Chris Horswell 1st December 2002 Determine the Gravitational Field Strength (g) at the Earths Surface, Using a Pendulum. Aim: – To determine the acceleration of free fall (g) using a simple pendulum. To achieve this I must research background information provided in the specification. In my specification I am provided with the Periodic Time equation: – L = Length – Measured in metres (m) g = Gravity – Measured in Metres per second (ms ) T = Periodic Time – Measured in second (s) Background information “This was produced by Galileo in 1632.
He was inspired by a swinging lamp, strung form the ceiling of the Cathedral in Pisa; using his pulse as a stopwatch, he calculated that the oscillations of the lamp remained constant even when the oscillations were dying away. ” It was acting like a Pendulum. A pendulum consists of a mass hanging from a string and fixed at a pivot point. Once released from an initial angle, the pendulum will swing back and forth with Periodic Motion. ‘a’ to ‘b’ to ‘c’ to ‘b’ to ‘a’ = 1 oscillation. at points ‘a’ and ‘c’ = greatest potential energy at point ‘b’ = greatest kinetic energy.
This is a diagram showing the motions of the pendulum and forces acting on it. It shows where the bob passes once swinging and the Simple Harmonic Motion it follows. Also shown above is where the greatest potential and kinetic energy is. We shall refer back to this later in our preliminary work. “Simple Harmonic motion is the movement of an object that moves with constant velocity back and forth with the same amplitude and period” “Ordinary Physics” Page 183 Simple harmonic motion is the vibrations of matter. Thus a important example of simple harmonic motion is the pendulum.
In the picture on the left we can see the forces acting on the pendulum. By considering the motions of the pendulum, and Newton’s second law the moment of inertia is: – and also the toque can be written as: – The torque is negative, as it is acting in a motion to decrease the angle at which the pendulum was released from. Once simplified: – This is the equation for simple harmonic motion (Non linear differential equation. ) Also I have come across another equation, although it is usable it only applies to pendulums when they are released with a small initial angle: –
But when the initial angle is too large of 30i?? or more, this equation is no longer valid. From the simple harmonic motion equation we can derive that the frequency and period of the pendulum are independent of the initial angular displacement amplitude. From this we can denote that a smaller initial angle would make the results more accurate. This is because… “For small angular displacements the error in the small angle approximation becomes evident only after several swings. Whereas when the initial angle is large the error between the simple harmonic solution and the actual solution becomes apparent.
” Internet : www. kettering. edu Preliminary Work As specified in my aim I must use this equation to work out gravity, but it is also possible to work out gravity in other ways. Using an Electronic timer. This is the use of an electromagnet, a ball bearing and a trapdoor. The timer starts when the ball bearing is released from the electromagnet and then stops when it enters the trap door. Using the Displacement equation : – S = Displacement t = Time taken U = Initial velocity a = Acceleration Substituting in common factors (Gravity, Height and initial velocity) : –
With this we can find out the gravity by adding the height of the ball bearing from the trap door, and the time taken. Using a ticker-tape timer. This method shows the acceleration of a mass with small dots. The dots increase distance apart as the mass falls. After this we separate the dots into 5 dot intervals, which represents 0. 1 of a second. If we then line these 5 dot intervals in a row it displays as a bar chart then by measuring the gradient : – We can find the acceleration of the mass (Gravity) Using a light gate. This is set up so the object falls through the set of light gates.
Setting the light gates at different lengths apart give more accurate reading. To use this method a computer is needed and the information can be converted into metres per second instantly but equations can be used also: – Putting this into the equation for Acceleration we can work out gravity: – But in relation to Galileo and the date at which he produced this equation, it would be impossible to recreate these experiments as electricity had not been discovered. So on choosing our experiment we first set out on finding what the constant factors would be and what we would be varying.
In order to do this we must firstly prove the equation experimentally, and from that we can see which factors we will be varying and which remain constant: – Using… 1. Square both sides to remove the square root: – 2. Divide both sides by ‘L’ to remove the fraction: – Therefore because we know what pie ( ) will be (3. 14159265358) and we are trying to find out gravity (g) The varying subjects will be “Periodic Time” (Ti?? ) and the Length (L). So from this we can derive that periodic time needs to be measured in seconds (s) for different sets of lengths in metres (m) to achieve the value of gravity in metres per second (ms ).
If this is true, then: – = Constant and The Gradient of the Line Analysis of Preliminary Experiment From our experiment we came across five problems:- 1. The angle at which the pendulum was released. The angle we used was far too large and at the moment it was dropped it would fall vertically for a split second. This would make our results inaccurate as this would be measuring Vertical Velocity not Periodic time. In order to eliminate this problem, we must use a shallower angle of around 10i??. This would make our results more accurate and reliable. This refers to Harmonic Motion mentioned in our background information earlier.
(Page 2) 2. The number of oscillations. The number of oscillations was far to low, this made our results inaccurate. In order to improve them I shall increase the amount of oscillations to 10. 3. The place at which we count the number of oscillations. We found counting the number of oscillations at the initial point was inaccurate because as the oscillations died away the pendulum would not reach the initial point anymore. So we decided that we would measure at the centre point using a ‘fiducial marker’ 4. The length of the pendulum was too short. Because the length was short the oscillations went too quickly.
This effected our results dramatically. We must use longer lengths to improve the accuracy. 5. The point at which the pendulum was attached, was slipping. This affected our results as this caused friction; thus affecting our results. In order to eliminate this I shall put the pendulum thread in between two pieces of wood. Rearranging the Equation In order to use the equation provided we must first rearrange it to make gravity the subject: – 1. Square both sides to remove the square root: – 2. Multiply both sides by ‘g’ to remove the fraction: – 3. Divide both sides by Periodic time to remove it from the subject side: –
This is the equation I must use to find out gravity with a simple pendulum Constant and Varying Factors Constant Factors: – * Pendulum/Bob Number of oscillations Fiducial marker Varying Factors: – Length of pendulum I shall be measuring the time it takes for 10 oscillations to occur using a stopwatch (seconds) Prediction. I predict that I shall get the value of 9. 81ms . I predict this because from research and common knowledge Galileo produced this almost 300 years ago and has become a legend since. But I cannot expect my results to show exactly 9.
81ms ,this is because of human error with the fiducial point, and air resistance with the bob. Safety The usual safety precautions should be enforced, safety goggles, long hair should be tied back and stools and bags under the desk. Extra precaution should be taken when walking about the lab as there are swinging weights about. Apparatus Diagram 1. Clamp and Stand 2. 2. 5 metres of thread 3. 2 pieces of slated wood 4. Pendulum / Bob 5. Stool 6. Stop watch 7. 2 metre sticks 8. Protractor 9. Card and felt pen Method 1. Set up as Diagram 2. Draw vertical line on card to make your fiducial point. 3.
Let pendulum hang vertically to elign the fiducial point. 4. Once prepared, hold protractor at the top of the string and measure the pendulum out to 10i?? 5. Let go of pendulum and start stopwatch when u see it pass the fiducial point. (you may want to let it swing back and forth a couple of times to regulate the oscillations) 6. Once the pendulum has oscillated 10 times, stop the stopwatch and record the time. 7. Repeat 2 more times and record results. 8. Change length to 1. 80 metres and repeat 9. Repeat for the other lengths until you have collected all your data. 10. Plot on Graph and work out the gradient.