Study of Circular Motion in a Conical Pendulum

Study of Circular Motion in a Conical Pendulum 1. Purpose The purpose of this experiment is to study the effect of the rotational motion of an object on the string attached to it. In this experiment, the object is an air-plane attached to a string and it rotates in such a way that the trajectory traces a right circular cone. Here the tension generated in the string can be decomposed into two resultant forces, one vertically upwards balancing the weight of the air-plane and the other horizontal component balancing the centrifugal force on the plane. 2. Tools Required The tools required for this experiments are: (a) A stop-watch (b) A linear scale (c) An inelastic string (d) A toy air-plane 3. Proedure First, an inelastic string is chosen. Length of the string is measured. Then to check the elasticity, different tensile loads are given on the string. If the string does not stretch or the stretch is negligible, the string can be used in our experiment. With this string, the toy air-plane is attached to the hook of a small disk connected to a motor. Then, just before starting the motor, the toy plane is slightly dislodged from its vertical position so that when the disk starts to rotate, the toy airplane will experience a centrifugal force and will start to rotate along the circumference of a circle. The trajectory of the air-plane traces a conical pendulum. This generates a tension in the string connecting the plane to the hook. One end of the hook of the disk is attached to a scale that measures the magnitude of the tension in the string. One member of the team would measure the time required to complete one 1 round around the circle. This time interval is called the time-period. For a precise measurement, a single trial could be repeated for 3-4 times and then take the average of the readings. To change the speed of the air-plane, speed of the rotating disk should be varied in different trials. The corresponding values of time-period, tension in the string, angle made by the string with the vertical to be noted and presented in a table. 2 4. Analysis Fig 01: Force Vector Diagram The required equations can be obtained by analysing the forces acting on the airplane in the horizontal and the vertical directions. We assume the mass of the plane is m. Horizontal Direction From the Fig 02, it can be observed that the radial component of the ~ sin θ balances the outward centrifugal force tension force in the string F m~v 2 . So the force equation in the horizontal direction is: ~r F~ sin θ = 3 m~v 2 ~r (i) Vertical Direction Analysing the Fig 02, it can be observed that the upward vertical component of the tension of the string balances the downward weight of the plane. So the force equation in the vertical direction is: F~ cos θ = m~g (ii) Now ~v is the velocity of the plane.The magnitude of ~v is obtained by dividing the circumference of the conic base-circle by the time period T . In the equation form: |~v | = 2πr T (iii) From the figure, we get: r = L sin θ (iv) Substitute this value of r in (iii), we get: |~v | = 2πL sin θ T Now, mv 2 F~ sin θ = r mg F~ cos θ sin θ v2 = cos θ rg Substitute the value of v from the equation (iii) to get: sin θ 4π 2 r2 = 2 T rg cos θ 4π 2 r sin θ = 2 cos θ T g Substitute the expression of r from (iv) to get: 4 sin θ 4π 2 L sin θ = cos θ T 2g Solving for cos θ, we get: T 2 g = 4π 2 L cos θ cos θ = cos θ = T 2g 4π 2 L  g  T2 4π 2 L (v) If we vary the value of θ and measure the corresponding values of T , we will get a graph of T 2 Vs cos θ that should approximate a straight line. Since we know the value of L, we can calculate the value of g, from the slope of the straight line. The value of the mass can be obtained from equation no. (ii) as: m= F cos θ g (vi) 5. Data & Calculation The data obtained are represented in the following table: Table 01 5 From the values of T and θ from the above table, we prepare a table containing the values of T 2 and cos θ to draw the T 2 Vs cos θ graph. air table.png Table 02 The graph is as follows: Fig 02 Equation of the best fit line as obtained from the graph is: cos θ = 0.42(T 2 ) Hence slope of the line from the graph is 0.42. Comparing with the equation (v), we conclude: g = 0.42 4π 2 L 6 g = 0.42 4π 2 ( sinr θ ) g sin θ = 0.42 4π 2 r g = 0.42 × g = 16.58  4π 2 r sin θ   r  sin θ (vii) Using the values of r and θ from the experiment, below we mention the table of different values of the gravitational acceleration g and the % error compared to its theoretical value 9.80 m/s2 Table 03 Mass of the airplane can be calculated from the equation (vi). Using the data from the first row of Table 01, we get: m= 4.9 cos 52◦ 9.80 m = 0.308 Hence, mass of the airplane is ≈ 300 gram. 6. Sources of error & Remedy Some discrepancies that can affect the outcome of the experiments are: 7 (a) Elasticity of the String The string chosen to hang and rotate the airplane should be as inelastic as possible. By inelastic, we mean that the string will be none or negligible elongation of the string. Elongation of the string under the tension would increase the value of L, length of the string which will alter other results also. (b) Recording Time Period Since the time period,T , appears as square of T 2 in the equation, extra caution should be taken while observing and recording the values of T . It is advisable to assign 2 or 3 people to note the time period individually to minimize the human error. Also a single trial should be repeated for 2 to 3 times to ensure proper recording of the value of the time period. (c) Proper Observation & Collection of Data While taking the readings of force, radius and the angle made by the string with the vertical, one should ensure that the eyes and the scale remain in the same horizontal plane to avoid any error in recording data. (d) Safety Like any other experiment, it is imperative to ensure a safe environment for the persons involved in the experiment and the experiment set-up. Since the airplane rotates in a conical pendulum,a prior knowledge of the maximum radius of the base circle of the cone is essential to maintain a safe distance from the rotating airplane. ********************** End of Assignment ******************* 8