website: http://www.dcu.ie/mechanical_engineering/technical/undergrad/mechanics/centrifugal_apparatus/cen trifugal_apparatus.shtml School of Mechanical and Manufacturing Engineering:speed control unit. (n.d.). Retrieved November 2, 2012, from School of Mechanical Engineering DCU Ireland Website: http://www.dcu.ie/mechanical_engineering/technical/undergrad/mechanics/speed_control_unit/speed _control_unit.shtml School of Mechanical and Manufacturing Engineering:tachometer unit. (n.d.). Retrieved November 2, 2012, from School of Mechanical Engineering DCU Ireland Website: http://www.dcu.ie/mechanical_engineering/technical/undergrad/mechanics/tachometer_unit/tachome ter_unit.shtml Young, F. (1996). University Physics. Massachusetts: Addison-Wesley Publishing Company Inc.